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TWO-PARAMETER STOCHASTIC PROCESSES WITH FINITE VARIATION 3Y CHARLES LINDSEY A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1988
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TABLE OF CONTENTS Page ABSTRACT i i i CHAPTERS I INTRODUCTION 1 1 .1 Notation 3 1.2 Fi It rat ions 1 1.3 Stochastic Processes 8 1.4 Stopping Points and Lines 8 1.5 Some Measure Theory 12 II VECTOR-VALUED FUNCTIONS WITH FINITE VARIATION 21 2.1 Basic Definitions and Some Examples 21 2.2 The Variation of a Function of Two Variables 26 2.3 Functions of Two Variables With Finite Variation 43 III STIELTJES MEASURES ON THE PLANE 69 3.1 Measures Associated With Functions 69 3.2 Functions Associated With Measures 85 IV VECTOR-VALUED PROCESSES WITH FINITE VARIATION 95 4.1 Definitions and Preliminaries 96 4.2 Measures Associated With Vector-Valued Stochastic Functions 112 4.3 Vector-Valued Stochastic Functions Associated With Measures 131 4.4 On the Equality |m| = p , , 1M V CONCLUSI ON 1^9 BIBLIOGRAPHY 150 BIOGRAPHICAL SKETCH 154 ii
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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy TWO-PARAMETER STOCHASTIC PROCESSES WITH FINITE VARIATION BY CHARLES LINDSEY April, 1988 Chair: Nicolae Dinculeanu Major Department: Mathematics Let E be a Banach space with norm |»|, and f: R + E a function with finite variation. Properties of the variation are studied, and an associated increasing real-valued function |f| is defined. Sufficient conditions are given for f to have properties analogous to those of functions of one variable. A correspondence f *-* \x between 2 such functions and E-valued Borel measures on R is established, and + the equality | u | = y . . is proved. Correspondences between E-valued two-parameter processes X with finite variation |x| and E-valued stochastic measures with finite variation are established. The case where X takes values in L(E,F) (F a Banach space) is studied, and it is shown that the associated measure u takes values in L(E,F"); some x sufficient conditions for y to be L(E,F)-valued are given. Similar results for the converse problem are established, and some conditions sufficient for the equality |y | = p. , are given. iii
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CHAPTER 1 INTRODUCTION Families of random variables indexed by directed sets have been objects of study for some time. The most common by far, though, has 2 2 been R , and especially R , the first quadrant of the plane, as this is considered the most "natural" extension of the usual indexing set R or R . One of the principal objects of study relating to such processes has been the stochastic integral of processes indexed by the plane. Stochastic integration with respect to two-parameter Brownian motion has been studied extensively; the focus of more recent study has been the more general problem of extending the one-parameter theory of stochastic integration with respect to a semima^tingale. In one parameter, this is done by writing a semimartingale X as X = M + A with M a locally square integrable local martingale and A a process of finite variation (see, for example, Dellacherie and Meyer [5]). The major problem in two parameters is with the notion of "local martingale": the theory of stopping is not sufficiently well-developed to permit a definition with all the necessary properties. Some preliminary work has been done, however, using the notion of an increasing path (see, for example, Fouque [9] and Walsh [18]). A definition of a stopped
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process has also been given for square integrable martingales in Meyer [12], but this too is rather limited. Another area that has seen renewed interest is the study of processes with values in a vector space, especially in a Banach space (see, for example, Neveu [1M]). In Dinculeanu [7] the correspondence given in Dellacherie and Meyer [5] between processes with finite variation and stochastic measures is extended to the case where the processes have values in a Banach space, and in Meyer [12] the correspondence is stated for two-parameter real-valued processes. In the one-parameter theory, this result finds its use in applications to projections, what in turn (at least in the scalar case) is relevant to the decomposition of supermartingales and to semimartingales . This correspondence X ** y is given by (1.1.1) u (*) = E(f* dX ) X ; z z for $ scala: valued, bounded process. We shall extend this correspondence to the case where X has integrable variation, with values in a Banach space E. Since the inner integral on the right side of (1.1.1) is computed pathwise, we begin by studying the properties of Banach-valued 2 functions f: R » E with bounded variation. In Chapter II, we develop the relevant properties of such functions, and in Chapter III 2 we show that to each such f we can associate a measure p : B(R + ) » E with finite variation, and we prove the equality (1.1-2) |p f | = M| f |
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(i.e., the variation of the measure associated with f is equal to the measure associated with the variation of f). In Chapter IV we prove that a stochastic measure can be associated with a process of integrable variation, in the same manner as in Dinculeanu [7], and we consider the converse problem: that of associating a function with a given stochastic measure. Unfortunately, the equality (1.1.2) is not preserved in general for processes and stochastic measures, so we end up by establishing some sufficient conditions for the equality to hold, 1 .1 Notation 2 The index set is R ; we shall sometimes consider functions and 2 processes extended by zero to all of R . In the rare cases where we look at points outside the first quadrant, we shall say so 2 explicitly. We denote points in R by z, u, w, v and their coordinates by the letters r, s, t, p. For example, then, we write z = (s ,t) , u = (p ,r) , etc. There are two notations commonly used in the literature for the 2 order relation in R : we adopt here the notation of Meyer [12]. For two points z = (s,t), z' = (s',t'), we have z£z' iff s^s', t£t'; z (s,t) by (2), F C O F , r #» s t , v s t (s,t)<(s ,t ) s >s t'>t Then A z F , for all s'>s, hence A c (\ F , = F . s * v s s s >s Similarly, A c F , for all t'>t, hence A e f] F , = F . 1 t's t'>t Putting the two containments together gives the equality.
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We say that the condition of commutation is satisfied if the 1 2 conditional expectation operators E( | F ) and E( * | F ) commute, 12 2 1 i.e., if E( I F |F ) = E( F F ). The product is then equal to 1 s' t ' t ' s i ? 1 E( I F HF ) = EC I F ). (In fact, denote, for f e L(P), E a Banach 1 s t ' st t 12 2 1 space with the RNP, g = E(f | F |F ) = E(f|F |F ). Then g is measurable 2 1 with respect to F and F , hence g is F -measurable. Also, for t s st A E F st , we have J A gdP = J A E(f | F^| F 2 )dP = / A E(f|F^)dP = J A fdP since A e F = F 1 C\ F 2 . Thus, g = E(f |F ).) All of the main st s t st results of the theory of two-parameter processes require the condition of commutation. The second (and more frequently used) way of describing filtrations on R is due to Cairoli and Walsh [2]. There, we begin + with a family If , z e R 2 } of sub-o-fields of F satisfying the z + following axioms: (F1) If zz Meyer . ) 1 2 We now define F =F =VF V , F. = F = V F . In place of s s 00 . st t *t st the condition of commutation, we impose the axiom 1 ? (Fit ) For each z, F and F are conditionally independent given z z F , i.e., for Y, F -measurable, integrable, Y_ F z 1 z d z measurable, integrable, we have
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(1.2.1) E(Y Y |F ) = E(Y |F )E(Y |f ). I c. Z I Z c. Z This condition is equivalent [4, 11.45] to (1.2.2) E(Y 1 |F 2 ) = E(Y 1 |F z ) for every F -measurable, integrable r.v. Y, . The condition (F4) can z l be seen to be equivalent to the condition of commutation as follows: (F4) => commutation : Let X be F-measurable, integrable. Then E(X|F 1 ) is F 1 -measurable, integrable. By (F4), E( E(X I F 1 ) I F 2 ) = 1 z z ' z ' z 1 2 2 E(E(X|F )|F ) = E(X|F ). Similarly, E(X| F ) is F -measurable, 1 z ' z ' z ' z z integrable; hence by (F4) E(E(X| F 2 ) | F 1 ) = E(E(x|F 2 )|F ) = E(X|F ). Thus E(E(X|F 1 )|F 2 ) = E(X|F ) = E( E(X I F 2 ) I F 1 ) , i.e., the operators 1 z ' z ' z ' z ' z 2 1 EC * I F ) and E( I F ) commute. 1 z ' z Commutation => (F4) : We shall assume commutation and prove that (1.2.2) holds. Let Y, be F -measurable, integrable. We have, from the 1 z commutation condition, E(YjF z ) = ECYjF^F 2 ) = E(Y., | F 2 ) , which is just (1 .2.2) . Although the difference between the two constructions is slight (the main difference being on the "border at infinity," where we do not necessarily have right continuity of the filtrations (F ) and p (F ) in the Cairoli-Walsh model), the condition most often imposed on the filtration in the literature is stated as the (F4) condition, although the notation used is more often that of Meyer.
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1 .3 Stochastic Processes Throughout this section, (Q,F,P) is a complete probability space, (F ) a filtration satisfying axioms (F1)-(F^) (in particular, one Z zcRj constructed by the method of Meyer), and E will denote a Banach space equipped with its Borel o-field, denoted B(E). The definitions In this section are taken from [12], [7]. Definition 1.3.1 . A stochastic process is a measurable (i.e., a 2 2 (B(R )xF, B(E) )-measurable) function X: R^xQ + E. We usually denote X(z,w) by X (w), and the mappings w * X (w) by X . Remark . It will sometimes be convenient to extend the index set to 2 all of R by defining X =0 for z outside the first quadrant, and F = the degenerate o-field for those z. When we wish to consider a process in this light, we shall say so explicitly. A brut or raw process is a process X such that X is F-measurable 2 for all z e R . X is called adapted if X is F -measurable for all + c z z 2 z e R . + 2 A process X is called progressive if, for every z z R , the map (z,w) X (w) from [0,z]xP, * E is (B([0,z])xF , B( E) )-measurable. z z ( Note : This definition comes from Fouque [9]; in Meyer [12] progressi vity is defined using half-open rectangles [0,z). The definition we give here is the one in common use today.) 1 . i) Stopping Points and Lines 1.11.1 Stopping Points The notion of stopping plays a fundamental role in the theory of one-parameter processes. The obvious extension of the definition of
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stopping time to two-parameter processes yields what is known as a stopping point . 2 Definition 1.^.1 . A stopping point is a mapping Z: fl * R such that for all v e R , the set {z < v} belongs to F . We often denote Z = (S,T). The components S,T are, it turns out, 1 2 stopping times with respect to (F ), (F ), respectively, but this by itself is insufficient to guarantee that (S,T) is a stopping point. We do, however, have the following characterization [12]: 1 2 Theorem 1.^.2 . Let S be an (F )-stopping time, T an (F )-stopping 2 time. Then Z = (S,T) is a stopping point if and only if S is F T ~ measurable, T F -measurable. Although stopping points have found some application recently (see, for example Walsh [18] and Fouque [9]), they are rather inadequate for the purpose of developing a theory of localization for two-parameter processes. To begin with, due to the partial order in R , we are not even assured that JZ>v} e F ! Also, if U and V are + ' v stopping points, UV may not be. Moreover, in one dimension, the graph of a stopping time T divides R xft into two components, namely the stochastic intervals [[0,T)) and [[T,°°)), whereas the graph of a stopping point Z does no such thing. Furthermore, an important realization of a stopping time is as the debut of a progressive set, and we have no analogous result in the plane.
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10 1.4.2 Stopping Lines p Let A C R xQ be a random set (the usual definition: 1 . is a + A measurable process). The open envelope of A, denoted (A,*), is the random set whose section for each w e P. is given by (A,-)(w) = \J (z,«). zeA(w) Some properties of (A,») (proofs can be found in Meyer [12]): 1) If A is progressive, (A, 00 ) is predictable. 2) The interior of a progressive set A is progressive, from which we obtain, by passing to complements, that the closure of A is progressive . We designate in particular by [A, 00 ) the closure of (A,*) (i.e., for each w, we define [A,°°)(w) = (A,«)(w)). [A,") is progressive if A is progressive, by the above properties; it is called the closed envelope of A. The random set D A = [A,«)\ (A,-) A is called the debut of A : it is progressive if A is progressive. Definition 1.4.3 . a) A random set Z is called a stopping line if it is the debut of a progressive set, i.e., if there is a progressive set A such that Z = D = [A,»)\ (A,-). A b) A stopping line Z is predictable if it belongs (as a set) to the predictable o-field P.
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11 Remarks . 1) The set A in the definition is not unique: for example, A and (A,°°) always have the same debut. 2) This definition has drawbacks: for example, D is not necessarily adherent to A. 3) For an alternate way of defining stopping lines (as a map from £2 into a certain set of curves on R ), see Nualart and Sanz [15]. H) Since (A,«) is predictable, to say that a stopping line Z = D is predictable amounts to saying that [A, 00 ) is A predictable, which is an alternate way of defining predictable stopping times in one parameter (cf. IH, IV. 69]). We introduce a partial order on the set of stopping lines by defining H < K if (H,-) 3 (K,»). We also make the convention H < K if (H,-) D [K,»). The set of stopping lines is then a lattice for £: HK is the debut of (H,») U (K,») , and HK is the debut of (H,») O(K,«0. We say that a stopping line H is the limit of an increasing (resp. decreasing) sequence (H ) of stopping lines if [H,») = [\[H ,») n (resp. if (H,») = I J(H ,»)). In the first case, if we have v y n n [H,») = 0(H ,°°), we say that the sequence (h ) announces (foretells) n
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12 H: if there exists a sequence (H ) announcing H, we say H is n announceable (foretellable) . In the oneparameter theory, this is equivalent to being predictable (and in fact many authors define predictability in this fashion). However, we only have one implication for stopping lines, namely, that every announceable stopping line is predictable. In fact, if H is announceable, then ex. (H ) such that [H,») = M(H .«). n n n Each (H ,*>) is predictable, hence the intersection is predictable. By remark (4), this implies that H is predictable. Unfortunately, the other implication does not hold. For a counterexample, see Bakry [1]. Finally, we note the existence of a predictable cross-section theorem for stopping lines. The proof is essentially the same as for the one-dimensional case. We denote by tt the projection 2 of R xQ onto Q. Theorem l.H.M (predictable cross-section theorem). Let A be a predictable set, and let e>0. There exists a compact* predictable set K satisfying the following: 1) K C A and PJK = 0, A * 0} < e 2) D is announceable and KCD . K ft 1 .5 Some Measure Theory In this section we collect some results from measure theory which we shall make frequent use of later. * That is, the section K(w) is compact for each w.
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13 5.1 Monotone Class Theorems There are several versions of monotone class theorems, both for families of sets and for functions. The two stated here are the variants we shall use later. The statements are taken from Dellacherie and Meyer [4]. Theorem 1.5.1 . Let C be a ring of subsets of a. Then the monotone class M generated by C is equal to the o-ring generated by C. If C is an algebra, then M = o(C). Theorem 1.5.2 . Let H be a vector space of bounded real-valued functions defined on Q, which contains the constants, is closed under uniform convergence, and has the following property: for every uniformly bounded increasing sequence (f ) of positive functions from H, the function f = lim f belongs n to H. Let C be a subset of H which is closed under multiplication. The space H then contains all bounded functions measurable with respect to the o-field o(C). The most frequent application of this theorem comes when we wish to prove that a certain property holds for all bounded F-measurable functions; it allows us to reduce to the case where f is the indicator of a set. We shall see numerous examples of this theorem at work. 1.5.2 Liftings Let (T,I,u) be a measure space, p>0. Material for this section comes from Dinculeanu [6, pp. 199-216]. We shall need these properties in Chapter IV.
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Ill Definitions 1.5.3 . * Let u be a positive measure. A mapping p: L°°(u) * L°°(y) is said to be a lifting of L (u) if it satisfies the following six conditions: 1) p(f) = f u-a.e. 2) f = g u-a.e. implies p(f) = p(g) 3) p(otf + 8g) = otp(f) + Sp(g) for a,S e R 4) f>0 implies p(f)*0 5) f(z) = a implies p(f)(z) at 6) p(fg) = p(f)p(g). We say that L°°(u) has the lifting property if there exists a lifting p of L°°(u). The following theorem affirms that, for probability measures P in particular , L°°(P) always has the lifting property. Theorem 1.5.4 . If it has the direct sum property, then L (p) has the lifting property. Let, now, E and F be Banach spaces, T a set, and Z a subs pace of F', norming for F, i.e., such that ----- 1 for every yeF. i y ^F = sup iff: r zeZ I »F' For every function U: T » L(E,F) ( continuous linear maps E -* F) , x: T -+ E and z: T + Z, we denote by the map t * __ and by |U | the map t * |U(t)|. * The definitions and theorems in Dinculeanu [6] are given in somewhat more generality; we restrict ourselves here to statements involving the measures and o-fields we shall be working with.
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15 For two functions U ,U : T * L(E,F) we shall write U U 2 if ____ = ____ u~a.e. for every xeE, zeZ. Let p be a lifting of L (p). Definition 1.5.5 . Let U: T > L(E,F) be a function. a) We shall write p(U) = U if for every xeE and zeZ we have e L°°(y) and p() = b) We shall write p[U] U if there exists a family A of subsets of T such that it has the direct sum property with respect to A, such that for every At A , xeE, zeZ, we have 1 A e L°°(u) and p(1 A ) = 1 p(ft) . ( Note : If u is o-finite (and in particular if u is a probability measure), the "elation in (b) holds for all A measurable.) The functions U with p(U) = U or p[U] = U have the following properties: 1) p(U) = U implies p[U] = U. 2) If p[U] = U, then is y-measurable for every xeE, zeZ. 3) If U » U , p(U ) = U, p(U 2 ) = U 2> then u i = Ug. H) If U U , pCU^ = U 1 and p[U 2 ] = Ug, then U 1 = U 2 p-a.e. 5) If U = U ? u-a.e. and p[U 1 ] = U 1 , then p[Ug] = Ug. We shall also make use of the following:
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16 Proposition 1.5.6 . Let U: T -> L(E,F). If p[U] = U, then the function t * |U(t)| is y-measurable. It is this proposition, along with property (5) above, that will be used most often later. 1.5.3 Radon-Nikodym Theorems We state here some generalizations of the Radon-Nikodym theorem to vector-valued measures with finite variation. These particular statements are taken from Dinculeanu [6, pp. 263-27^]. Throughout this section, T denotes a set, R a ring of subsets of T, E and F Banach spaces, and Z a subs pace of F', norming for F. Theorem 1.5.7 . Let m: R » L(E,F) be a measure with finite variation p. If \i has the direct sum property, then there exists a function U : T * L(E,Z') having the following properties: m 1) |U (t)| = 1 y-a.e. * m 2) For all f e L^(m) and zeZ, ____ is y-integrable and we have E m <[fdm,z> = [____dy. J ' m 3) If p is a lifting of L (y), we can choose U uniquely so that p(U ) = U (cf. Definition 1.5.5). K m m k) We can choose U (t) e L(E,F) for every t e T, in each of the m following cases: a) F = Z' b) There exists a family A covering T such that y has the direct sum property with respect to A , such that for every A e A , x e A, the convex equilibrated cover of the
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17 set I J \Jjxdm: i|) R-step function, {j^dy < 1 } is relatively compact in F for the topology o(F,Z). c) For every x e E, the convex equilibrated cover of the set {/ijjxdm: i> R-step function, J|i|i|du < 1 } is relatively compact in F for the topology o(F,Z). Note : If m is defined, on a o-algebra F, which will be the case in our uses of this theorem, then the condition that tt have the direct sum property is automatically satisfied, and we can replace the family by F in part (Mb) of the statement. These remarks also hold for the following generalization of the Radon-Nikodym Theorem. Theorem 1.5.8 (Extended Radon-Nikodym Theorem). Let v be a scala" measure on R and m: R * L(E,F) a measure with finite variation y. If \> has the direct sum property and if m is absolutely continuous with respect to v, then there exists a function V : T * L(E.Z') having the K m following properties: 1) The function |V | is locally* v-integrable and m \m\i = f|V Udlvl for i> e L (u) ; j « rn (here | \> | denotes the variation of v) . 2) For f z li(]i) and z e Z, is v-integrable, and we have :[fdm,z> = [dv(t), * As before, if the measures are defined on an o-algebra, this can be dropped .
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3) If p is a lifting of L (p), we can choose V uniquely v-almost everywhere such that p[V ] = V (cf. Definition 1.5.5). Tf, m m In addition, there exists a>0 such that u i a|v|, then we can choose V uniquely such that p(V ) = V . m y m m 4) We can choose V (t) e L(E,F) for every t e T, in each of the m following cases: a) F = Z' b) There exists a family A covering T such that v has the direct sum property with respect to A such that for every A e A , x e E, the convex equilibrated cover of the set {Jijttdm: 4jR-step function, L|*|d|v| £ l} relatively compact in F for the topology o(F,Z). b') The same statement as (b) , with A such that y has the direct sum property with respect to A and with ||^|dp <> 1 instead of J A |*|d|v| £ 1. In this case we may not have p[V ] = V . m m c) For every x e E, the convex equilibrated cover of the set Ijtjjxdm: \\> R-step function, ||^|d|v| ^ 1 } is relatively compact in F for the topology o(F,Z). c') The same condition as (c) with J|ip|dy ^ 1 instead of J|\|»|d|v| 5 1. In this case we may not have p[V ] = V . Theorem 1.5.7 gives a "weak density" of a vector measure m with respect to its variation y, whereas Theorem 1.5.8, more generally, gives such a density of m with respect to a scalar measure v not obtained from m. The former is a particular case of the latter, but
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19 we shall make good use of it in its own right, and so we take the time to state it separately here. The final result we shall need is a "converse" of these theorems. Theorem 1 . 5. 9 . Let v be a scalar measure on R and U: T * L(E,F) a function such that |u J is locally v-integrable and the function is v-measurable for every x e E and z g Z. Then the function is v-integrable for f e L 1 (|uJ|v|) and z g Z and there exists a measure m: R -* L(E,Z') such that = /____dv for f e L (|ufl|v|) and z e Z, and J|i|/|du £ J |U 1 1 ^ | d | v | for E be a function. a) We say that f is right continuous (in the order sense) if, for 2 all z e R , we have f(z) = lim f(u), or equivalently lim |f(u) f(z)[ = 0. u-*-z u-»z u^z uSz 21
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22 ( Note : We shall use sometimes the notation u + z for u + z, u > z.) b) We say that f is incrementally right continuous if, for all 2 z e R , we have (denoting z' = (s'.t')) lim |a ,(f)| = lira |f (s' ,t ')-f (a' ,t)-f (s ,t')+f (s ,t ) z'-*z (s ' ,t ')-»(s ,t ) z'Sz s'£s t'>t 0. Remarks. 1) The limits are path-independent: in particular, in (a), this limit includes the path where u + z along a vertical or horizontal path. 2) In (b) if s' = s or t' = t, Ba ,(f)[ = 0, so we can take the 11 zz " inequalities s' > s, t' £ t to be strict. The chosen definition is simply to preserve symmetry in the limits in (a) and (b). 3) When we say simply, "f is right continuous," without further specification, it will always mean in the sense of (a). 1) If f is right continuous, then f is incrementally right continuous. To see this, note that |a ,f| = |f(s',t')f (s',t)-f(s,t')+f (s,t)| = |f (s'.t')-f (s,t)+f (s,t)-f(s',t)f (s,t')+f (s,t) I (adding and subtracting f(s,t)) < flf(s',t')-f(s,t)l + ||f(s',t)-f(s,t)U + ||f(s,t')-f(s,t)|. Then f right continous implies each of the three terms on the
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23 right tends to zero as (s',f) + (s,t), hence |& ,fj > 0, which is (b) . Unfortunately, we do not have the converse implication in general. To see this, we give the following example, which we shall refer to later in pointing out further weaknesses of using increments alone. Example 2.1.2 . Let g be any E-valued function defined on [0,»). 2 For (s,t) z R , we then put f(s,t) = g(t). Then, for any (s ,t ) $ (s' , t ') , we have |f(s',t')-f(s',t)-f(s,t')+f(s,t)| = |g(t')-g(t)-g(t')+g(t)f = 0. The function f is then evidently incrementally right continuous for any f so defined. If we take, however, g to be a function which is not right continuous, we have lim |f(s',t')-f(s,t)j = lira |g(t')-g(t)J * 0, (s'.t'Ws.t) t'+t (s',t')>(s,t) hence f is not right continuous. Later on, we shall establish some additional conditions on f sufficient to have (b) => (a). Another basic notion for one-parameter functions with regard to Stieltjes measures is that of increasing function, as we reduce functions of finite variation to this case via the Jordan decomposition. Again, we have two definitions, the first the natural extension of the one-variable definition (order sense), the second
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2iJ more closely related to measure theory: namely, a condition sufficient to generate a positive measure. o Definition 2.1.3. Let f: R -> R be a (real-valued) function. a) We say f is increasing (in the order sense) if z < z' => f (z) < f(z'). b) We say f is incrementally increasing if A ,(f) i for all z £ z*. The scalar-valued functions we shall typically consider are defined using the variation of vector-valued functions: these (as we shall see later) are increasing in both senses. In general, however, the two notions are distinct neither implies the other, as the following two examples show. In these, we focus our attention on the unit square [( 0, 0) , ( 1 , 1 ) ] for simplicity, but we can extend them (by constants, say) to give a perfectly good counterexample defined on all of R 2 . + Examples 2.1.H . i) We define here a function satisfying definition 2.1.3(a) but not (b). The particular function we shall give is defined on the unit square; we could extend it arbitrarily outside [( 0, 0) , ( 1 , 1 ) ], but we shall not give an explicit extensionthe square is sufficient to indicate how things can go wrong. The idea consists of writing A ,f = f (s ' ,t ' )-f (s ' ,t )-f (s ,t ' ) + f (s ,t ) zz as f (s',t ')-f (s',t)-[f (s,t ')-f (s,t)], so if the second difference is larger than the first, the increment will be negative even if f is incresing in the sense of 2.1.3(a). Accordingly, for (s,t) in the
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25 unit square, we define f(s,t) t + 3(1 t) Each one-dimensional path, for fixed t, is a straight line connecting the points (0,t,t) and (1,t,1). Thus, for t < t' f the slope of the section f(«,t) is greater than that of f(«,t'), and so the second difference is larger than the first, so A ,f < for any z < z' in zz the unit square. The following computations bear this out: 1) For (0,0) < (s,t) < (s'.f) < (1,1), f (s',t')-f(s,t) i 0. In fact, f(s',t')-f(s,t) = t'+s'(1-t')-(t+s(1-t)) = t '+s'-s't '-t-s+st = (t '-t) + (3 '-s)+3t"S't' = (t '-t) + (s '-s)+st-s't+s't-s't ' = (t'-t)+(s'-s)-t(s'-s)-s'(t'-t) (1-s')(t'-t)+(1-t)(s'-s) > 0, since each of the four factors is nonnegative. 2) Denoting z = (s,t), z' = (s ' ,t ') , (0,0) < (s,t) < (s'.f) $ (1,1). ,f = f(s',t')-f(s',t)-f(s,t')+f(s,t) =t'+s'(1-t')-(t+s'(1-t))-(t'+s(1-t'))+t+s(1-t) = t'+s'(1-t')-t-s'(1-t)-t'-s(1-t')+t+s(1-t)
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26 = s'[(1-t')-(1-t)]-s[(1-t')-(1-t)] = (s'-s)Ct-t') < 0. ii) If we set g(s,t) = -f(s,t), we get a function g satisfying Definition 2.1.3(b), but not (a): g(s',t')-g(s,t) = -f(s',t')-(-f(s,t)) = -[f(s',t')-f(s,t)] $ 0, and similarly A ,g = -(A ,f) £ 0. We could even create a zz zz nonnegative g (g = 1-f) with these properties. As can be seen, these two definitions of Increasing are not nearly so closely related as the definitions of right continuity we have given. Later, however, we shall give sufficient conditions for a function f of two variables to have a "Jordan decomposition" f = f -t , where f and f are increasing in both senses of the word. 2.2 The Variation of a Function of Two Variables In this section we define the variation Var r ,-,(f) of a [z,z J 2 2 function f: R * E on a rectangle [z.z'J (closed) in R + and establish some of its properties. Throughout this section, by "rectangle" we 2 shall mean a closed, bounded rectangle in R (but everything goes 2 equally well for such rectangles in R ), unless otherwise specified. Definition 2.2.1 . Let R = [ (s ,t ) , (s ' ,t ' ) ] be a closed, bounded 2 rectangle in R .
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27 a) A partition P of R is a family of rectangles (R . ) . , J J J c J finite, satisfying the following: i) for j.j'eJ, j * j', R . C\ R . = (i.e., any two distinct rectangles in (R . ) are either disjoint or intersect only on their boundaries) ii) R = \J R . (This is a straightforward extension of the notion of partition of an interval [a,b](ZR). b) Let P = (R ) ,, Q = (R ) , be two partitions of R. We say J jeJ i iel that Q is a refinement of P if, for each R. c P, there exists J a family of rectangles from Q forming a partition of R . . ( Note : It is evident from the definitions that for j * j'. The two families from Q forming partitions of R. and R., must J J be disjoint.) We show next that any two partitions of a given rectangle R have a common refinement, as is the case in one dimension. The main step in this, and a result we shall use again in its own right, is the following: 2 Lemma 2.2.2 . Let R » [ (s,t) , (s ,t ) J be a rectangle in R , and P = (R.). T be a partition of R. Then there exist partitions J JeJ o: s = s^ < s, < ... < s = s' of [s,s'] and t: t * t n < t, < Dim u i ... < t = t' of Ct.t'] such that the family Q of rectangles of the n form [(s ,t ),(s . ,t . ) ], £ p < m, % q < n. Is a refinement of P. p q p+1 q+1 Remark . A partition of R constructed from partitions o,t of [s,s'] and [t,t']> respectively, as Q is above is called a grid on R. We
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28 often use the notation oxx to denote the set Q of rectangles as defined above as well as the vertices of these rectangles. There is rarely any danger of confusion and where there is we shall be more explicit. We shall use this notation in the p^oof of the lemma and afterward. Proof of Lemma. For each jeJ, denote R, = [ (s . ,t . ) , (s ' ,t ') ]. The J J J J J construction of Q is straightforward: we take o to be the set of all the s. and s ,, ordered appropriately, and t to be the set of all t. and t.,. put in ascending order. We need to show that oxt is a J J refinement of P_. Let, then, R. = [ (s . ,t . ) , (s ' ,t ' ) ] be a rectangle in _P. Let o ' be a partition of [s.,s '] obtained by taking the points from o between s. and s' (inclusive), and t ' a partition of [t.,t'] J J J J obtained from t in the same manner. Then o'xt' is evidently a partition of R , B Proposition 2.2.3 . Let P,P' be two partitions of R. Then P and P' have a common refinement, i.e., there exists a partition S of R that is a refinement of both _P and P'. Proof . Let Q = oxx be a grid refining _P (Lemma 2.2.2), and Q' = o'xt' be a grid refining P". Denote by p the partition of [s,s'] formed by oo' (put in ascending order), Y the partition of [t,t'] formed by putting tt ' in ascending order. Then S = pxY is a common refinement of Q and Q', hence S refines _P, and S refines P', so S = pxY is our common refinement. (In fact, we have proved that any two partitions have a common grid refinement.) B For a given rectangle R, we can define an ordering on the class Pp of partitions R as follows: we define P S Q for two partitions P,Q of
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29 R if Q is a refinement of P_. Prop. 2.2.3 says, then, that the class P D of partitions of R is directed under this order. We are now ready to define the variation of a function on a rectangle . Definition 2.2.4 . Let R = [z,z'] = [ (s ,t) , (s' ,t ') ] be a closed, 2 2 bounded rectangle in R , and let f: R -» E be a function. a) For P = (R.) a partition of R, R = [ (s . ,t . ) , (s ' ,t ' ) ] , we J JeJ J J J J J define Var rz z'i (f; ^ = l K f ' Lz.z j jeJ Kj = X |f(s' t')-f(s' t )-f(s ,tp + f(s ,t )J. 4_T JJ JJ JJ JJ b) We define the variation of f on R, denoted Var r ,-,(f), by Lz,z J Var r ,,(f) = sup Var r ..(f;P) S + ». Lz,z J [z,z ] ? C R Remark . The supremum in part (b) always exists (finite or infinite), since the map P -> Var^ ,, (f;£) is increasing for the order defined above on P . To see this, consider a rectangle [z,z'] partitioned R into two rectangles R and R , as in Figure 2-1, R 2 I R l Figure 2-1 A partition of [z,z']
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30 We ha ve |A (f)| = |f(s',t')-f(8',t)-f(s,t')+f(3,t)| = |f(s',t')-f(s',t)-f(s ,t')+f(s ,t)+f(3 ,t') -f(s ,t)-f(s,t')+f(s,t)l < Jf(s',t')-f(s',t)-f(S ,t')+f(S ,t)l + |f(8 ,t')-f(B ,t) -f(s,t')+f(s,t)| = |A R f| + |A R f|. We can do a similar calculation (only longer) for any partition of R, by adding and subtracting values of f at all the additional vertices of the refinement, and applying the triangle inequality. In the next result, we give some properties of the variation. 2 Proposition 2.2.5. Let f: R + E be a function. E + 2 i) For any rectangle [z,z ] C R , Var,,-,(f ) £ 0. ii) Var r ,-,(f) can be computed using grids, i.e., partitions of the form Q = oxt. iii) Additivity : For < s < s' < s" , £ t < t' < t" we have [ (s,t) ,(s",t ) ] L(s,t) ,(s ,t ) J + Var r . , t , , ,. ,(f ) [(s ,t) ,(s",t ) ] and similarly
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Var [(s,t)(3',t")3 (f) = Var [(s,t),(3',t')] (f) +Var r( S( t') )( s' ( t")] (f) (See "igure 2-2.) 3 1 t".t 't Figure 2-2 Additivity of the variation iv) If R,C R_, Var D (f) < Var D (f). 1 2 R 1 R 2 v) If f is right continuous (order sense) , we can compute Var> ,-,(f) using grids consisting of points with rational Lz,z J coordinates (and hence we can take the suprernum along a sequence of partitions). Proof . i) Since Var r ,,(f;P) £ for any partition P, we have [z,z ] far r (f) = sup Var r ,,(f;P) ^ 0. [z,z ] D [z,z J ii) If Var r , n (f) = + », then for every N>0, there is a Lz,z J partition P such that Var (f;P ) > N. B Y Lemma 2.2.2, there N [z,z'] N
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32 exists a grid Q refining P by the remark following Definition 2.2.4, we have Var r , n (f;Q) £ Var r , n (f;P) > N. Thus, if L z , z J L z , z J N Var r ,n(f) = + °°, for any N>0 there exists a grid Q such that L z , z ] N Var Cz,z'] (f; V > N ' Ue " Var [z,z'] (f) = *£ p Var rz,z'] (f;Q) ' Qer R Q=OXT Similarly, if Var r ,-,(f) = ctO, then for every e>0, there is a [z,z ] partition P such that Var r ,-,(f,P ) > a-e. Again, taking a grid -e [z,z J ' e Q refining P , we have Var r ,-,(f;Q ) > Var r ,-,(f;P ) > a-e. e -e [z,z J e [z,z J -e e arbitrary => sup Var r , n (f;Q) S a = Var r , n (f). The other p Lz.zJ Lz.zj Q=OXT inequality is evident, so we have Var r ,-,(f) = sup Var r ,-.(f;Q) Lz.z j Qe p R Lz.z J Q=OXT ( Note : From now on, we shall compute variations using grids.) iii) We shall prove the first equality; the proof of the second is completely analogous. Denote R = [ (s ,t ) , (s ' ,t ' ) ] , R 2 = [ (s ' ,t ) , (s" ,t ' ) ], R = R \J R^ For any grid Q = oxt on R, we can add the point s' to a to get a refinement Q' = Q \J Q , where Q is a grid on R , Q is a grid on R . Then Var R (f;Q) < Var R (f;Q') = Var R (f;Q ) + Var R (f;Qg) < sup Var D (f;Q.) + sup Var D (f;Q.) Q 1 R l 1 Q 2 R 2 2 = Var (f) + Var D (f). R 1 R 2 Taking supremum on the left, we get
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33 Var (f) < Var (f) + Var (f). n n R For the other inequality, if Q ,Q ? are any grids on R R respectively, then Q U Q is a grid on R, and we have Var (f;CL) + Var (f;Q.) = Var(f;Q) n 1 n_ d n < sup Var D (f;Q) = Var_(f). _ n n Q=OXT Since this inequality holds for any grid Q on R , we have sup Var (f;Q,) + Var D (f;Q ) < Var(f), Q =oxt 1 2 i.e., Var (f) + Var D (f;Q.) < Var_(f). Similarly, Q being arbitrary, we have on taking supremum for Q . Var D (f) + Var D (f) < Var_(f). R 1 R 2 R Putting the two together, we have the equality: Var R (f) + Var R (f) = Var R (f). iv) Assume R. = [ (s ,t) , (s' ,t ') ] is contained in R = [ (q ,r) , (q' ,r ') ]. Then we have q £ s < s' i q', and r £ t < t' £ r' (see Figure 2-3). By the additivity property, we have Var D (f) = Var D (f) + Var D (f) + Var D (f) + Var D (f) + Var D (f). R 2 R 1 R 3 R U 5 R 6 Since each term on the right is nonnegative, we have Var (f) > Var (f). R 2 R 1
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3U t -t R 5
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35 rectangle Rj = [(s't' ) , (s' ,t ' ) ] e Q', and we have I IK f 1 " IKf 11 IK f A, f 1 R k,l ' \,1 " " \,1 \,1 " = If c» k+1 .t 1+1 )-f o k+1 .V-^vVi^W-^ 3 ^ '^1 e e , e e _ s 4mn 4mn 4mn 4m n mn" Summing up, then, we obtain I I K f| E |A f|| | I (|A f| |A f|)| R eQ i R.eQ' J 00, there is a grid Q with Var r ,-,(f;Q) > N + 1. By the above, there exists a grid Q' with Lz ,z J N N rational coordinates such that Var r ,-,(f;Q') > N + > N,* hence Lz ,z J N 2 sup Var r ,-.(f;Q) = + « Var r , n (f). Similarly, if Q=axx [Z ' Z ] [Z ' Z ] Q rational Var r ,-,(f) = a < °°, for every e > there exists Q = oxt such that L z , z J e Var r , n (f;Q ) > a =, and there exists Q' = o'xt' with rational Lz ,z J e 2 E * Again, Q' is not a partition of R, so this must be interpreted directly as the sum given in Definition 2.2.4(a).
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coordinates such that Var r ,-,(f;Q ) Var r ,-,(f;Q') | < ~, Lz,z J e Lz,z J e ' 2 hence Varv , n (f;Q') > Var r , n (f;Q ) § > (a §) § Lz,z J e lz.zJ e 2 2 2 (f). e arbitrary => sup Q=oxx Qrational Var r ,,(f,Q) = Var r ,[z ,z J [z ,z J An important property of the variation in one dimension is that a function of finite variation f is right continuous if and only if 11m Var r ,-,(f) = for all s in the domain of f. Unfortunately, [s,s J s + + s we do not have this equivalence in two dimensions without additional assumptions about f. We do have one implication, however. Theorem 2.2.6 . Let f: R -* E be a function with Var_(f) < » for every bounded rectangle R. If f is right continuous, then for every 2 z, z , u in R with z < z < u, we have Var r ,,(f) = lira Var r ,(f). [z.z ] u++z , [z,u] (Note: The notation u+ + z' means u * z', u > z'.) Proof . We divide the region outside [z,z'] and inside [z,u] into three parts, labeled R , R , R (see Figure 2-4). The proof consists R 3
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37 of showing that, as u decreases to z', the variation on each of the three rectangles R , R , and R vanishes. We shall give separate proofs for R and R , and the proof for R is identical to that for R p . Note first of all, that for u' such that z' < u' < u, the corresponding rectangles R", R', R' satisfy R ' CT R , R'CH,, R'CR_, so by Proposition 2.2.5(iv), Var D ,(f) < Var D (f), etc., and 3 3 R 1 R 1 so each of the limits lim Var (f), lim Var D (f), lim Var D (f) *R. K~ , n _ u-i-4-z 1 u + + z 2 u + + z 3 exists and is nonnegative. a) Denote z' = (s',t') u = (p,r). We show that lim Var r . , . ,. . . ,(f) 0. p^s' [(s » t Mp.D] r + + t' Assume not: then there exists a>0 such that, for all u>z', we have Var r , ,(f) > a. Let u > z', g > 0. Denote u. = (p.,r.). [z ,u] Since Var r , i > a, there exist partitions o . s' = s _ < s_ . < [z ,u-J 0,0 0,1 S 0,2 < < S 0,m = P 0' V fc ' = fc 0,0 < fc 0,1 < ' < fc 0,n = r SUCh that i} l |A [(s t ) (s t )] fS > a 0* Km US 0,i 0,j '^ S 0,i + 1 ,l 0,j + r J and 0£j a §. V°0 XT 6 V°0 XT 6 V°0 XT 8 R C I R CII R C III B 6 8 (See Figure 2-5.) Since each of the three sums is less than the
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33 '''O,!* U l,l (s',t') (s o,i >r o ) ( Po» r o> ^0^0, 1 } I
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39 Denote u = (p ,r ) (s ,t .). By assumption, Var r , ,(f) > a, etc. Continuing In this manner, we construct a lZ ,u_J sequence u , u , u, with u, > u, , > z' for all i such that, for i 1+1 "0' "1 ' 2 all 1, the total variation of f on the rectangles making up [(s',t'),(p,r)]\ [(s',t'),(p.,r)] is greater than (a |) + (a -|) ... + (a *y) ia ,1, *>*' hence we have Var r , -.(f) > ia £ for all i, i.e., [z ,u Q ] Var r -,(f) » + », a contradiction of the hypotheses that [z,u ] Var (f) < « on every bounded rectangle. Hence, for any a<0, we have R < lim Var r . , ,, , ,,(f) < a => lira Var D (f) = 0. p ++ s' [(s ' l ) ' ( P' r)] u*+z' R 1 This takes care of R . b) We show now that lim Va~ R (f) = 0, or, more precisely, that u++z' 2 lim Var r , , . . , . ,. -,(f ) p^s' C(s M*^'* )] 0. (See Figure 2-6) . We proceed as before, by contradiction. Assume there exists a>0 such that (s,t') z(s,t) (P,t0 R 2 __~~ a for all p > 3'. Let, then, p^ > s', let Li.3,t;,ip,t;j e>0. There exist partitions n : s' = s < s < ... < s = d , 0,0 0,1 0,m y 0' T : l = Vo < '0,1 < '" < \n ' t# 5UCh that E ' A R f 5 > °" VV T 6 Now, consider the rectangle [(s',t),(s. .,t')]. For each i, u, 1 i = 1,2,..., n, there is r. , s' < r < s n , r < r such that for each i [(3 ''Vi-i ) '^r t o,i )] f l <-£ by right continuity at each (s',t ) (r n ,t') (s 01 ,f) (p ,t') t, \ 0,2 '0,n-l (s',t) r n -r 2 v 1 (s Q1 ,t) (p ,t) Figure 2-7 Breakdown of o n xt Subdividing each of the leftmost rectangles of xt in this manner creates a refinement F_ of o n XT n , hence I \h f | > o. Now, we also have that R y eP Y n E |A i-1 C(3 '' t 0,i-1 ) '^i' t 0,i )]f| ' i = 1 2 i+1 "~ 2 '
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41 n 1 K f l " E S A [(s' t ) (r t )l f| > a ' \ * The rectangles in this sum form a partition of the n rectangles [( V t),(p,t 0J )], [(r 2 ,t 0J ),(p,t 0>2 )], ... [(Vt^Mp.t')] (shaded rectangles in Figure 2-7). Thus n 1-1 [(r i' t 0,i-1 ) » ( P' t 0,i )] > E |A f| E |& f| > a -§ R y eP Y 1-1 L ^ S '^0,1-1 j,lr i ,t 0,i ; d In particular, then, Var r , . . , . ,., > a % . Let p = r . By L [r ,t ) , (p ,t ) J 2 H n ssumption, Varr, , > ,.-,(f) > a, so we can repeat this as; ..US . t i . U n procedure and get another point s' < r' < r such that n n Var r , , , » , . M1 (f) > a T7 . Continuing in this manner, we can L(r ,t),(p ,t ) J n construct a sequence p , p , p , such that i) Pj > P l+1 > s, p ++s ii) Var r , .. , k ^i(f) > a -frr for alL i > L(p i+1 ,t),(p i ,t )] 2 i+1 We have, then, by additivity, 1-1 [(p. ,t),(p ,t )] L(p J+1 ,t),(p ,t )] 1-1 i-1 > I (a . . ) = ia E > ia e j-0 2 J j-0 2 J i-1 Then Var (f) > E Var r , , , N -,(f) > ia-e. R 2 j=Q [(p. +1 ,t),(p.,t )]
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42 e arbitrary => Var (f) > ia for all 1, hence Var (f) = + », again a R 2 R 2 contradiction. Hence lira Var r . , , , ,,.(f) £ a for any a>0, so P++S ' [(s -t).(P.t )] lira Var r . , , . ,.,(f) 0. p++s , [(s ,t),(p,t )] By the same argument, lim Var r , ,_,. , , N -,(f) = (the R_ ,,,.C(s,t ),(s ,r)J 3 r + + t case). Putting everything together, we have lim Var r .(f) = lim [Var r , n (f) + Var,, (f) + Var (f) + Var (f)] , , , Lz.u] , , , [z,z ] R R. R_ u++z u++z 1 2 3 = lim Var r ,,(f) + lim Var (f) + lim Var (f) + lim Var n (f) [z,z J R R R_ u++z u++z 1 u++z 2 u++z+ 3 = Var r .At) + + + 0) = Var r ,,(f), [z,z J [z,z J which is what was to be proved. Remarks . 1) As we stated above, the converse is not true. However, if lim Var r ,(f) = Var, ,-,(f), then lim Var r , -.(f) = 0, and from u**z' Cz ' u] Cz ' Z ] u**z' [Z « U] ' A [z',u] f l " Var [z',u] (f) WS get that lirn ,' A [z',u] f ' = °» i ' e ** f iS Incrementally right continuous. 2) Returning to Example 2.1.2 , for f as defined there, we have Var f ,,(f) = on any rectangle [z,z'], since I Ja r (f)f = Lz,z J R eP a a I = for any partition P of [z,z'], so lim Var r -.(f) = R cP " u**z' [Z ' U] a Var r ,-,(f). However, if f is constructed from a function g that is Lz,z J not right continuous, then neither is f. In fact, for all e>0,
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H3 f(s+e,t+e) = g(t+e), so lim f(s+e,t+e) = lira g(t+e) * g(t) = f(s,t), so f is not right continuous. In the next section, we shall give sufficient additional conditions on f for the converse to hold. 2. 3 Functions of Two Variables with Finite Variation As the example in the previous section (at the end) shows, the requirement that Var (f) < » on bounded rectangles R is by itself n insufficient to give all the properties necessary to associate a Stieltjes measure to it. In order to deduce properties of f from its variation, we need some extra conditions. It seems natural to require that each of the one-dimensional paths also have finite variation, but we do not need quite that much. In fact, if Var (f) < » on all bounded R rectangles R C R , and if the one-dimensional path f(«,t n ): R E has finite variation for some t , then the paths f(«,t) have finite variation for all t. More precisely, for any s>0, we have Var [0>s] f(.,t) ~~`s] f(.,t ) Var [( § (g ^ ] (f (Note: We replace the second term by Var r . , . ..(f) if [(0,t) ,(s,t Q ) ] t < t.). To see this, let o: = s^ < s, < ... < s = s be a 1 n partition of [0,s] (Figure 2-8): We have, for each i, < i < n-1, |f(s. +1 ,t)-f( Si ,t)| = |f(s i+l ,t)-f(s i ,t)-f(s. +1 ,t ) + f(s i ,t ) f(B 1+1 ,t )-f(8 lf t )| ^ |f( Si+1 ,t)-f(3 l ,t)-f(S l+1 ,t ) + f(8 l ,t )| + l f(s 1+ rVf(s i'Vl'
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H'4 \ 1 S l S 2 'n-1 Figure 2-8 Partition of bounding Var ro ,f(',t) L0,s J Summing over the i's, and denoting R = [ (s ,t ) , (s ,t ) ], we have n-1 n-1 Z |f(S i+1 ,t)-f(3.,t)I < E (|A R f| + |f(3 l+1 ,t )-f(3 lf t )|) n-1 n-1 Z |A f| + Z |f(s ,t )-f(3 ,t )| i=0 i i-0 Var C(O f t ),( S ,t)] (f) + Var -C0,s] f( -'V ( ° r Var [(0,t),(3,t )] (f) + Var [0, S ] f( -' t ) lf ts] f(.,t) < Var [(Q Stt)] (f) + Var [0>g] f (. ,t Q ) . By the same proof (using partitions of [0,t]) we see that if the one-dimensional path f(s ,): R + > E has finite variation for some
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^5 s , then the paths f(s,») have finite variation for all f, and in fact (same proof) Var ro ,f(s,0 S Var r , , , ...(f) + Var ro ,f(s n ,«). [0,t] C(s ,0),(s,t)] [0,t] 0' Up to now, we have avoided using the phrase "f has finite variation" because of the weakness of the condition Var (f) < °° for R K bounded. We shall reserve this term for functions with the additional conditions described above. We will see that this is enough to give the additional properties we need to associate useful measures. 2 Definition 2.3.1 . Let f: R * E be right continuous, with 2 Var_(f ) < for bounded rectangles R C R . We say that f has finite R + variation if the real-valued function |f|(s,t) = lf(0,0)l + Var [0>s] f(.,0) + Var [0>t] f(0,.) + Var [(0>0)j(Sit)] (f) < 2 for every (s,t) e R . We say f has bounded variation if there exists M>0 such that 2 |f | (s,t) < M for all (s,t) g R + . 2 The map |f|: R -* R is called the variation of f . (Note that we use the single bars to distinguish it from the norm in E.) Remarks . 1) Henceforth, the phrase "f has finite variation" will be 2 understood to mean that |f|(s,t) < °° for all (s,t) e R . 2) We extend |f| by outside the first quadrant to get a 2 function defined on all of R .
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3) The "jump at zero," |f(0,0)|, will play a role later, similar to that of the jump at zero in the theory of one-parameter processes. When we associate measures with |f|, we shall need this term to get some compatibility between these measures and those associated with f. 4) The function |f| is increasing in both senses of Definition 2.1.3First of all, if z = (3,t), z' = (s',t*), and z < z', then |f|(s',t') |f|(s,t) = if (s',t') lies outside the first quadrant; |f|(s',t') |f|(s,t) = |f|(s',t') > if z' > 0, z outside FT. If S z < z', then |f|(s',t') |f|(s,t) = |f(0,0)| Var [Q a ,.f(. f 0) + Var r , n f(0,-) + Var r . , . , ,,,(f) [0,t'] L(0,0),(s',t')] [|f(0,0)| Var rn ,f(«,0) Var r ,_ .,f(0,« 1 ' [0,s] [0,t] +Var [(o,o),(s,t)] (f)] Var r ,,f(-,0) + Var r . . o f(0,v [3,3 ] [t,t ] +(Var [(0,0),(s',t')] (f) " Var [(0,0),(s,t)] (f)) > 0. As for the other sense, we have A r ,, f = if z' lies outside the [z,z J ' ' first quadrant. We then deal with the case where S z'. If z is in the third quadrant, i.e., if s<0, t<0, then A r ,,|f [z ,z J ' = |f|(s',t') |f|(s',t) |f|(s,t') + |f|(s,t) = |f|(s',t') > 0.
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^47 If z is in the fourth quadrant, i.e., if s>0, t<0, then A r z z'] (f) = |f| (s'.t*) |f|(s,t') + = If (0.0) | Var [0jS , ] f(-,0) Var [ Q ^ #] f (0, ) +Var [(0,0),( S ',t')] (f) " ( l f( °' 0) ! + Var ro,s] f( -' 0) Var [0ft#] f(0 f O + Var [(0)0Ms>t , )]( f)) = Var [s,s'] f( -' 0) +Var [(s,0),(s',t')] (f) °« Similarly, if z is in the second quadrant, i.e., if s<0, t>0, then A r ..(f) = Var r . . , 1 f(0,O + Var r , n . , . fc .x-,(f) i0. Lastly, if [z,z ] Lt ,t ] [(0,t) , (s ,t )] £ z < z', then we have A (f) = |f|(s',t') |f|(s',t) |f|(s,t') |f|(s,t) L Z 9 Z J = If (0,0) | Var [0jS . ] f(.,0) Var [0tt<] f(0,.) +Var [(0,0),(s',t')] (f) " ( H f( °' 0) l +Var [0,s'] f( -' 0) + Var [0)t] f(0,.) + Var [(0(0)>(s . (t)] (f)) (|f(0,0)| Var [0i8] f(..0) * Var [0>t<] f(0,.) + Var [(0,0),(s,t')] (f)) + « f(0 ' 0) l +Var ro,s] f( '' 0) Var [0ft] f(0,.) + Var [(0>0Ms>t , )]( f) = (Var [(0,0),(s',t')] (f) Var [(0,0),(s',t)] (f)) " (Var [(o,o),(s,t')] (f) " Var C(o,o),(s,t)] (f))
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iJ8 =Var [(0,t),(s'.t')] (n " Var [CO,t),(s,t')] (f) Var C(3,t),(s' f t')] (f) "^ This definition allows us to recover many results analogous to those of functions of one variable, as we show in the next few theorems. Theorem 2.3.2 . Let f: R > E have finite variation |f|. Then f is right continuous if and only if |f| is right continuous. Proof . Assume, first, that f is right continuous. We write, for (s.t) > (0,0), |f|(s,t) |f(0,0)| Var [0(S] f(.,0) Var [Q>t] f (0, ) Var [((J>0) , (s>t)] (f ) The first term is constant; to show that |f|(s,t) = lim |f|(s',t'), it s '+s t'+t suffices to show that i) lim Var f(-,0) = Var rn -f(.,0) s'++s [0 ' s ] [0 ' 3] ii) lim Var f(0,-) = Var f(0,O t'+n L ' J L ' J (s',t')++(s,t) ^°'°Ms »t )] [(0,0), (s.t)] We proved (iii) in Theorem 2.2.6 (taking z = (0,0), z' = (s,t), u = (s*,t')). As for the other two, f right continuous implies f(«,0), f(0,«) right continuous: in fact, taking t' = t = 0, we have
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149 lim f(s'.t') = lim f(s',0) = f(s,0! (s',t') + (s,t) s' + + s (s',t ')*(s,t) (Recall that the definition of right continuity allows us to take limits along vertical or horizontal paths as well, unlike left limits.) Hence f(«,0) is right continuous, so the variation is right continuous, i.e., Var Pn ,f(-,0) = lim Var_ ,,f(»,0). Similarly, [0,s] s% + 3 [0,s ] taking s' = s = 0, we have f(0,t) = f(s,t) = lim f(s'.t') = lim f(O.t'), (s',t') + (s,t) t' + + t (s',t')*(s,t) so f(0,-) is right continuous. The variation is then right continuous, so Var r -f(0,O = lim Var r f(0,O. Then each of LU.tj t% + t LU.t J the terms of |f| is right continuous; hence |f| is right continuous on R 2 . + Conversely, assume |f| is right continuous at each point 2 (s,t) e R . Taking t 0, and letting s'++s along the path t = 0, we have Var [0,s'] f( *' 0) " Var [0,s] f( '' 0) = l f l (s '» 0) " |f|(s,0). In fact,
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50 |f|(s',0) |f|(s,0) = |f(0,0)| + Var [Q g ^f(«,0) + Var [Q 0] f(0,-) + Var [(0,0),(s',0)] f " ( l f(0 '°' +Var [0,3] f( -'° : Var [0i0] f(O f O + Var [(o>0)i(g>o)] f) " Var [0,s'] f(> ' 0) " Var [0,s] f( -' 0) Then Var r _f(»,0) = lim Var. ,,f(-,0) since Ifl is right CO,s] s , ++s [0,s ] continuous, i.e., the variation of f(«,0) is right continuous; hence f(»,0) itself is right continuous. A similar computation taking s = shows that f(0,«) is right continuous. Then, writing Var [(0,0),(s,t)] (f) = l^ 3 ' ' ' f( °' 0) « Var [0,s] f( '' 0) -Var [0>t] f(0,.), we see that each term of |f| is right continuous. Now, let (s',t') £ (s,t), (s',t') * (s,t). We have |f(s',t') f(s,t)| = |f(s',t') f(s',t) + f(s',t) f(s,t)| £ |f(s',t') f(s",t)| + |f(s",t) f(s,t)|. We note now the following inequalities (cf. Figure 2-9): 1) |f(s',t') f(s',t)| = |f(s',t') f(s',t) f(O.t') + f(0,t) + f(O.t') f(0,t)| < |f(s',t') f(s',t) f(O.t') + f(0,t)f + |f(0,t') f(0,t) |
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51 = ' A [(0,t),(s',t')] f l + I" '*'' f( °' t) l Var [(0,t),(s',r')] (f) + I'* '*'' -f(0.t)|. 2) |f(s',t) f(s,t)| = |f(s',t) f(s,t) f(s',0) + f(3,0) + f(s',0) f(s,0) | < |f(s',t) f(s,t) f(s',0) + f(a,0)| |f(s',0) f(s,0)| ' ,A [(s,0),(s',t)] (f) l + > f(3 '' 0) f(3 ' 0) B Var [(s,0),(s',t)] (f) + l f(5 '' 0) -?(*>Vl Putting everything together, we have |f(s',t') f(s,t)| S |f(s',t') f(s',t)| + |f(s',t) f(s,t)| * Var [(o,t),(s',t')] (f) + I" '*' 5 " ^'^l + Var ;cs,o),(s',t)] (f) + l f(s '' 0) f(s « 0) " = [Var [(0,0) f (s',t')] (f) Var [(0,0),(s,t)] (f)] + |f(0,t') f(0,t)| + |f(s',0) f(a,0)| (cf. Figure 2-9). As we showed above, each of the three terras on the / Figure 2-9 Rectangles used in (1 ) and (2)
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52 right tends to zero as (s',f) decreases to (s,t) , so we have lim |f (s'.t') f(s,t) | = 0, (s ',t ') + (s,t) i.e., f is right continuous. I Remark . We still have the same result if we extend f, |f| by zero 2 outside R . + The next result concerns the existence of "one-sided" limits and "limits at infinity." o Theorem 2.3.3 . a) Let f: R + E be a function with finite variation |f|. Then each of the following limits exists* at each z = (s,t) e R 2 : 1) f(s +f t + ) = lim f(s',t') s ' + + s t'++t 2) f(s_,t + ) = lim f(s'.t') s' + t s t' + + t 3) f(s_,t_) = lim f(s'.t') s ' + + s t' + tt H) f(3 + ,t_) = lim f(s',t'). s'++s t'+tt Moreover, if f has bounded variation (i.e., if there exists M>0 such that |f|(s,t) < M for all (s,t) c R^), then each of the following * Of course, on the axes, not all these limits make sense. It will be understood that at each point we take limits from quadrants where f is defined.
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53 "limits of infinity" exist: 1 ') f(s + ,«) lira f(s'.t') s '+ + s t 'too 2') f(s_,°°) = lira f(s'.t') s 't+s t 'too 3') f(*,t ) = lira f(s',t') t'++t s ' + 00 4') f(",t_) = lim f(s',t'), and especially t't+t s '*°° 5') f(») = lim f(s',t') exists. b) If, moreover, f is right continuous, then the one-sided limits along the vertical and horizontal paths f(s,-), f(*t) are equal to the following. i) lira f(-,t) = lim f(s,-) = f(s+,t+) (right limits) s' + + s t'++t ii) lim f(. ,t) = f (s-,t + ) , and s '++3 lim f(-,t) = f(»,t+) if |f| is bounded, s'+m iii) lim f(s,-) = f(s+,t-), and t't+t lim f(s,«) = f(s+,») if |f| is bounded. t't* p Remarks. 1) Here f is defined on R : if we wish to use an f defined + also on the "boundary at infinity," the above limits will be denoted with the symbol »in place of °°.
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54 2) In general, a function of finite variation can have eight different limits at a point: the four "quadrantal" limits from part (a) of the statement, plus the four one-sided limits along the vertical and horizontal paths. Part (b) of the statements says that if F is right continuous, the one-sided limits can be incorporated into the quadrantal limits, so there are only four distinct limits at a point (s,t) (at most). The first part of (b) says that both right limits along the vertical and horizontal paths are equal to limit (1), the second part says the left limit along the horizontal path is equal to limit (2), and the third part says the remaining left limit is equal to limit ( H) , giving the division of the plane shown in Figure 2-10. There are analogous considerations for the "limits at infinity." (s,t) (1) (3) ObO (s.t) (2) (s,t) (4) Figure 2-10 The four "quadrants"
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55 Proof , a) Assume first that f has finite variation |f|. The proofs of limits (1 )-(*) are similar; we treat (1) first. We shall show that for any sequence (s ,t ) with s ++S , t + + t, the M n' n n n sequence |f(s , t )} ,, i s Cauchy in E. We shall do this by denial: n n ncN Let (s ,t ) be a sequence as above and assume that |f(s ,t )} is n n n n neN not Cauchy--we shall reach a contradiction. Since f(s ,t ) is not Cauchy, there exists e>0 and a subsequence n n (n, ), such that, for all k, we have k keN If (3 ,t ) f(s ,t ) | > e n . n, ' n, , n, n. k+1 k+1 k k We will henceforth denote this sequence by |f(s ,t )} .,# so we have n n 'neN the inequality for all n. For j given, consider the subdivisions s > s > ... > s. 3 = s of [s.s^ and t 1 > t 2 > ... > t. > t. . = t of [t,t,]. For i = 1,2,...,j-1 denote R = [( s ,0) , (s , t ) ] and R' = [(0,t ) ,(s ,t )] (Figure 2-11). For each i, we have |A R f| = l«VW ' «« H .t 1M ) " f ( s i'°) + «» 1+1 .0)|. so u R n >i«vW ^im'Vi'I ! f(s i« 0) f(s i + r 0) i => |f(s.,0) f(s i+1 ,0)| |A R f| > |f(s lf t l+1 ) f(8 1+1 ,t 1+1 )|. We hav similarly |A R ,f j = IfCSj.tj) f( Sl ,t. +1 ) f(0, ti ) f(0,t. +1 )| 1 > Ifts^tj) r(s 1 ,t 1+1 )| Bf(o,t i ) f(o,t i+1 |
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56 t . J s s .--s , , .1 J 1+1 ) s j+l s i s s 1 I Figure 2-11 Partition of [ (0, 0) , (s ,t) ] > |f(0,t.) f(0.t l + 1 )| |A R .f| i Ifls^t.) f(s if t i+1 )|. Puttlm the two together we have, for each i |f(s.,t.) f(s l+l .t l+1 )= If(s.,t.) f( Si ,t. +1 ) f(s.,t i+1 ) f(s l+1 .t l+l )| < !f(s. >ti ) f(S i ,t. +1 )l !f(s.,t. +1 ) f(3 i+1 ,t l+1 )| < |A R ffl + |A R .f| + lf(s.,0) f(s 1 + 1 ,0)| ||f(0,t.) f(0,t. + 1 ) Now, upon summing over the i's, we have
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57 J-1 Z |f(s.,t ) f(s. ,,t. J " 1 i i + 1 i + 1 J-1 l (|A R . f ' + 'V fl + l f( V 0) " f(s 1+1 ,o) J i = 1 i + |f(o,t i ) f(o,t i+1 )f) j-1 j-1 E (|A f| + |A_,f|) Z |f(s.,0) f(s. ,,0) 1-1 R i R i 1-1 X 1+1 j-1 + Z |f(0,t ) f(0,t. ,) 1-1 i i+1 L (0,0; , (s ,t ) J [s.,s J [t.,t J (since the union of the R , R' is contained in [( 0,0) , (s , t ) ] ) * Var [(o,o), ( s l ,t 1 )] (f) + Var [o.,s/ ( -' 0) + ^[o.t/^ (J-1)e , hence If^s^t,) > (J-l)e The left hand side does not depend on j , so letting j -» , we obtain |f|(s ,t ) = + °°, a contradiction on the assumption that |f| < »,
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58 Thus, for any sequence (s ,t ) decreasing to (s,t), the sequence n n |f(s ,t )} is Cauchy in E complete; hence lim f(s ,t ) exists. We n n neN n n show now that we get the same limit for any sequence decreasing to (s,t). Consider two sequences (s ,t ) and (s'.t"), both decreasing to n n n n (s,t). We construct a new sequence (p ,r ) as follows. * n n We set (p.tO = (s ,t ), (p 2 ,r ) = the first term of (s^.t/) smaller than (p ,r ), (p ,r ) = the first term of (s ,t ) after (s ,t ) smaller than (p ,r ), etc. The sequence (p ,r ) then decreases and converges to (s,t) since both the evenand oddnumbered terms do. Then L = lim f(p ,r ) exists from above. Looking n n n at the odd-numbered terms, we have L = lim f(p ' r 2k+1 ^ " But the k '" '' odd-numbered terms form a subsequence of (s ,t ), and we know f(s ,t ) converges, so L = lim f(s ,t ) as well. Similarly, since n n n n n the even-numbered terms form a subsequence of (s',t'), we obtain n n L = lim f(s'.t') as well. The limit is then independent of the n n particular sequence, so lim f(s',t') exists, and it is (s',t')*+(a,t) this limit we denote by f(s ,t ). The proofs of C 2 ) ( M ) are similar, and we will omit some computational details where they are identical to the ones for (1). Proof of (2). We consider a sequence (s ,t ) * (s,t) , with 3 tts n n n t + + t, and show that the sequence ff(s ,t )} ,, is Cauchy in E, which n n n 'neN we again do by denial.
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59 As in the proof of (1), we extract an e > 0, and (s ,t ) such that (s ,t ) + (s,t), and for each n we have s , > s , t , < t , n n n+1 n n+1 n and If (s , ,t ,) f(s ,t ) I > e.. As in (1), for given j, conside1 n+1 n+1 n n " the subdivisions s < s < ... < s. < s = s of [s ,s] and t > t > ... > t. > t = t of [t,t ]. For i = 1,2,...,j-1, denote R, = [(s.,0),(s. ,,tj], R; = [(0,t. ,),(s. ,,t.)]. (See Figure 2-12.) l ii+1i i i+1 i+1 l We have, for each i (similar to before): ' a r f ' = ' f(s i + i'V " r^i^S ~ f(s i+ i' 0) + f(s i' 0) l i > |f(S i+1 ,t.) " f(3 1 ,t 1 )l |f(S 1+1 ,0) f(s.,0)|, hence t.
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60 |f(s. +1 ,0) f(s lf 0)| |A R f| Z !f(s 1+1 ,t.) f(s.,t.)| |A R jf| If(3. +1 ,t.) f(3. +1 ,t i+i ) f(0,t.) f(0,t l+1 )I * |f. |f(8 .t ) f(s i + 1 ,t i + 1 )|. i Here we diverge from the proof of (1), since the rectangles R., R' 11 overlap (see Figure 2-12). From the first inequality we have, upon summing over i, J-1 J-1 I |f(a ,t ) f(s ,t )| i I (|A f| |f(s ,0) f(s 0)|) 1=1 1*1 i j-1 J-1 = * K f| + E |f(s .0) f(s 0)J 1=1 11-1 v ai°r/^ ^ , ,,(f) + Var r -,f(',0) [(0,0) ,(s,t )r [s ,s] as in the proof of ( 1 ) f|(s,t ) (as in the proof of (1)). Also, by a similar computation, we have
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61 j-1 Z |f(s. ,,t.) f(s. .,t. , 1+1 i 1+1 i+1 i-1 j-1 J-1 < z |a ,f| + z |f(o,t ) f(o,t ) 1-1 i i=1 [(0,0) ,(s,t. ) J Lt.t.J * |f|(8,t ). Putting the two together, we have (as in (1)): j-1 Z 1 = 1 I !f(S i ,t i ) -f(3 i+1 ,t i+1 ) j-1 z i-1 = Z |f(8 ,t ) -ft^.V '<» 1 + 1'V -f(3 i + 1 ,t i + 1 ) `*S .) < M, so as before j-1 (J-1)e < I |f(a 1 ,t 1 ) f(s l+l .t l+1 )| S 2M, and we conclude as above. For (2'), (4'), and (5'), we follow the same computation as in the proof of (1) and obtain j-1 (j-1)e < I IfCSj.tj) f(s i+1 ,t l+1 )| < |f | (Sj.tj) < M and conclude as in the proof of (1). This completes the proof of (a). Proof of (b) . Assume, now, that f is right continuous (order sense!). We shall deal with (u) and (ui) first).
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65 Ad (ii) . Denote L = f(s-,t+), let z > 0. There exists 5 > such that for all (s',t') with s'*~~t, s-s ' < 5, t '-t < 6, we have |f(s',t')-L| < . Let, then, s' < s with s-s' < 5: since f is right continuous, there exists a point (s",t') with s' < s" < s, t' > t, t'-t < 6, such that |f(s',t) f(s",t')| < . But we also have " 2 jf(s",t')-L| < | , hence |f(s',t)-L| < |f(s',t) f(s",t')| |f(s",t')-Lfl < | + | = e . Thus, lim f(s',t) = L = f(s-,t+). s ' 1 1 s The proof of the limit at infinity is much the same, except that instead of s-s' < 6, there is N such that for all s' > N, the conditions hold. We then take s' > N, s" > s, and the remainder is the same. Ad (iii) . The proof of (iii) is the same as that of (ii), with the roles of s and t being reversed: for e > there exists 6 > such that for all s' > s, t* < t, etc. The remainder is the same. Ad (i) . This follows immediately from the definition of right continuity: all three limits are equal to f(s,t). We should remark, however, that it is the same to define right continuity using the open quadrant: the limits along the horizontal and vertical paths are then the same as the "quadrantal" limit. In fact, for e > 0, choose 6 > so that for s' > s, t' > t, s'-s < <5 , t'-t < 6, If (s, t)-f (s ',t ') J < ^ . 2 Then for any s' > s with s'-s > 6, there is a similar " neighborhood" for the point (s',t). Pick any point in the intersection of these " -neighborhoods," and apply the triangle inequality as before. Our next result concerns the existence of a "Jordan decomposition" for functions of two variables with finite variation:
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66 in two variables, we have two distinct definitions of "increasing," but our decomposition satisfies both. Proposition 2.3.^ . Let f: R * R have finite variation (again, in the sense of Definition 2.3.1). Then we can write f = f f , where f and f are increasing in both senses of Definition 2.1 .3, namely a) for (s.t) £ (s',t') we have f (s,t) < f (s'.f) and f (s,t) S f (s'.t') and b) for (s.t) < (s'.f), A r , .. , , , N -,(f,) ^ and L(s,t) , (s ,t ) J 1 [(s,t) , (s ,t )] 2 Proof . For (s,t) e R^, set f (s,t) = |f|(s,t), fgCs.t) = f (s,t) f(s,t) = |f|(s,t) f(s,t). In remark (4) following the definition of |f| (Definition 2.1.3), we showed that |f| is increasing in both senses. We then have only to deal with f . a) Let (s,t) £ (s'.t'). We have f (s'.f) f (s,t) = |f|(s',t') f(s'.t') (|f|(s,t) f(s,t)) = ( | f | (s'.t ') |f|(s,t)) (f(s'.t') f(s,t)) We shall show f(s'.t') f(s,t) ^ |f|(a',t') |f|(s,t). Denote by R the rectangle [(0,t) , (s'.t ')], by R the rectangle [ ( s,0) , (s ', t) ] (Figure 2-1*0 . We have f(s'.t') f(s,t) < lf(s',t') f(s,t)|
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67 If(s'.t') f(s',t) + f(s',t) f(s,t) Jf(s'.t') f(s',t)| |f(s',t) f(s,t)| = |A f + f(O.t') f(0,t)| + |A . f f(s'.O) f(s,0)| 1 2 < |A f| ||f(0,t') f(0,t)| | A_ f| + |f(s',0) f(s,0) < Var_ (f) + Var r . . ,-,f(0,O Var D (f) + Var r , n f(«,0) n. Lwt J Kp LS,SJ = (Var r , . . , ..-.(f) Var r , . . n (f)) [(0,0) ,(s ,t ')] . [(0,0) ,(s,t) ] + Var [t.t'] f<0 ' + Var [s,s'] f( '' 0) = |f| (s'.f) |f| (s,t) (cf. Remark H following Definition 2.3.1). t"
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66 f (s'.f) f (s,t) = (|f|(s',f) |f|(s,t)) (f(s'.t') f(s,t)) > 0, i.e., f (s,t) £ f (s',t'), b) Let, (s,t) < (s',t'): we have A [(a i t>,(s\t')] Proof . We give the proof in several steps. 2 1) Let 6 be the family of rectangles of R" of the form (z,z'] t z0, there exists a finite family (J ), h = 1,2, ...n of disjoint rectangles from 6, J C j for all h, such that h n n I |o(J ) | > a, i.e., I |A (f+) | > a. h-1 n h-1 n Denote J = ( (s. ,t. ) ,(s ',t ')] for all h. We may, of course, assume h h h h h n that J = \^J J (so that [^J j c 5). Let e > 0. Since f has n=1 h
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72 right-limits everywhere, there exists a number p > common to all the vertices of all the J, such that h |f f(s h+P , V p)i <. h h= 1 closed, bounded rectangle in R , and the family P_ = jl : h = 1...n} forms a partition of I according to Definition 2.2.1. Also for each h, we have |A (f+) A (f)| h h |f(s'+,t'+) f(s'+,t +) f(s +,t'+) + f(s +,t +) 1 v h ' h h ' h h ' h h * h (fCSjJ+p.tjJ+p) f(s^+p,t h +p) f(s h +p,t^+p) + f(3 h +p,t h +p)) (f(s^,t h +) f(s^+p,t^+p)) (f(s h '+,t h +) f(s^+p,t h +p)) (f(S h +,t^) f(S h + p,t^p)) + (f(S h +,t h +) f(3 h + p,t h +p)) ,f(s h +,t h +) " f(s h +p>t h +p) l * l f(s h +>t h +) " f(s h ' +p 'V p) l + ,f(s h +,t h +) ' f ^v p,t h +p) l + l f(s h +,t h +) " f( V p *V p)1 e e e e + + -r + ^=r
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73 In particular, we have |A. (f)| > |A (f+)[ E. h h Upon summing over h, we obtain Var (f;P) = I |A (f)| > I lA (f + )| nc > a ne. h h h Now, denote J = ( (p,r) , (p ' ,r ') ] : we can take p<1 , and decreasing with E, so for any e, we have {J I = IC:[(p,r),(p' + 1,r' + 1)]. Denoting h this latter rectangle by K, we have K Dl for all I (in general, I depends on e) , so Var (f) > Var (f) > a nc . c arbitrary => K 1 Var (f) > a. Now, the collection J depends on a, but they all have union equal to J, so we can repeat the above procedure for any a and keep I C K. Thus, Var (f) > a for any a => Var ..(f) = + , a K K contradiction on our assumption of finite variation of f. Then o has finite variation on 6. 4) o is inner regular on 6. We observe first that, f^om the 2 fact that the right-limit of f exists at each point of R , it follows 2 2 that for each z e R , e > 0, there exists z e R with z0 such that for any points u,v>z, with | u-z I |f(z+) f(z')| < | . Then |f( u +) f(z+)| S |f(u+) f(z')| + |f(z') f(z+)| $ Z + I = e.
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74 Now, let J e 6, J = ( (s, t ) , (s ', t ') ] with s~~~~0. There exists a point (p,r) e J such that for any point (h,k) with (s,t) < (h,k) < (p,r) we have |f(h+,k+) f(s+,t+)| < | , |f(s+,t'+) f(h+,t'+)| < | , ||f(s'+,t + ) f(a'+,k+)| < | , as in Figure 3-2. (We can do this for each by the above, and we use a common n in choosing our (p,r).) ~~~~1 ,t >1 ) ] belongs to 5: clearly A C Int B for each i; hence n n n A = \^J A C \^J Int B G Int ([^J B.) = Int B, denoting i=1 i=1 i=1 x B = [J B. t C. Take A' = B. We have now an extension o of o to C satisfying: 1 ) o is additive on C 2) o is regular on C 3) o has finite variation on C. Then, by a standard theorem of measure theory, o can be extended uniquely to a Borel measure m of finite variation. This measure clearly coincides with o on 6 , so the theorem is proved. I Remarks . 1) The theorem proved by Radu is for R : we have restricted ourselves to R to enhance the clarity of the proof, but R n presents no additional difficulties (except with notation!!).
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77 2) The theorem holds in particular for the situation we use: that where f is defined on R , with |f| bounded, and f extended by zero outside the first quadrant. 2 Suppose, now, that f is defined on R , right continuous, with |f| finite, and extend f by zero outside the first quadrant. As an 2 exercise, we shall compute explicitly the measure of some sets in R using the limits developed in Theorem 2.3.3. More precisely, we will compute the measure of points, intervals, and some rectangles in terms of the "quandrantal" limits of f. 2 i) Let (s,t) e R . Denoting by m the measure associated with f, we compute m ({(s,t)}). We can write |(s,t)} = I ) A , n=1 where A denotes the rectangle ((p ,q ),(p',q')l with n ° F n ,M n F n ,M n (p ,q ) < (s,t) < (p',q'), p + + s, q + + t, p'+s, q'+t (see Figure n n *n n n n n n 3~3). If we decompose A into four parts, labeled I-IV in Figure 3-3, we see that as (p',q') * (s,t) , parts I, II, and IV vanish. We may, n n ) = f(s +t O -f(s_,t;) -f(s + ,t_) + f(s_,t_). As for the open interval, we have {s} x (t,t') = |s} x [t,t') \ l(s,t)}, so m f ({s}x(t,t')) = m f ({s}x[t,t')) m f ({(s,t)}) = f(s + ,t;) f(s_,0 f(s + ,t_) + f(s_,t_) (f(s,t) f(s + ,t_) f(s_,t + ) + f(s_,t_)) f(s + ,0 f(s_,t_') f(s,t) + f(s_,t + ). We use the same method for the intervals with t fixed. We give the results: m ([s,s')x{tj) = f(s^,t+) f(s^,t_) f(s_,t + ) + f(s_,t_)
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82 m f ((s,s']x{t}) = f(s',t) f(s + ',t_) f(s,t) + f(s + ,t_) m f ((s,s')xft}) » T(s'_,t + ) f(s_',t_) f(s,t) + f(s ,t_). iii) We now compute the measure of certain rectangles in R If we allow the possibility of each side being open or closed, this gives 16 different rectangles, and we do not give explicit computations for them all. We shall go into detail for only a few, and indicate the procedure for the remainder. First, we shall give the measure of an open rectangle R = (s,s')x(t,t'). We write R -( )r , where R = ((s ,t ),(s',t')] V n n n n n n n with (s .t ) + (s,t), (s ,t ) +* (s'.f) (see Figure 3-6). n n n n r l_ _ (s,t) (s ,t ) "1 (s',tO (s'.t 7 ) Figure 3~6 Approximation of open rectangle We have, then, in (R) = lim m fR ) f f n n lim (f(s'.t') f(s',t ) f(s ,t') + f(s ,t )) n n n n n n n n
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83 lim f(s'.t') lim f(s',t ) lim f(s ,t') + lim f(s ,t ) nn nn nn n n n n n n lim f(s'.t') lim f(s',t ) lim f(s ,t') + lim f(s ,t ) '.. » n n , , n n n n n n s t+3 3 t+3 s +s s +3 n n n n t'ttt' n t n n t't + 1 n t +t n f(s'.t') f(3',t ) f(s ,t') + f(s,t) We next compute the measure ra f of a closed rectangle R = [(s,t),(s',t')] = [s,s'3 x [t,t']. We can write R =C > \R , where n n R n = ((s ,t ),(s',t')] with (s ,t ) + (s,t), (s',t') + (s'.t') (see Figure 3"7). As before, when n * », the rectangles labeled I-III vanish, and so by o-additivity of m_ we can take (s'.t') = (s'.t') f n n without loss of generality. (s',0 n' rr Figure 3~7 Approximation of closed rectangle
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8^ We have m_(R) = lim m_(R ) i in n = lim(f(s',t') f(s ,t') f(s',t ) + f(s ,t ) _ n n n n = f(s',t') lim f(s ,t') lim f(s',t ) + lim f(s ,t ) n n n t t+t n f(s'.t') f(s ,t') f(s',t ) + f(s ,t ). With these in hand, to obtain the measure of other rectangles it is simply a matter of adding or subtracting the appropriate intervals that comprise the sides of the rectangle. We illustrate this procedure (as well as check our work!) by using R and some intervals to compute m ( ( (s,t) , (s ',t ')]). (Note that the rectangle ((s,t),(s',t ')] = (s,s'] x (t,t'].) We have, denoting this rectangle by A (Figure 3-8), (s,t') i (s,t) (s',f) 1s',t) Figure 3-8 The rectangle ( s,s ']x( t , t ']
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A = f([s,s']x[t,t']) \ C{s}x[t,t'])} \ {(s,s']x{t}}. (Note that the points (s,t ') , (s ', t) do not belong to A!) Then m f (A) m f ([s,s']x[t,t']) m ( {s}x[t,t ']) m ( (s,s ']x| t) ) = f(s'.t') f(s_,t + ') f(s + ',t_) + f(s_,t_) (f(s.t') f(3_,t + ') f(s f ,t_) + f(s_,t_)) (f(s',t) f(s + ',t_) (s,t) + f(s + ,t_)) = f(s'.t') f(s.t') f(s',t) + f(s,t), which is how m (A) was originally defined. The measure of other rectangles can be computed similarly using the parts already explicitly given. 3.2 Functions Associated With Measures In this section we consider the converse problem, namely, given o an E-valued measure m on R with finite variation, is it possible to associate a function with finite variation such that m = m in the f sense of Theorem 3.1.1? The following theorem provides a partial answer to this question. 2 Theorem 3.2.1 . Let m: B(R ) * E be a measure with finite variation i i ? |m|. There exists a function f: R > E with Var D (f) < on bounded R
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86 rectangles R such that m is the measure associated with f by Theorem 3.1.1, i.e., such that for all rectangles R = ( (s,t) , (s ' ,t ') ] we hav= m(R) = A (f) = f(s'.t') f(s',t) f(s.t') + f(s,t). Proof . Define f: R 2 E by f(s,t) = m( (-», (s,t)])« for 2 (s,t) e R . We show first of all that A D (f) = m(R) for bounded R rectangles R. We have, denoting R = ( (s, t) , (s ',t ')] : A R (f) = f(s'.t') f(s',t) (f(s.t') f(s,t)) = m(( ,(s',t')]) m(( ,(s',t)]) [m(( f (s,t')]) m(( ,(s,t)])] = b(( ,(s',t')] \ ( ,(s' f t)]) m(( ,(s,t')] \ ( ,(s,t)]) = b({( ,(s',t')] \ ( ,(s',t)]} \ {( ,(s,t')] \ ( ,(s,t)]}) = m(R). We now show that f has finite variation on bounded rectangles, i.e., that Var (f) < » for bounded rectangles R = [ (s,t) , (s ',t ') ] . Assume note: suppose there exists R = [ ( s, t) , (s ',t ') ] such that Var (f) = + °°. Denote by R the half-open rectangle ( (s,t) , (s'.t ')]. n Let o: s = s < s < ... < s = s ' be a partition of [s,s'], t: t » t. < t, < ... < t t ' be a partition of [t.t'L and let 1 n * ' £ = oxt be the corresponding partition of R (cf. Prop. 2.2.5). Note : ( ,(s,t)] = {z e R 2 : z < (s,t)
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37 Now, for any a>0, there exists such a partition _P such that Var (f;P) > a , i.e., E [a (f)| > a . 0~~* a . a arbitrary => |m|(R) = + », a contradiction since m has finite variation. Hence Var (f) < » for R bounded, and R the theorem is proved. I Remarks . 1) We have said nothing about uniqueness of f. In the case of functions on the line, f is determined within a constant by m (i.e., any other associated function g is determined by adding or subtracting a constant from f ) , but here this is not the case. In fact, as we have seen before (Example 2.1.2) that many completely unrelated functions can have zero as its associated measure. 2) The o-additivity of m implies that f is incrementally right continuous, but as we have seen in Chapter II this is insufficient to imply right continuity in the order sense without imposing finite variation on the one-dimensional paths f(s;«) and f(-,t).
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We return now to the situation with f defined on R . right continuous, with finite variation |f|, both extended by zero outside the first quadrant. We have an important equality we shall make use of in the next chapter, which is given in the following. Theorem 3.2.2 . Let m be the E-valued measure associated with f, and let m | be the real-valued measure associated with |f|. Then m has finite variation |m | and we have the equality m |f| = ivProof . We showed in Theorem 3.1.1 that m has finite variation. The real thing to be proved here is the equality. Let S be the semiring of rectangles of the form R = ( (s,t) , (s ',t ')]. We shall show first that m, = |m | on S. We must consider various cases. First of all, if (s',t') < (0,0), then m, ,(R) = |m | (R) = 0. We will assume, then, in what follows, that (s'.t') lies in the first quadrant. There are four cases, according to what quadrant (s,t) lies in. 1) Assume (s,t) lies in the first quadrant. Let o: s = s,_ < s, < ... < s = s ' be a partition of [s.s'l, t: t = t < t 1 m ' ' 1 < ... < t = t ' be a partition of [t,t']. Denote P = oxt the corresponding partition of R: P = R. .|R. . = C(s.,t,),(s. ,,t. J], < i < m-1, < j < n-1 Denote by R. . the corresponding half-open rectangles
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89 ( (s. ,t ) ,(s. + 1 ,t )]. (We shall use this notation in the other cases as well. ) We have E |m (R )| = E |A (f)| < Var (f) = A_(|f|) i,j ,J l.j H i,j R R (cf. Remark 4 following Defn. 2.3.1), and the right hand side is just mi f i(R). The family (R .) forms a disjoint cover of R with {J R = R, so taking supremum we obtain |m |(R) < m, ,(R). For the 1,J 1,J -J LU other inequality, let e>0. There exists a grill oxt such that E |A (f)| > Var (f) e = m |H (R) e. I.J i,J ! ' But the left hand side is equal to Z |m (R )|. and the R i,j f iJ iJ forms a decomposition of R, so we have |m f (R)| > Z |m f (R 1 )| = Z |A R (f)| > m |f| (R) e , l.j ' ? 1,J i.j i.e., |m (R)| > m. ,(R) e. Letting e -» (neither side now depends on the corresponding grill), we obtain |m (R) | > m, ,(R); hence |m f (R) | = m, f ,(R). 2) Suppose now (s,t) lies in the second quadrant, i.e., s<0, tSO. For any grill oxt, we can refine o by including zero if it is not already there, so that we may take oxt with zero included in o, and compute variations with these grills (Figure 3-9). Denote by k the index where s = 0: For i < k 2, |m,(R, .)| = 0; w e also have * 1 f i , j '
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QO (s',f) Figure 3 _ 9 The grid oxt K'Vij'I = rCB kt j*i^ " f( VV " ''VrVi 5 + f( Vi'Vl = |f(0,t j+1 ) f(0,t j )|. Putting everything together, we get n-1 I |m (R )| = I |f(0,t. +1 ) f(0,t,)| + Z |A B (f)I l.j f 1,J j = 1 J 1 J i>k R i,j SVar^^ftO.O + VaP [(0>t)((s . tt , )](n = m, f , (R), since m. f i(R) = |f|(s',t') |f|(s',t) |f|(s,t') + |f|(s,t) = |f|(s',t') |f|(s',t) = [|f(0,0)I Var rQ s . ] f(',0) Var [Q t . ] f(0,O + Var [(0,0),(s',t')] (f)] " Uf(0.0)| Var^^fC-.O) Var [0>t] f ( 0, . ) + Var C(0,0),(s',t)] (f)] " [Var [(0,0),(s' ( t')] (f) " Var [(0,0),(s',t)) (f)]
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91 + [Var [0,t'3 f(0 '° ' Var CO,t] f(0 '' )] Var, [(o,t),(3',t')3 (f) + Var r t t .'-i f (°.')Taking supremum over grills oxt on both sides, we get (as before) |m (R) | < m, ,(R). On the other hand, for any c>0, there exists a partition t ' of n-1 Ct.t'] such that z |f(0,t ) f(0,t.)| > Var r ,-,f(0,O § , and * _g J i J L t , t J 2 a grill o q xt of [0,s ']x[t, t '] such that i|a (f)| > i , J Var T(0 t) (s' t')1^ " 2 ' We ctloose a common refinement x of t' and i , and extend arbitrarily to get a partition o of [s,s']. Then, for the grill oxt, we have n-1 £ |m (R )| = I |f(0,t ) f(0,t )| I |a (f)| (as before) i.J ' J-0 J J o q xt K i,j >Var [t,t'] f( °'* -! +Var [(0,t),(s',t')] (f) "I = m, , (R) e. Since the left hand side is bour ded above by |m |(R) for any grill, we have |m f |(R) > m . f , (R) e. Letting z * again, we get |m |(R) 2, m, ,(R), hence equality. The next case proceeds similarly. 3) (s,t) in the fourth quadrant: s£0, t<0. This time, we refine t by including zero if necessary, and compute variations along such grills. Denoting t = a: before, we have (same computation as before) :
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92 m-1 £ h^CR, .) f (R ifj )| = l |f(3 1+1 .0) f(s.,0)| I |A (f)| i=0 i>0 ' i ,j j^k < Var r , n f(0,O + Var r , ., . , ..-.(f) Ls,s ] [(s,0) , (s ,t )] m, f ,(R) (same as before). Taking supremum we get |m (R)| < m, ,(R). The proof of the other inequality is the same as that for case (2). 4) Finally, assume (s,t) < (0,0). We proceed similar to the above, but this time we add zero to both o and t, and use these partitions in our figuring of variations (Figure 3-10). Denote s k = 0, t : = 0. For i < k 2 or j < 1 2, we have |m f (R. .)| = 0. (s',t*) Figure 3-10 The grid axx For i = k 1, j = 1 1, we have fm f (R. )| = |f(0,0)|, for
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9? i = k 1 , j > 1, we have |m f (R. )| = |f(0,t ) f (0,t ) |. For j = 1 1, i > k we have |m f (R. )| = |f(s i+1 ,0) f(s.,0)|, all as before. Putting everything together, we have m-1 I |m (R .)| = |f(0,0)| + I |f(s. ,,0) f(s.,0)| 1,J ' J i = k 1+1 1 n-1 I |f(0,t ) f(0,t )| I |A (f)| J-l J 1 J i^k R i,j JSl < |f(0,0)| Var^^fC-.O) + Var^^^fCO,-) + Var [(0,0),(s',t')] (f) = |f|(s',t') = m, f ,(R). Taking supremum again, we obtain |m (R)| < m, ,(R). The proof the other direction is similar to the ones before: for e>0, we choose a common o,t so that E|f(s 1+1 ,0) f(s 1>0 )| > Var^^-jfC-.O) | l|f(0.t J+1 ) -f(0,t.)| >Var [0)t , ]f (0,.) | , ^ o lJV^(f)|>Var [(0>0)f(s , it , )]( f)-£. We extend o and t arbitrarily to partitions o'.t' of [s,s']
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94 and [t,t']» respectively. The same computation as before gives * IV R i 1>l = l f(0 '°H + S|f(3.. 1f 0) f(3,,0)| X|f(0,t. + 1 ) i»J a l x T J f(0,t )| I |A R (f) axx l,j > l«o.o)| + v«. [0i8 , ] f(. i o)-f *v«. -| + Var [(0,0),(s',t')] (f) "I f|(s',t') e = m. f .(R) e. Hence |m f (R)| > m, (R) c. Letting e 0, we obtain |m (R) | > m, .(R); hence equality. This takes care of all the possibilities, so we have |m I = m, , 1 f 1 |f| of S. Moreover, both are o-additive on S; the first since m is by Theorem 3.1.1, the second since |f| is right continuous by Theorem 2.3.2. Since |m |, m, , are equal and o-additive on S, they are equal on R is a function of finite variation in the sense of Definition 2.3.1. We define, for X a process of finite variation, a real-valued process |x|, called the variation of X by the following: 2 for we n, z = (s,t) e R , |X| (w) = |X > (w)|(s,t) |x (0t(J) (w)| Var [0(S] |x > (w)|(.,0) + Var [Q>t] |X. (w) | (0, ) + Var [(o,o M s,t)] ( l x »l>. b) If the random variable Ixl = lim Ixl, . < + » (which s +. '(s.t) exists since |x| is increasing in the order sense) is P-integrable, we say X has lntegrable variation . In this chapter, we shall concern ourselves with processes of integrable variation. We will consider them extended by zero outside 2 R , as we did for functions earlier. Remark . In the book by Dellacherie and Meyer [5], processes of finite variation are defined as differences of Increasing processes. In two parameters, it seems we might have a problem with this, as we have two
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98 distinct definitions of "increasing." However, we have shown (Prop. 2.3. 4) that a process of finite variation, as we have defined it here, can be written as a difference of two processes (apply Prop. 2.3.4 to each path) that are increasing in both senses, thus removing the ambiguity. We give now one more result concerning functions, which will be used extensively in later theorems. Proposition ^.1 .3 . If g: R + L(E,F) is a function with finite variation |g| (Defn. 2.3.1), then for every x e E and z e F', the 2 2 functions gx: R * F and : R * R have finite variations |gx| and ||. (For the real-valued functions we shall use double bars for the absolute value to avoid confusion.) Moreover, if f: R * R + is dg-integrable (i.e., d | g| -integrable) on a set ICR , then f is d(gx)and d integrable on I, and we have (4-3.1) x /jfdg = Jjfxdg = JjfdCgx) and also (4-3.2) = = / I fd. 2 Proof . For the first assertion, let z = (s,t) e R . We have, from Definition 2.3.1 , |g|(s,t) = |g(0,0)| + Var [Q a] g(-,0) + Var f Q fc] g(0,O + Var (a) < <*> a c(o,o), (s,t)y s)
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99 We have |(gx)(0,0)| = |g(0,0)x| < flg(O.O) | |x |, and from the onedimensional case proved in Dinculeanu [7], we have Var [0,s] (gX)( '' 0) N ' Var [0,3] g(, '° ); Var [0tt] (gx)(0,.) < |x| Var [0jt] g(0,.). In fact (we prove the first; the proof of the second is identical) for = s < s < ... < s = s a partition of [0,s], we have |(gx)(a 1+1 .0) (gx)(3 i ,0)| = |(g(s. +i ,0) g(s ,0))x| * |8(s l+1 ,0) g(5.,0)|Ixl for i=0, 1 , 2, . . . ,n-1 . Summing over i, we obtain n-1 n-1 I |(gx)(s 0) (gx)(s 0)| £ I |x|-|g(s. .0) g(s.,0)| 1=0 1 ] 1 i=0 * 1 l n-1 Il x l l |3(s i+1 ,0) g(s 0)| 1 = x " i X |Var [0,s] s( *' 0) Taking supremum over partitions of [0,s], we get Var -,(gx)(-,0) L U f S J S x l' Var [0>s] g(-»0). The same proof gives Var [Q t] (gx)(0,-) £ |x|«Var r t -.g(0,«). We obtain a similar inequality for the remaining term of |gx|: For any rectangle R = [(p,q) , (p',q ') ] C R 2 we have |A (gx)| = |(gx)(p',q') (gx)(p',q) (gx)(p,q') + (gx)(p,q)
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100 |[g(p',q') " g(p',q) g(p.q') + g(p,q)]x| |(A p g)x| * MBy Then, for any grill axi on [( 0,0) , (s, t) ] , we have 1 K( S t ) (s t )i (gx) ' " l! x B * K (s)| l|x| ' Var [(0,0),(s,t)] (g) ' Taking supremum over grills of [ (0,0) , (s,t) ] , we obtain Var [(0,0),(s,t)] (gx) " X 'Var [(0,0),(s,t)] (g) Putting everything together, we get |gx|(s,t) = |(gx)(0,0)[ + Var [Q a] (gx)(.,0) + Var [Q t] (gx)(0,-) + Var [(0,0),(s,t)] (gx) S |x||g(0,0)| |x|.Var [0jg] g(.,0) + |x|.Var [0>t] g(0, ) + " X " Var [(0,0),(s,t)] (g) « |x||g|(s,t), so |gx|(s,t) £ flx||g|(s,t) < « for all (s,t) e R 2 , i.e., gx has finite variation. The same argument works for : in fact, for (p,q) e R 2 , we have (p,q) = < f (gx) (p,q) J |z[ |g(p»q) I [x| Jz J . The same proof as above then gives
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101 ||(s,t) £ ||x||z||g|(s,t) < for all (s,t) e R + , i.e., has finite variation for all x e E, z e F'. Now to prove equalities (4.3.1) and (4.3.2): Let I e B(R 2 ), 2 f: R + R be dg-integrable. Assume |g|(I) < °°. We shall use the monotone class theorem 1.5.2 to prove the equalities for bounded f, then extended to f-integrable. Let H denote the set of bounded, real-valued, dg-integrable (on I) functions f satisfying (4.3.1). Then: i) H is a vector space (evidently). ii) H contains the constants: let f = a constant. Then x/jfdg = xjjctdg = x/aljdg = x(ag(D) = (ax)g(I) (g(I) is the measure of I for the measure dg), Jjfxdg = |(ctx)1 dg = (ax)'g(I) , and /jfd(gx) = JaljdCgx) = a-(gx)(I) = a(g(I)-x) = (ax)g(I) . Hence xj^fdg = J^fxdg = / fd(gx), which is (4.3.1). iii) H is closed under uniform convergence: suppose f * f n uniformly and (4.3.1) holds for each f . Then n xLf dg * xLfdg, since by Lebesgue dominated convergence we have that f is dg-integrable over I and Lf df » f fde. ; I n J i &> hence xj f dg » xLfdg. (In fact, from some index n on we ' I n J I
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102 have If -fl < 1 => for n > n A we have If I S If | + 1. " n " 2 " n ' n ' which is dg-integrable on I.) Similarly, f x + fx n uniformly, so from some index n^ on, If x-fx| < »> " n "2 for n>n , If xl < If x| + 1 < ff llxl 1, integrable since H is a vector space. By Lebesque, then, /j^xdg Jjfxdg. Finally, since | gx | < |g|.|x|, f R are d(gx)-integrable, so J f d(gx) + Lfd(gx) by Lebesgue as before. For each n we have x| T f dg = Lxf dg = Lf d(gx), ; I n ; I n 'In so on passing to the limit we get xj fdg = Lxfdg = J fd(gx), which is (4.3.1 ). iv) Let (f ) be a uniformly bounded increasing sequence of positive functions from H, and denote f = lim f . Show n n f e H. Let M > f for all n. Then we have n J x f n d|g| £ J I Md | g| = M«|g|(I) for all n. By Lebesgue, f is dg-integrable on I, and Lf dg + C-fdg, hence 'In 'I x /l f n d 8 " xjjfdg. Similarly, Jf^J < |f n ||x| £ M* |x | e L (d|g|), so fx is dg-integrable on I by Lebesgue, and J f xdg -* J fxdg. Also, f is d(gx)-integrable since |f n | < M and J Md(|gx|) < M|x||g|(I), so by Lebesgue again
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103 we have f is d(gx)-integrable on I and Lf d(gx) -* J I n J fd(gx). Now, (1.3.1) holds for each n, so passing to limits as before, (4.3.1) holds for f as well. Now, let S be the family of sets of the form R = ( (s,t) , (s ', t ') ] () R . Since the rectangles ( (s,t) , (s',t ')] form a semiring 2 ? generating the Borel o-field on R , S is a semiring generating B(R ). Let, then, C be the family of indicators of sets of S. To complete the monotone class argument, we must show that CC H and that C is closed under multiplication, as H then contains all bounded functions p measurable with respect to o(C) = B(R ). C is closed under multiplication, as 1 -1 = 1 , and S is R. n R O R_ a semiring, so R C\ R e S => 1 e C. Now we show that (1.3.1) I £ K i i H holds for f = 1 , R e S. We have H xj^pdg = x/l Rni dg = x(g(ROl)) (Again g(-) refers to the measure dg.) Since |x(g(ROl))| £ |x|-|g| (ROD S |x|»|g|(I) < », 1_ is dg-integrable and d(gx)integrable. Also, JjXl R dg = Jx1 Rn;[ dg = x(g(RPlI)), and f I 1 R d(gx) = Jl Rni d(gx) = (gx)(RplI) = xtglRfil)), hence x/j1 R dg = JjX^dg = / 1 d(gx), which is (4.3.1), so C H. This completes the proof for f bounded, |g|(I) < °°. Assume now, |g|(I) < *, f dg-integrable on I (not necessarily bounded). There exists a sequence (f ) of bounded functions n
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ion converging to f a.s. and in L 1 (d|g|) on I, with (4.3.1) satisfied for each n. In the first integral, we have ft dg » Lfdg, hence x/j^dg x/jfdg. Also, I/ I xf n dg JjXfdgl = |/ I x(f n -f)dg|| I x lJ I l f n ~ f l d ls| * °; hence x| I f n dg » xjjfdg. Finally, f is d(gx)integrable [5, Theorem 4, p. 172], and we have |f f d(gx) / fd(gx) | = l/^^-DdCgx)! < /jIVflcKlellxl) ||x|J I |f n -f |d(|g| ) as n » «. Then J^dKgx) + Jjfd(gx). Since (4.3.1) holds for each n, we have it for f as well by passing to the limit. 2 Finally, let I e B(R + ) , f dg-integrable on I. There exists a sequence (I ) of sets from B(R 2 ) with |g|(I ) < », and I t I. n + ' ' n n Then f-^ -> f-1 a.s. and in L (d|g|) and L 1 (d( |g| |x J ) ) by Lebesgue n (since |ri x [ < [f.1 | E L 1 (d|g|) and L 1 (d | g| |x[ ) ) ; (4.3.1) is n satisfied for each I , so we pass to limits exactly as above. This completes the proof of (4.3.1). The proof of (4.3.2) is completely analogous (since || < |x||z||g|). I We conclude this section with the theorem establishing the correspondence between stochastic measures (P-measures) with finite variation and processes of integrable variation for the case of realvalued processes and measures. Although this is a special case of the more general result we will establish later, it cannot be deduced from that, since our proof for the vector-valued case will make use of the real-valued result. Theorem 4.1.4 . There is a one-to-one correspondence X +» \i between 2 raw processes X: R + xti * R with raw integrable variation |x| and
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105 stochastic measures u with finite variation |p v |, given by the A ' X ' equality (H.l|.1) u Y (A) = E(J A dX ) for A bounded, measurable. X R 2 z z + Remark . We shall later prove the equality p. . = |p | for X with |X | X values in a Banach space, from which the equality follows for realvalued X as a special case. Proof. We remark first that the correspondence is one-to-one in the sense that we identify processes that differ only on an evanescent set . 1) Let X: R + xQ > R be a raw process with raw integrable variation ]x|. For any bounded measurable process A, |a| < M, the map w * J A (w)dX (w) is in L 1 (P). In fact, we have R ^ z z + |J A (w)dX (w)| < / |A (w)|d|x| (w) < Ml X I (w), and by assumption R R^ Z z + + |X| e L (P). Then, for any M e M = B(R 2 )xF, E(f 1 w dX ) exists. + ; 2 M z R + Set, then, u Y (M) = E(f 1 u dX ). Then, for step functions X J 2 M z rv + n n B = I a 1 , M e M, a e R, we have y (B) = JBdy = I a.y(M ) = 1-1 i * * x ' i=i x l n = E(J ( I a 1 ) dX ) = E(f B dX ). Now, suppose A is bounded, R* i=1 l n i z z R^ z z ) dX )) z z
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106 measurable. Let A be a sequence of step functions, converging uniformly to A except on an evanescent set. For each n, we have u (A ) = E(J A n dX ). Since A is bounded, EC f A dX ) exists, as W X 2 z z J z z + have shown. Moreover, |E(J 2 A^dX z ) E(J 2 A z dX z )| = |E(J 2 (A n -A) z dX z ) R . R R S E(J |A n -A| d|x| ) R z z S (sup |A^-A z l)|x| oo zeR 2 + as n * « by uniform convergence. Thus E( (A n dX ) * E((a dX ). For ' z z ; z z each n, the double integral is equal to u Y (A n ), and u (A n ) -> u v (A) (since y has finite variation, which we shall prove in a minute), hence we have equality in passing to the limit, i.e., M X (A) E(| R2 A z dX z ). Now, we must show that (i) y is a stochastic measure, and (ii) u x has finite variation. The first is easy: if M e M is evanescent, then M(w) = for almost all w, hence { 1 M (w) dX (w) = P-a.s., so E(f(1J dX ) 2 M z z **' i M z Z' 0,
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107 i.e., n x (M) = 0. As for (ii), let M e M, (M.) i = 1,2,...,n disjoint n sets form M, with (JM.C M. We have then E |y (M ) | 1 i-1 X l l ! 1 ,E(/ H 2 (, M 1 ) . d vi * .= * = e( .V r 2 (1 m >, 11 a + 1 1-1 Kl 1=1 R 1 + + + E M ?' 1 n m ) 7 d l x l ) since all sections are disjoint = E(f (1 ) dlxl ^ ^ M i z ; R 2 M z I 'z + + The last integral is independent of the family (M ), so by taking supremum we get |u x |(M) < E(J 2 (1 M ) z d l X U ) " E( I X U < "» so ^ x haS R + finite variation (in particular |y | is bounded by E(|x| )). Uniqueness of y is evident: this completes one half of the correspondence. Next we prove the converse. 2) Let p be a stochastic measure with finite variation |y|. Assume, first, y>0 (then |u| = y). We will associate an increasing process X (increasing in both senses) satisfying (4.14.1); then the final result is an easy consequence of the decomposition of measures with finite variation. o For each bounded r.v. Y and u e R , consider the raw process Y (w) = Y(w)«I ,,-,(z). The map A defined by A (Y) = y(Y ) = " L U , U J U u y C Y*IjQ u -j) is a bounded, positive measure on (ft.F). In fact: for B e F, A (B) = M CI -1 r ) = y(Bx[0,u]) < y(Qx[0,u] ) . Also, A is PU B LU,UJ u absolutely continuous; if P(B) = 0, then A (B) = y(Bx[0,u]) = since y is a stochastic measure. Then A has a density a with respect to u u ^ P.
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108 Now, if u a.s. In fact, for B e F, E(1_A D (a)) = E(1 a , fc ,) n D n D S t E(1 B a s't ) " ^Vst' 5 + E0 B a st ) " Vt' (B) " Vt (B) " A sc' (B) + X (B) = y(Bx[(0,0),(s',t')]) u(Bx[(0,0),(s',t)]) y(Bx[(0,0),(s,t')]) + p(Bx[(0,0),(s,t)]) = y (Bx( (s, t) , (s ', t ') ] ) > 0. Since the fou" functions that make up A D (a) are F-measurable, we have A R (a) ^ a.s. (i.e., for w t N) for R with rational coordinates. Now, we show that for w t N (see note preceding page), we have . 1 1 2 A n (a (w)) Z for all R = ( (s,t) , (s ',t ') ] C R . Let w t N and suppose * Outside an evanescent set: for u£v, u,v rationals, a £ a a.s., u v so for each pair u,v there is a negligible set N outside of which a (w) £ a (w). We put these together into a common u v negligible set N outside of which a (w) 2 a (w) for any u,v 2 u v rationals in R .
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109 A (a (w) ) < t < 0. We can find, since a^(w) is increasing (on the rationals) on this set, nationals p*~~ If A ((s,t),(s',t')] (a1(w)) < e, then we have A.. . , , ,.,(a(w)) = a , ,(w) -a , (w) ex ,(w) + a (w) ( (P»q) .(P ,q )J P q p q pq pq £ a ,. ,(w) a. (w) a ,(w) + a (w) st p q pq st (by definition of a ) Vt' (w) (a st (w)+ ^ +a st (w) ( (s,t) , (s ,t )] 2 2 2 < 0, i.e., A., . , , , N -,(a(w)) < 0, a contradiction. Then a' is ( (P.q) . (P .q ) J increasing in both senses for w i N. 2 1 1 Now, for each z e R , we have a.s. a £ a £ a . Also, the + z z z+ map z * X (Y) is right continuous. In fact, lim X (Y) = z u u+z lim p(Y(w) I ,(v)) = u(Y(w) I_ ,(v)). Then uiz [ °' U] C °' z] X = a is also a density of X^ with respect to P. In fact, on one z z+ t hand, since a < a , for B e F, X (B) = E(1_a ) ^ E(1 D a 1 ) = z z + z B z B z+
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110 E(1 X ). On the other hand, E(1 X ) = E(1 D a 1 ) = E(1 Him a 1 )) o z a z ti z+ 3 u u+z = lim (E(1 a )) (by monotone convergence) £ lim (E(1 a )) = , l^ H u B u U+2 U+z lim A (B) = X (B) by right continuity. Putting the two together, u + z A (B) = E(1 X ) for B e F, i.e., X is a density for X . z B z z J z Then for processes of the form A (w) = Y(w) I r -,(z), z [0,u] u(A) = E(J A dX ), since E(J J dX ) = E(f _Y(w) I. ,(z)dX ) a d z z r 2 z z d 2 [0,UJ Z = E(Y(W) '^[O.u^V = EU(W) ' X u ) " V Y) * ^ (Y ' I [0,u] ) = Hlil+ Moreover, a = lim a , so a is also (same proof) z+ z z+ Vi u+z u rational incrementally increasing for w t N; hence X = a is an intestable z z+ increasing process: more precisely, |x|. . (w) = X. n <,( w ) + X (s,0) (w) + X (0 t) (w) + X (s t) (w) " 4X (s t) (w) f0r W * N ' SO outside an evanescent set |x| < 4u(R + xft) (we shall use this later). To prove the equality (4.4.1) for A bounded, we use a monotone classes argument. Let H be the set of all bounded, M-measurable processes A (w) for z which (4.4.1) holds. H is clearly a vector space. Also, H contains the constants: If A = c constant, then A = lim c I r , . (z), n+0B [(0,0) ,(n,n)] and we have shown (4.4.1) for these processes already. We get the result in the limit by monotone convergence. More precisely, denoting A (w) = c I-,,. .. . s-i(z), we have, for w outside a negligible z L (0,0) , (n,n)j s B set, sup] A (w)dX (w) = supX (w) < *>, so by monotone convergence ' z z z (n,n) ° K
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11 1 we have J A (w)dX (w) * J .A (w)dX (w) P-a.s. Then, the maps >2 z z ' n 2 z z K R + R + w + J .A (w)dX (w) increase to w * [ A (w)dX (w) P-a.s. and the r; z z V z latter is integrable as we saw above, so by monotone convergence we have that E(f A n dX ) + E(f A dX ). Since u (A n ) u(A) and (4.4.1) R Z Z R Z z + + holds for each n, (4.4.1) holds in the limit as well. Also, H is closed under uniform convergence: let A be bounded, measurable, A a sequence from H with A * A uniformly. By Lebesgue y(A ) * y(A). Also, for almost all w, X(s) is a bounded positive measure, so J S (w)dX (w) * I A (w)dX (w) P-a.s. (by Lebesgue again) so by R R + + Lebesgue the map w -» | _A (w)dX (w) is P-integrable (since each R 2 Z + A n z H) and E({ _A n dX ) ^ E(f J dX ). Since (4.4.1) holds for each R z z R z z n, it holds in the limit as well. Finally, let A be uniformly bounded, A" +A, A e H for all n. As above, using a double application of the monotone convergence theorem this time, we have E( [ A' dX ) R 2 Z Z + E(J A dX ) , and p(A ) » y(A), and we conclude as above. R ^ z z + To complete the monotone class argument, let C be the class of processes of the form Y(w) I -,(z). We already know C H; it is easy to see (taking Y indicators of sets of F) that o(C) M. Finally, C is closed under multiplication; in fact, U\w) I [0iU] (z))(Y 2 (w) I [QtV] (z)) = (Y 1 (w)Y 2 (w)) I [0>uv] (z) e C, and the monotone class argument is done. This completes the
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112 proof for \i £ 0; if p has finite variation, we write y = y u + + and associate X with y and X with y . We have, then, for A bounded, measurable: y(A) = y (A) y (A) = E( f A dX + ) R 2 Z Z ' E(J .A dX _ ) = E(J A dX + f _A dX~) = E({ A d(X + X _ )). Setti ' . ? 7. 7. J ? 7 7 J ? 7 7 J P 7 7 7 ng X = X + X", y(A) = E(J A dX ), and |x| = |X + X~| < |X f | + |x"|, R ^ z z + so X has integrable variation, and the theorem is proved. B Remark . In Meyer [12] a version of this theorem (without proof) is 2 given for P-measures and random measures on R xft. 4.2 Measures Associated With Vector-Valued Stochastic Functions In this section we shall show that, starting with a stochastic function, we can associate a P-measure, with finite variation if the function has integrable variation. Our first theorem is for measurable processes with integrable variation. Theorem 4.2.1 . Let E be a Banach space, X an E-valued, raw, right 2 continuous process such that X is integrable for every z e R , with raw integrable variation l x l(s,t) = l x (o,o)S +V8r co i 8] (x (.,o) ) + Var [o,t] (x (o,.) ) + Var [(0,0),(s,t)] (X) 2 There is a stochastic measure (P-measure) y : B(R )xF -* E with finite variation satisfying the following.
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113 If $ is any scalar-valued measurable process, we have * e L (y v ) if and only if E(( Ai> Idlxl ) < «. In this case, n + E(( dX ) is defined, 'r 2 Z z (1.2.1) y (*) E(f $ dX ), and X J R 2 z z CI2.2) |y x |(») = E(J 2 * 2 d|x| z ), R i.e., |y x | = p |x| . Proof . For M e M, the integral E([ 1 tJ dX ) is defined; to see V M Z + this, we use a monotone classes argument. More precisely, let H = {M e M: E(J 1 w dX ) is defined}. We will show that H is an V M z + algebra, is closed under monotone convergence, and contains a semiring generating M; we then conclude from the monotone class theorem that H Z5 M, hence H = M. H is an algebra . Let A,B e H; show AVJB, APlB, A c c H. First of all » / o 1 A (w ) dX Cw) exists P-a.s. as well as J 1 (w)dX (w). r; A z R 2 B z Then 1.(w) 1 R (w) is dX (w)-integrable almost surely, i.e., J 2 1 A (w) 1g(w)dX (w) = J 1 n (w)dX (w) exists P-a.s. Moreover, R + L Z R + I/ 2 1 AnB (w)dX z (w) l ' / 2 1 AOB (w)d l X !z (w) ' / 2 1 A (w)d|x| z (w); hence R , R R + + + w * J o 1 A^ Q Cw)dX (w) is P-integrable, i.e., E(f J s ^ n dX ) exists, ' 2 Add z J 2 AflB z
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m so JIObs: H, We get A (J B e H by writing 1 =1+1-1 W ' 6 AUB A B AHB and A C by 1 = 1 i (E(fldX ) = E(X )). ,0 A ' z " A H is closed under monotone convergence. Let A be a sequence of sets . n from H with A increasing to A. For almost all w, f 1 . (w)dX (w) -* a 2 A R n J 2 1 A (w)dX z (w) by Lebesgue (since 1 < 1 e L 1 (|x|(w))). For each n, R + n the map w * Jl (w)dX (w) is bounded by ^^(w) e L 1 (P) , so these n converge to w Jl (w)dX (w) a.s. and in L (P). In particular, E(Jl (w)dX (w)) exists. The proof for decreasing sequences is the same. H contains a semiring generating M . Let S be the semiring of halfopen rectangles in the plane from before, and denote P = sDn. Then U = {AxF, A e P, F e j} is a semiring generating M. For a set B e U, not only is E([l dX ) defined, but we can compute it 1 B z explicitly. There are four types of sets in P (cf. Theorem 3.2.2): 1) A = ((s,t),(s',t')]: then E(/l, ^.dX ) = E(1_fl,dX ) " AxF z F J A z = EM (A.(X))). F A 2) A = (s,s'] x [0,t']: then we get E( [l . dX ) = E(1 (X , X, ,..)). F (s ,t ) (s,t ) 3) A = [O.s'l x (t.t']: we get E(Jl dX ) = E(1(X , X. . .)). F (s ,t ) (s ,t)
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115 1) A = [0,s'] x [0,t ']: we get E((l dX ) = E(1 X, , ,J. J AxF z F (s t ) ( Note : See Theorem 3.2.2 and preceding example for computations of the measure dX(w) on these rectangles.) By the Monotone Class Theorem, H contains o(U) = M, so E( fl dX ) J M z is defined for all M e M. Set p v (M) = E(( _1dX ). Then y v : M > E X ' 2 M Z X K + is a o-additive stochastic measure: p is evidently additive. If M + , then for each w, ]1 (w)dX (w) + by a-additivity of the n ' M z n integral. Then E(/l dX ) * by Lebesgue; hence p Y is a-additive. n Z X Also, if M is evanescent, then M(w) is empty P-a.s. => fl (w)dX (w) = ! ' M z a.s. => u x (M) = E({l M dX z ) = 0. Now, y>; also satisfies Ju x (M) | < U| X |(M), since | u (M) | = |E(/l M dX z )| < E(|/l M dX z |) < E(/l M d|x[ z ) = M| x j CM) ( U | X | is the measure associated with |x| by Theorem 4.1.4); hence p has finite variation |u x l U| x | (since the variation is the smallest positive measure bounding the norm). We shall prove this is an equality. Each X , being measurable, is almost-separably valued, so we can find a common negligible set N rt outside of which X is separately z valued for z rational. By right continuity, X = lim X , z u u+z u rational so for w t N , X takes on values in a separable subspace EC E. Let Z C E' be a separable subspace norming for E n . Since |u x | £ u I x I which is finite, we have p << u . . By the extended Radon-Nikodym Theorem (Theorem 1 .5.8) , there exists a stochastic function jf: R 2 xJ2 > Z' ( = L(R,Z')) having the following properties:
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116 1 ') |H| is y, .-measurable and |h| < 1. In fact, Theorem 1.5.8 says that Jh|| is y , , -integrable and that for \\i e L (|u v |) | X | X we have Ji^d|y x | = J |H|^dp i x | Taking tj; = 1 A , A e M, we obtain |u x |(A) = /|h| 1 A dy x . < W| X |(A); hence |h| < 1 on A, so |h| £ 1 except on a y, ,-negligible set (on which we l x ! modify H appropriately, say by setting H = 0). 2') is y, | -integrable for every z e Z, and we have = J dy, , for every M e M. In fact, taking f ! 1 in Theorem 1.5.8 (2), we get y, , -integrable for all z. Also, for z e Z and M e M, we have, taking f = 1 , = Jdy| x |, i.e., ~~__ = /l M dy | x | (since = 1 ) = f dy, ,. M M ^M | X I 3') |y |(M) = J |H |dy I | for M e M. We showed this in proving (T). Now, taking M = [0,u] x A, A e F, in (2'), we deduce that (4.2.3) EM ) = E(1 L ,d|x| ) A u A J [ 0,u] w ' ' w In fact, on the left hand side of (2'), we get = = = = ; R 2 M w A J [0,u] w A u J A u + | A dP = E(1 A ), which Is the left hand side of (4.2.3). As
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117 for the right hand side, Ldy , v , = E([l dlxl ) (by Tneorem J M [X| ' M w w 4.1. i», since || < |H||z| S \z\) = E(1 A / [0>u] y d |X \J , which is the right hand side of (4.2.3), thus proving the equality. Now, since (4.2.3) holds for all A e F, there is a P-negligible set (depending on u and z) N(u,z) C Q outside of which = J r , dlxl . Since both sides of this equation are U ; [0,u] v ' ' v right continuous, there is a negligible set N9x) = [) N(u,z) u rational outside of which = \ rn , dlxl for all u e R 2 . u ; [0,u]v li v + Let S be a countable dense subset of Z and set N = (J N(z); zeS N is negligible. Also, since |x| is integrable, there is a third negligible set N outside of which Ix^U) < ». Let, now, w t NgU^U^ be fixed. The function X # (w) = X(w) is an E -valued function defined on R + , having bounded variation ^(w)^ = [ X | _ ( w) . Then (Theorem 3.1.1) it determines a Stieltjes measure y . on X ( w) B(R ) with finite variation |u , J. By Theorem 3.2.2, we have + i x(w) ' \v vf \| = V\ v , \i« Then by Theorem 1.5.7, there exists a function mw; |mwj| 2 G : R. > Z such that 1M) i G w« 1 ^ixwr a e 2") is \i, . . -integrable for every z e Z and W | A(,W j | = f M d M|x(w)| for M e B(R 2 ) . In fact, taking f = 1 in 1.5.7(2), we obtain (as for (2')): M
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113 ____ = K 1 M' Z>d ^|X(w)| = / 1 M dy |x(w)| ** /m ^|x(w)|Taking M = [0,u], we obtain = , and X(w) u ^ dy |x(w)| = /[0,u] . d l X l. (w); henCe <\^>*> = MO ul d l X l ^ * Puttin 8 this together with what we had 2 earlier for , we now have, for Z e S, u t R , J[Q,u] . d l X !. (w) = <^ u (w),z> J [0|U] j d ]x| > (w). 1 _ 2 There is then a m^,. , -negligible set N (w,z) C R + outside of which we have z> = . In fact, the two integrals above form measures on B(R + ). By taking differences, we have, for u____d l x l (w) = !r .-,d|x| (w), and these rectangles, along with those [0,u] generate B(R ); hence = + w y |x(w)] -a.e. Now, the set N (w) = [) N (w,z) is y . , ..-negligible, and for 1 zeS l X(w) l u t N (w) we have = for all z z S; hence for all u w z e Z since S is dense in Z. Since Z is norming, we have H (w) = G (u) for u t N 1 (w). u w Let A = {(u,w): |H (w) I < 1 } . A is then y -measurable (in fact, A = A U N with A Q e M, N y| x , -negligible. Then N is y x -negligible so A is y -measurable). For each w, consider the section A(w) = {u||H u (w)| < 1}; since for w t N JjN.ljN we have |G wl = l p |x(w)|" a e ' and H u (w) = G w (u) y |x(w)|" a ' e " we deduce that A(w) = {u: |H u (w)| < 1} = |u: |G w (u)| < l} (a.e.) is V| x(w) |negligible. Then U| X |(A) = E(J g 1 A(w) (u)d |x | (w) ) = since R
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119 J 2 1 A(w) (u)d l X 'u (w) = ° P_a S * Then l H I = 1li| x i-a.e.; hence by (3') we have |w x |(M) = J M |H Jclu i x = / M 1<*M| X | ^| X | (M) for M E M > i.e., | vi | = u i i The remainder of the theorem ($ e L (u v ) iff E(l | Jdlxl ) < «) will be proved in the next theorem in more n + generality. I Proposition 4.2.2 . Let E,F be Banach spaces and V: R xQ -» L(E,F) p be a process with V integrable for every z e R and with raw integrable variation |v|. If X is an E-valued measurable process, then x e L(y.,) iff X is u. . -almost separably valued and E V |X| E (J 2 l X z l d l V l z ) < "* In this case » E(J X dV ) is defined, and R + R + U(X) = E(fx dV ). V J Z z Proof. One way is easy: if X e l1,(h ), i.e., X e L^(y,,), then X is t, v b V M| v |-almost separably valued (being measurable) and E({ |x| d|v| ) R + < ». In fact, if X e Lg(u| v |), then |x| 1 = U i v i C |x | ) = l e Cm |v| } E U ?l x l d l v l ) b y Theorem 4.1.1 (which holds for positive measurable R^ z z + processes as well by monotone convergence), and this is finite by assumption. For the other implication, let X be an E-valued, measurable process, ui .-almost separably valued and satisfying the condition E(J ol x l d l v l ) < " Let (X n ) be a sequence of u ,, -measurable step ' _2 z * ' z Mm + processes such that x n > X u, ,-a.e. and [ x n J < |x [f for every n. Let
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120 A be a \i, , -negligible set outside of which X is separably valued and X * X pointwise; then u.,.i(A) = E( [ J.dlvl ) = => I v I J 2 A ' ' z J 1 (w)d | V | (w) = P-a.s. Denote the exceptional set by N. For IT A z + w i N, the section 1.(w) is d | V| # (w)-negligible, so X (w) X (w) d|v| # (w)almost everywhere, and |x"(w) | < |x (w) J. Now, since E(J|X z |d|v| ) < », there is a P-negligible set N 1 e F such that for w t N 1 , / |x (w)|d|v| (w) < », i.e., |X(w) | is d|v| (w)-integrable. Then for w t N {J N we have |x n (w)| S |X(w)| e L 1 (d|v|(w)) and ||x n (w) [ > |X(w) |d|v| .(w)-a.e. so by Lebesgue / |x n (w)|d|v| (w) + / |x (w)|d|V| (w) for R^ z z R* z z w ef N^JN 1 and in particular J X n (w)dV (w) -» f X (w)dV (w) ' 2 z z J 7 7. v. R 2 Z R 2 Z for w «f N VJN 1 . Repeating the procedure, since the map w * J |x (w)|d|v| (w) R 2 " Z is P-integrable, J|X^(w) |d | V | (w) < /|X (w)|d|V| (w) P-a.s. and /|xj(w)|d|v| z (w) + /|X z (w)|d|v| z (w) P-a.s., we can apply Lebesgue again and deduce that we also have convergence in L (P) j in particular, E(J |x n |d|v| ) -> E(J |X |d|v| ). Moreover, for each n R^ Z z R^ Z z we have E(l/x^dV z |) < E( J|x£|d| Vj ) < E(J|X z |d|V| z ) < ; hence JX z (w)dV z (w) e L (P) for each n. ( Note : We must show this map is measurable; this will come out of a later computation.) We already showed that Jx n (w)dV (w) -> fx (w)dV (w) P a.s. so by Lebesgue the z z J 7. 7.
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121 limit is P-integrable and we have E( [x n dV ) * E( fx dV ). (in ' z z ' z z particular E(/x dV ) is defined). Next, we show that x e L 1 (u ) : z z g ^Y for each n, M |v , < !lx n || ) = E(J|xJj|d|v| z ) < E(J|X z |d|v| z ) < -, so by Fatou we have M .. (lira inf|x n fl) < lira inf M | v |(jx n [) < -, in particular lim inf Jx n | is M , y , -integrable. But [x| = lim|x n I a.e. so |X| is y , y , -integrable. Also, x = lim X n is y -measurable, II n V so X e L E (u v ). Moreover, y v (X n ) -> u (X) by Lebesgue again, since |x n B S |X| e L 1 (u |v| ). Finally, we show that, for each n, we have u (X n ) = E( fx n dV ) V ; z z ' so we get the desired equality by passing to limits. Being a step k process, we can write x n = E 1 X. , M. e M, X, e E. Then we have 1-1 M i i i V^1 = V E1 M V = Zx iW = Lx i (E( / 1 M dV z )) (by Theorem ^-2.1) i i = SE(xJl M dV z ) = EE(/1 x dV ) (by Theorem K.I .3) = E(/(E1 x )dV ) i i M. i z = E(Jx"dV ) (and in particular the map [x n (w)dV (w) is Pz z J 2 i measurable). Letting n * », we obtain u(X) = E((x dV ), and the V ' z z theorem is completely proved. I Remarks . 1) By taking E = R in the statement, we have the following: If X is any scalar-valued measurable process, then X e L 1 (u ) iff
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122 E (J I x l z d l v l z ) < " ( x is automatically separably valued.) Then E( J' X z dV z ) iS defined ' and M y (X) = E(Jx dV ). Finally, equality (4.1.2) is proved the same way as (4.1.1), by taking step processes and passing to limits. 2) The correspondence is not one-to-one: as we shall see in the next section, a stochastic measure with values in L(E,F) is generated by a stochastic function(not necessarily measurable) with values in a subspace of L(E,F"). We can also generate stochastic measures from stochastic functions (not necessarily measurable) with raw integrable variation, as the next theorem shows. Theorem 4.2.3 . Let E,F be two Banach spaces and Z C F' a subspace 2 normmg for F. Let B:R + xQ L(E.F) be a right-continuous stochastic function satisfying the following conditions: i) B has raw integrable variation |b|. ii) For every x e E and z e Z, is a real-valued process (measurable!) with raw integrable variation ||. Then there exists a stochastic measure m: M * L(E,Z') with finite variation |m| satisfying the following conditions: 1) If X is an E-valued process and if X is u ,_ , -integrable, then 1 l B l X e L (m) , the integral E() is defined for every fc J d u u J H + z e Z, = E(<[x dB ,z>), j ii ii
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123 and |m| C |X1> S E(/|xJd|B| u ). 2) If, in addition, Bx is separably valued for every x e E and if X is y, ,-integrable, then the integral E([ x dB ) is defined, and |B| J R 2 u u + m(X) = E([ X dB ). V U U + 3) The measure m has values in L(E,F) in each of the following cases: a) F = Z\ b) For every x e E and v e FT, the convex equilibrated (balanced) cover of the set (b (w)x: w e flj is v relatively o(F,Z)-compact in F. c) For every x e E and v e R , the function B x is F+ v measurable and almost separably valued; in particular, this is the case if F is separable. Proof . Let u, be the measure generated by |3| via Theorem 4.1.4. For every x e E and z e Z the variation of the process satisfies || < |b| |x||z| (cf. proof of Prop. 4.1.3). Let m be the stochastic measure generated by : m Y 7 (M) = E( / 5 1 M d) for M e M. The mapping (x,z) > m (M) A » » D^ x , z + is linear in each argument: in fact, m (M) = X + X 7 1 2 ,Z E(J 2 1 M d__**) = E(Jl M d) = E( Jl M d( ****+) )
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121 E (/ l M d+/l M d) = E(Jl M d)+E(/l M d) = m (M) + m (M). The computation for z is completely analogous. x i » " x i z Also, we have |m (M) | = |e(J 1 M d****)J < EC if 1d****|) < x , z. ' £ nv ^ £ n V R + R + B(J 2 1 M dl****|) < E(J 2 1 M |x||z|d|B| v ) = |x|.|2|.E(Jl M d|8| ) R + R + = 1 X |-B Z 1-U| B |(M) , so |m Xj2 (M)| < |x||z|ui B i(M). Then for given M e M, the map (x,z) m (M) is continuous, bilinear. Then there is a continuous linear map m(M) e L(E,Z') satisfying = m x (M) = E(/ 2 1 M d**** ) , (More precisely, any continuous, bilinear function f(x,z): ExZ » R is continuous and linear in each component. Then the map x » f(x,«) is a continuous linear map from E into Z'. In our situation, we have m(M)x = m x ^(M), so = m (M), i.e., for M e M, we have m(M) e L(E,Z').) We also have |m(M) I < p. ,(M), since |B| |m(M)| = sup |m(M)x| = sup |m (M) | = sup ( sup |m (M)|) < |x|S1 |x|<1 x «* fx|<1 Jz|<1 x ' z sup ( sup ]x| |z|p . , (M) ) = y.|(M). This gives us a map |x|<1 Jz|<1 l B l l B l m: M L(E.Z'). We now verify that m is o-additive and has finite variation |m| (in particular |m| < \i. .): First of all, m is additive . Let M,N e M be disjoint; we show m(MlJN) = m(M) + m(N) , i.e., that m(M\JN)x = m(M)x + m(N)x for all
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125 x e E. This amounts to showing that = for all x e E, z e Z. Now, = E(f 1 d****) R 2 MUN v E( -f ? (l M +1 N )d****) " E( / 2 1 M d< V' 2> + / 2 1 N d< V' Z> E( I 2 1 M d****) + E( J" 2 1 N d****) = z> + = R + R + ' , which shows that m is additivie. If, now, A n +0, |m(A n )| since Jm( A n > | < ji. .(A ) and the latter is o-additive. Then m is o-additive as well. As for the variation, \i. , is a bounded, positive measure |B| satisfying |m| < u , B ; hence |m| < u i B since the variation is the smallest positive measure dominating the norm. In particular, m has finite variation. Now we prove assertions (1)-(3). Ad (1 ) : Let X be an E-valued process. From the inequality 4(u |B| ), then x E 4< 4"|B| Cauchy in L^(|m|); hence X e Lg(m) since X is M-measurable, and |m|(|x|) < U | B |(|xJ) = E(J 2 |xJ v d|B| v ), which is the second part of (1). The first part of (1) is satisfied for any M-measurable step process X = I 1 x , M e M disjoint, x. e E. In fact, for 1-1 i 1 z e Z, we have = < I m(M )x ,z> 1-1 M i i 1=1 i i |m| < Uj B | it follows that if x e ^(U| B |), then X z L^(|m|) (since any sequence of step functions Cauchy in ll\i, , is then also
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126 n n t n = I = 1 E(J 1 d<8 x z> ) = EE« 1 dBx.,j»( 1=1 1=1 fT M i v 1 i=1 R 2 M i v x + + n Prop. 4.1.3) = E( E ) = E() = i=i v 1 J M v l by r t n E( n. l v ; . , M, l v i=1 i E(****). Now, let X e L^y^,) and let X n be a sequence of measurable step functions such that x" X u. .-a.e. and |x n || < j X || everywhere. Let A be a y.. -negligible set outside of which X is separably valued, / |X^ jd | B | v < » (more precisely, for (w,u) i A, / [0 , u ]l X v (w) ! d l B lv (w) < °° ; 3ince E( / 2 l X v I d l B ly ) < ". R + / 2 l X v (w) i d l B l v (w) < " p a s -: hence J [0(U] |X y (v) |d|B| y (w) < « + y I B I ~ a * e * since i,: is a P-measure), and X n > X. There then exists a P-negligible set N t F such that for w i N, the section A(w) is d|B| (w)-negligible (in fact, E(fl , .dlsl (w) ) = y, .(A) = => J A (w; ' ' v |3| J 2 1 A(w) d ' B l. (w) = ° p ~ a « s -)» so for w ^ N we have d | B | (w)-a.e.: R + i) Xjw) is separably valued ii) |X%)| < |X.(w)| iii) x"(w) * X/w). Since / |X (w) |d |B | (w) < » for w t N, X (w) is d|3| (w) R v + integrable, and x"(w) X (w) in L.J,(d|B| (w)), so by Lebesgue
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127 Jx"(w)dB (w) -> JX (w)dB (w). Then, by continuity, for w ^ N, z e Z, we have * < f X (w)dB (w),z>; moreover, R 2 v 'R 2 v v + + |l * NI*l/x"(w)dB v (w)| < |z|./|xJ(w)|d|B| y (w) i ll z l/l x v ( w )I d l B l v ( w )« Now, the function w / |X (w)|d|B| (w) is PR 2 v v + integrable, so by Lebesgue is P-integrable and R V V + E(< / ? x"(w)dB (w),z> E() for all z e Z. For each R R V v + + n, = E(<[ _X n dB ,z>) as we saw above. Finally, X is J d v v J H + M-measurable by assumption, and flx| e L (\i> i)CL (|m|); hence X e L £ (m), and |m(X n ) m(X) | < |m|(|x n X|) ^ by Lebesgue (since |x n | < |x|, X n X |m|-a.e.), i.e., m(X n ) m(X). Then * for all z e Z. Passing to limits, we obtain = E() for all z e Z, which completes the proof of (1). R v v + Ad (2) : Suppose, now, that Bx is separably valued for every x e E. Let X c L (pi ), let X be step processes convering to X yj.-a.e. with [X | < Jx|| for all n. We shall show first of all that the map w * J X (w)dB (w) is integrable for all n; write R v X = I 1 x , M . e M disjoint, x e E. For each i, Bx. i=1 i l 1 is separably valued. Also, by (ii), is measurable for z e Z. Since Z is norming, Bx is weakly measurable, so Bx , being separably valued, is
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128 strongly measurable, with integrable variation (in fact, | Bx | S |B||x | by Prop. 4.1.3). By Theorem 4.2.1, E(f 1 d(Bx.)) exists. 1 ' 2 M. l R i n n Then I E(J 1 d(Bx )) = I E(/ (1 x. ) dB ) (by Prop. 4.1.3) 1-1 R i 1-1 R^ i * u u + + n n E( I Jl x dB ) = E(J( I 1 x.)dB ) = E(fx"dB ) exists. We proved i=1 M i : u 1-11 u u in (1) that Jx"(w)dB (w) * Jx (w)dB (w) P-a.s. Moreover, for each n, |/x"(w)dB u (w)| < J|x"(w)|d|B| u (w) < /|X u (w)|d|B| u (w). By assumption, the latter is P-integrable, so by Lebesgue Jx (w)dB (w) is Pintegrable, and we have E(fx n (w)dB (w)) * E( fx (w)dB (w)). We have J u u ' u u from before that m(X ) » m(X). It remains to prove that m(X n ) = E(/x n dB ) for all n. For all z e Z, we have = k = *** = Km(M.)x.,z> = Im (M ) = 1 = 1 i i l X1 x.,zi *E(Jl M d***) = ZE() = EE( z>) = E(Z****) = E«/(Z1 M Xi )dB u ,z>) = E() = . ( Note : We can now do this last step since Jx"(w)dB u (w) is P-integrable; it was not in part (1)!) Both m(X n ) and E(Jx n dB ) are Z '-valued, so this means that m(X n ) = E(J 2 X u dB u^ for a11 n " Pa3sin 8 t0 the limit, we obtain R + m(X) = E(J X dB ) ; in particular, the double integral on the right is R + defined.
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129 Ad (3) = (a) is trivial. (b): Let x e E, v c R . Since the set C = coJB (w)x: weftl + y J ( balanced closed convex hull) is o(F,Z)-compact , the natural embedding of C in Z», the algebraic dual of Z, is a(Z*,Z)-compact (see Dunford and Schwartz [8]). There is then a family (z.). , of elements of Z such that l id C = ( ){yeZ*: || < l} (any closed convex set is an intersection iel of half-planes; we can use balls since C is equilibrated). Then we have |****| < 1 for all i e I, w e R. Let M = [0,u] x A, A e F; we have | \ = |E(/l { ^ u]xA d**** ) | = |E(1 A ****)| < E(1 A |****|) < 1; hence m([0,u]xA)x e F, i.e., m([0,u]xA) t L(E,F). By taking differences, we have m((u,u']xA) e L(E,F), and also finite disjoint unions of such sets. We shall use the monotone class theorem to prove that m(M) e L(E,F) for all M e M. Let M = {M e M: m(M) e L(E,F)}. We show first that is a notone class: Let M e M , M t M. Then m(M )x e FC Z'. We have n n n (M n )x m(M)x| < |m(M n ) m(M)||x| by o-additivity of m. Hence m(M n )x + m(M)x in the metric topology of Z'. Since F is closed in Z' for the metric topology, m(M)x e F as well, i.e., m(M) e L(E,F). The proof for M + H is exactly the same, n Now, let C be the algebra generated by sets of the form (u,u']xA, A e F. C consists of finite unions and complements of such sets. We have shown that if M is a finite union of such sets, then mo lm
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130 m(M) e L(E,F). As for complements, m(M°) = m(R + XQ) m(M), and 2 m(R + xft)x = lim m([(0,0), (n,n)]xfl)x e F by closure of F in Z' as before. Thus, if m(M) e L(E,F), then m(M c ) e L(E,F). Then CC M , C is an algebra, so M = o(C)Cf/ by the monotone class theorem, i.e., m takes values in L(E,F). c) Suppose that for x e E, v e R , the function B x is F+ v measurable and almost separably valued (in particular, if F is separable, then B x is separably valued and weakly measurable by (ii); hence B x is F-measurable) . Then for every A t F, x e E, the function 1 B x is integrable; in fact, B x is F-measurable, by assumption and we have |B xj <, \s | Jx| e L (P). We also have, as before, = E(/l [0>v]xA d**** x,z>) = E( ) = = (again, we can move the expectation inside since 1 B x is integrable). Since this holds for all z £ Z, we conclude m([0,v]xA)x = E(1.B x) e F. Then A v m([0,v]xA) e L(E,F), and we conclude by the same monotone class argument as in (b). I Remarks . 1) This theorem shows that if B has values in L(E,F), then m B has values in a subspace of L(E,F"). Moreover, we do not have in general |m | = Uigi. Later we will establish some conditions sufficient for equality. 2) The correspondence B » m is not infective . For an example involving measures associated with functions, see Dinculeanu [6, p. 273].
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131 4.3 Vector-Valued Stochastic Functions Associated With Measures In this section we consider the converse; starting with a stochastic measure m with finite variation, we will find a stochastic function B with integrable variation such that m is associated with B in the sense of Theorem 4.2.3. The precise result is the following: Theorem 4.3.1 . Let E,F be two Banach spaces and ZC F' a subspace norming for F. Let m: M -» L(E,F) be a stochastic measure with finite variation |m|. Then there exists a right continuous stochastic function B: R"~x8 -» L(E,Z') satis r ying: i) B has raw integrable variation |s|. ii) For every x e E and z e Z, is a real-valued raw process with integrable variation ||. Moreover, we having the following: 1) If X is an E-valued measurable process we have X e L (m) E if and only if X e L (p, ,). In this case the integral E() is defined for every z e Z, V u u = E(), and R 2 u u |m| C |X|> = E(J 2 |xJd|B| u ), i.e., |m| = U( 2) If F is separable (or more generally if Bx is separably valued for every x e E), then Bx is measurable for every x e E. If B is separably valued, then B is measurable.
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132 3) We can choose B with values in L(E,F) in each of the following cases: a) F is the dual of a Banach space H and we choose Z = H; hence F = Z'. b) For every x e E, the convex equilibrated cover of the set {Jijixdm: $ simple process, J|$|d|m| < 1 } is relatively o(F,Z)compact in F. c) E is separable and F has the Radon-Nikodym property (we say F c RNP) ; in this case 3 can be chosen such that Bx is measurable and separably valued for every x e E, hence m(X) = E(Jx dB ) for X e l1(id). J u u E d) The range of m is contained in a subspace G L(E,F) having the RNP: in this case B can be chosen measurable, with separable range contained in G; hence m(<{0 = E(J<{> dB ) for * e L (m). ; u u 4) If p is a lifting of P, we can choose B uniquely up to an evanescent set, such that p[B ] = B for every v e R (see Definition v v + 1.5.5(b)). Proof . Let V be the integrable increasing raw process associated with | m | via Theorem Jl.1 .4: I ml (M) = E(f 1 M dV ) for M e M. II ; D 2 M u
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133 Denote the rectangle [0,z] by R ; for z e R 2 set m Z (A) = m(R xA) z + z for A z F. We verify that m : F -» L(E,F) is a o-ad;Jitive measure with finite variation |m |, and that m Z is absolutely P-continuous : i) m is o-additive : First of all, mz is additive. Let A,B z F, disjoint. Then R xA and R xB are disjoint, so we have z z J . m Z (A(JB) = ai"(R x(AUB)) = m((R xA) [J (R xB)) = m(R xA) + m(R xB) z z z z z = m (A) + m (B). Now, let (A ) z F, A +0. Then (R xA )+(R x 0) n n z n z = 0, so lim m"(A ) = lim m(R xA ) = m(0) = 0, so m Z is indeed n z n n n o-additive. ii) m has finite variation : we show in particular that |m | < |m| z , where |m| z (A) = |m|(R xA). Let A z F, and let (A.), z i n i = 1 n be disjoint sets from F with \JA C A. We have, since i = 1 n n n n R x((JA ) = M(RxA)CRxA, E h Z (A.)| = I |m(R xA . ) j < z i=i 1 fii z l z i=r l i-i z i n E |m|(R xA. ) < | m | ( R xA) = Iml (A). Taking supremum, we obtain i-1 z 1 z |m Z |(A) < |m| Z (An iii) m Z << P : In fact, we have |m Z (A) | = |m(R xA) | ^ |m|(R xA) = E(J 1 A dV ) = E(1.V ); hence |m| Z << P, so z '2 R xA u A z ' ' ' R z + z m << P as well. Applying the Extended Radon Nikodym Theorem (1.5.8), we get, for 2 ° each z z R , a function B : Q + L(E,Z') satisfying: 1) |B z | is P-integrable, and for ty z L (|m Z |), we have f i z, r ° J4id|m | = J |B | is P-integrable for all f E ll(\m Z \) and z e Z, z u E ' ' and = [****dP. 1 ' z 3) If p is a lifting of L (P), we can choose (B ) uniquely P-a.s. such that p[B ] = B , i.e., for all A e F, x e E, z e Z. z z * we have ****1 A e L°(P) , and p ( ****1 ft ) = ****1 . If, in addition, there exists a>0 such that |m Z | < aP, then we can choose ° ° ° ° oo B uniquely everywhere such that p(B ) B . i.e., <3 x.z > e L (P) z z z z for all x e E, z Q e Z, and p(****) = **** for all x,z . 4) If one of the conditions in 3(a) or 3(b) is satisfied, then B takes values in L(E,F). z ' ' Now, in particular, taking i|/ 1 , A e F In (1 ) we obtain 1 ') |m Z | (A) = E(1 |B ||) for A e F. Also, taking first f » x, x e E, and then f = x1 , A c F, we get 2') **** is integrable for x e E, z n e Z, and z = = /****dP = E( 1 A ****) , for A e F, x e E, z e Z. (Notice also that from (1'), if B is bounded, the condition |m Z | < aP in (3) is satisfied.) From (1') and the inequality |m Z | < |m| z we obtain EdjBj) = |m Z |(A) < |m| Z (A) = |m| (R^A) = E(/l R xA dV u ) = E(1 V ), z o i.e., E(1 A |B |) S E ^ 1 A V Z ) for a11 A c F; hence |B | S V P-a.s. Let z = (s,t), z = (s',t') be points in R , z < z'. Denote by D , the set R , \ R , and by R , the rectangle ( (s,t), (s',t ')]. zz z z z z
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135 We have m(D xA) = m((R , \ R ) X A) = m((R # xA) \ (R xA)) = zz z z z z m(R ,xA) m(R xA) = m (A) m Z (A). Likewise, m(R ,xA) = * zz m (A) m (A) m (A) + m (A). Then for x e E, z e Z, we have <(m Z -m Z )(A)x,z > = = E(1 A ****) " E(1 A ****} = E(1 A <(B z'" B z )x,Z >) Tne Same computation gives <(m S t m S t m St + m St )(A)x,z > = O E ( 1 A <(A R (B°))x,z >). Now, since p[B , ,] = B , ,, etc., we hav? zz ' s t s t o P[B Z' " B Z ] = B z' " B z' and p[A R (B )] = A R (B K In faCt (we z z ' z z ' give the proof for the first; the second is the same), for A e F, x e E, z e Z, we have «B z , B z )x,z >1 A = ****l A ****1 A e l"(P) since each term is. Also, p(<(B . B )x,z >1) = p(****1 z z A z A ****1 A ) . p(**** lA ) P«B z x,z >1 A ) = <3 z .x,z >1 p(A) z V-VVA) = <( V V^V 1 p(A) » S ° p[ V " B z ] " V and the same for A D (B ). Then, by Proposition 1.5.6, both H , ZZ i ° ° I . ° . B , B I and A D (B ) I. are P-measurable. Also, |B . B I z z R zz , " z z' I B Z 'I + |B | < V , + V < 2V .; hence |b . B j is P-integrable; . ° similarly, |a r (B ) | < ^^.\ hence |a r (B ) \ is P-integrable as zz' '" zz' well. Also, by properties of liftings, <(B , B )x,z> and z z <(A R (B ))x,z Q > are measurable for x e E, z e Z (see property 2 77 ' U
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136 after Defn. 1.5.5). By the "converse" of the generalized RadonNikodym Theorem (Theorem 1.5.9), there exist measures m D : F * L(E ' Z ' ) and m R : F + L (E,Z') (the measures have the same values as the function B since Z' is a dual (cf. part 3(a) of the statement of this theorem) with finite variation |m I and |m I 1 D 1 ' R 1 such that: i) = E(1 A <(B . B z )x,z Q >) for A e F, x e E, z Q e Z (by taking f = x1 ft in 1.5.9), and = E(1 A <(A R (B ))x,z Q >) likewise. Also, zz ' ii) |m D |(A) E(1 A |B°. B°|), and |ni R |(A) E(1JA R (B°)||) zz' for A e F (we take ip = 1 in 1.5.9). From (i) we have = E(1.<(B , B )x,z>) = u U A z z z z <(m m )(A)x,z > from earlier. Likewise, = u R ,. S 't ' S 't St ' St . , . , <{ m m m + m )(A)x,z >. Both these hold for all A, x, z Q so we have m D (A)x = (m Z m Z )(A)x, and m R (A)x = m 3 m st + m st )(A)x for all A, x; hence m n = m Z m Z (m s and m m K s't' s't St' St . . . , m m + m (and in particular m , m have values in L(E,F)), By (ii), we have |m Z m Z |(A) = |m_|(A) = E(1,|B°. B°|). i S 't ' s't and similarly m m st st + m |(A) E(1 |A (B )|) for A e F. On the other hand, we have l(m Z m Z )(A)| = |m(D ,xA) I ^ 1 zz " |m|(D zz ,xA); hence |m Z m Z |(A) < |m|(D ,xA) = |m| Z (A) |m| Z (A) = E(1 V ) E(1 V ) = E(1 (V ,-V )); hence E(ljB°, B°l) < AZ AZ AZZ A'z Z "
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137 E(1.(V ,V )) for all A e F, so |3 , B I < V , V P-a.s. for a z z " z z z z each z < z'. The same computations for the rectangle yield |A D (B )J < A d (V) P-a.s. If we take z,z' with rational t\ , n , ZZ ZZ o coordinates, we can find a common negligible set and modify B on it to get the inequalities everywhere for all z,z' rational. Next, lee z be fixed. We show that for any sequence r + z, r n n rational, the sequence (B (w)) is Cauchy for all w. In fact, the n sequence (V (w)) is Cauchy for all w since V is right continuous, n i.e., for any e>0, there exists n such that n,m > n implies w w K |V^ (w) V (w)| < e. Then, for n,m > n , we have |b (w) B (w)j n m n m I v ( w ) " v r ( w )| < e « Thus, for any w, the sequence (B (w)) is n m r n Cauchy in L(E.Z') complete, so lim B (w) exists. n n Now, let (r ),(s ) be two sequences of rationals decreasing to n n ° z. We can construct a sequence (v ) decreasing to z, containing subsequences of both (r ) and (s ). Then lim B (w) exists; moreover, n n v J J since lim B^ (w) and lim B (w) exist, and subsequences of both are n ' n n n contained in (3 (w)), all three limits are equal. In particular, lim B (w) = lim B (w) , so we get the same limit for any sequence of n n n n p rationals decreasing to z. Then, for every w c Q, z e R , B (w) = lim B (w) exists. The stochastic function B thus z r r + z r rational defined is right continuous.
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138 In fact, let e>0: there exists a neighborhood to the right of z so that if r is rational and lies inside the neighborhood, then |B (w) B (w) | < . For any z* in this neighborhood, there exists a similar neighborhood for it, and the intersection of these has nonempty interior. Let r be in their intersection, r rational. We have |B (w) B ,(w) | = |B (w) B (w) + B (w) B ,(w)| < |B z (w) B r (w)J + |B r (w) B ,(w)| < | + | = e. Thus 3 is right continuous. Some more properties of B are the following: a) For wz, r>w. Then |B 3 I = 1 q r IB -B +3 -B +B B I < V -V. Letting q+z, we get q z z w w r 1 q r =>m» b o |B -3 + B B | < V -V by definition of B and right continuity of V. Letting r+w likewise, we get |3 B I < V V . We ' z w 1 z w similarly have |A_ (B) I ^ A n (V). zz zz o 2 8) For each z, B = B a.s.: In fact, for z e R , r rational, z z ' +' ' r>z we have JB B I S V V a.s. (in fact there is a common r z r z negligible set outside of which this holds for all r>z) letting r+z, we have |b B I = liralB B I < lim(V V ) = a.s., i.e., 1 z z 1 ,'r z 1 r z r+z r+z B = B P-a.s. z z Next, we show that B has raw integrable variation |B|. Since B = B P-a.s., B , B = B , B a.s., and A (3) = z z z z z z R , zz A (B ) a.s.; hence p[B , B ] B* B , and p [a (B)]
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139 = A R (B) (property (5) following Definition 1.5.5), so by 1.5.7, zz ' |B Z ' " B | and |a (B)[ are measurable; hence the finite sums we z z ' use to compute Var^^^B^ § Q) ) , Var [0>t] (B (0 ^ # } ) , and Var r,-A m / vnt( b ) are also measurable. L (0,0) , (s,t) J Moreover, since B is right continuous, we can compute the variation using partitions consisting of rational points: the first two terms from the one-dimensional result, the third by Proposition 2.2.5. Each of these limits can then be taken along a sequence, so Var [o.s] (B (.,o) ) ' ^[o.ti^o.o*' Var [(o,o),(s,t)] (B) area11 measurable; hence Ib^ |B ((J>0) | Var [0>g] (B (# >Q) ) Var [0,t] B (0,.) + Var [(0,0),(s,t)] (B) iS measu rable, i.e., |B| is a raw process, evidently increasing. We show now that B has integrable variation. We shall show, in fact, that |b| £ V. We proceed with each term of |B| separately: 1) Since Ko,o)l K v (o,o) we have l B (o,o)l * v (o,o) a s 2) Let o: = s. < s < ... < s = s be a partition of [0,s], 1 m m-1 m-1 We have I |B. . B, ,.h I (V, , V, J 1-0 (S i*1' 0) (3 i' 0) 1-0 ( V 1 ' 0) (s i' 0) (V ( S1 ,0) " V (0,0) ) + (V (s ? ,0) " V (s 1 ,0) ) + " + (V (s,0) " V (s V i £ i m-i = v ^o m " V /a n\» p_a 'SNow, let (a ) be a sequence of partiIs, 0) (0,0) n neN tions of [0,s] such that Var rn ,(B, _,) = lim Var r ,(B, .;o ). LO.sJ (',0) [0,s] (',0) n Each term is dominated by V^^ V (Q>0) a.s., and y { ^ Q) V (0>0) does not depend on n, so taking limits we have Var r ^ ,(B, .) £ LP, s J (',0) \s,0) " V (0,0) a S '
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140 3) The same argument gives Var f (B, .) < V V 4) Let oxx be a grill on [ (0,0) , (s,t) ] : we have I |a (B)J Rcoxt R^ A R (V) = \s,t) V (0,t) V (s,0) +V (0,0)To see this last equality, consider for each w the measure m , V(w) 2 on B(R + ) associated with V(w). Consider the half-open rectangles R associated with the closed rectangles R of oxt. Then r ^ V v) . ., ( , ]( ui) Vw) «(o.o).(..t)]) v (5>t) v (0>t) ~ V (s 0) + V (0 0)" This holds a,s * Taking supremum along a sequence of grills as before, we have Var r . . . N -,(B) < V, V [(0,0) ,(s,t)] v (s,t) (0,t) " V ( Si o) + V (0,0) 3,S " Addin 8 U P (D-Wi we obtain |bL l B (0,0)l + Var [0,s] (B (.,0) ) + Var [0,t] (B (0,.) ) + Var [(0,0),(s,t)] (B) " V (0,0) + (V (s,0) " Vo)' + (V (0,t) " V (0,0) ) + (V (s,t)" V (s,0) " V (0,t) + V (0,0) ) = V (s,t) a ' S US ' f0r (s '^ e R + 2 ' ! 3 l (s ,t) V (s,t) a ' s Since I 8 ' is ^creasing, and |B| (n>n) < V ( ^ n) a.s., |b| is finite outside an evanescent set*; hence |b| is right continuous outside an evanescent set. Then, since both |b| and V are right continuous, |b| S V outside an evanescent set. In particular, l B L V oo' 30 B has integrable variation, and this completes the proof of (i). More precisely, for example is a negligible set N such that n w t N => |3| (w) < V. (w); hence |b| < V, n '(n,n) (n,n) ' ' (n,n) outside N n . Then N Q = {J N is negligible and |B| < outside this set. n
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mi Now for (ii). For x e E, z n e Z we ha ve **** = **** z z a.s., so by (2') **** is integrable; moreover, **** is right z u z continuous since B is, and has integrable variation (we showed in the proof of 14.1.3 that | ****(w) | < |b| (w) fx | |z | ) . Also, for each 2 x e R , **** is F-measurable since B is; hence **** is a + z u z z real-valued raw process with integrable variation, which is (ii). We now turn to assertions (1)-(4). Proof of (1 ) : For M = [0,z]xA, A e F, we have, for x,z : z ° = = E(1.**** (by (2')) = E(1.<3 x,z >) u uazu AzO E(1 A / 2 1 co, 2 ] d< V'V ) = e). Then l^w^.Vl = H + |E(JV**** ; hence |m|(M) < E(Jl d|B| ). On the other hand, from |B| < V we have E ( / 1 M d | B | w D < E(/l dV ) = |m|(M); hence |m|(M) = E(/l M d|3| w ) for M = [0,z]xA. By additivity, then, as usual, we have |m|(M) = E ( /l M d | B | w ) = jj . (M) for M = RxA, A t F, R a 2 rectangle of the semiring generating B(R ) (there are four kinds; cf. Theorems 3.2.2 and 4.2.1). As |m|, y, , are both a-additive, and sets of the form M = RxA form a semiring generating M, we have H = u |3i Now, let X be E-valued, measurable. If X e L_(m) , then E |X| e L (|m|) = L (vi B i) => X e L E ^| B |^ Conversely, if X E L E y (|B|)* then IM E l1( ^ j |b| ) => H X I E L^hl): hence X e L(m) since X is measurable. E
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142 We already showed that |m| = p.,, which establishes the second equality in (1). As for the first, we note that by Theorem 4.2.3, E(< J 2 X w dB w' Z > ' ) is defined for z E z To prove the equality, we R + note first, that by Theorem 4.2.2 there exists a stochastic measure m: M * L(E,Z') corresponding to B satisfying = E(< J 2 X w dB w' Z > ' ! * ^ Note that since Z' is the dual of a Banach space, R + m has values in the same space of operators as B; cf. statement (3a) of this theorem.) We shall show that m = m; as both are o-additive it will suffice, as above, to prove for sets of the form M = [0,z]xA, A e F. Let M = [0,z]xA, A e F; let x e E, z e Z (which is a norming subspace of (Z')'). We have = = MO E() = E(/l M d****) (by 4.1.3) = E(1 A **** ) E(1 A ****) = . Then, for x e E, z e Z, = ; hence m(M) = m(M) for M = [0,z]xA. As p before, we conclude that m = m on all of B(R )xF. Then for X e L (m) we have = = E(), which t J w w completes the proof of (1). Proof of (2) : If Bx is separably valued for x e E, then Bx is measurable for x e E since it is weakly measurable by (ii). If B is separably valued, then B is measurable since is measurable for all x e E, z Q e Z, by (ii) [6, Proposition 24, p. 106]. Proof of (4) : If p is a lifting of P, we can choose B z uniquely a.s. for all z; in particular, outside an evanescent set for
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1^3 all z rational. Then B is determined uniquely outside this evanescent set; we already showed that p[3 ] = B for everv v e R 2 v v ' J ' + Proof of (3) : a ) . b) If one of these is satisfied, then B takes values in L(E,F); hence B takes values in L(E,F) since L(E,F) is closed in L(E,Z') for the metric topology (recall that B = lim B ). z r r + z r rational c): Assume E is separable, F e RNP. Let x e E, z e R 2 . The + measure m:F + F defined by u(A) = m Z (A)x for A e F is o-additive (since |m Z (A)x| < |m Z (A)|-|x| so A +0 => |m Z (A ) | + => |u(A n )| + 0; u is evidently additive) and has finite variation |u| S |m | |x | £ |m| Z flx|. Then u << P since |m| Z << P, as we showed already. Since F e RNP, there is a Bochner-integrable function B^ e L p such that u(A) = E(1 B^J ) for A e F. We can choose B' separably valued; therefore, we can consider F separable. More precisely, let S be a countable dense set in E. For x e S, we have lim B J ., = B v -7 a ' 5 ' In fact » for A &> E ( 1 « B ' ) = u(A) = u+z x,u x,z A x,u u rational m (A)x = m([0,u]xA)x * m([0,z]xA)x = m Z (A)x = E(1,B' ); hence A x,z B x,u "* B x,z a,S * Then ^ B x z (w): x e s » z national} is separable; hence {b^ z (w): x e E, z rational} is separable. For any z, B J _ = lim B' a. s., and we modify B' on the exceptional set; x,z ,j.^ x » u x,z u + z ' u rational we can do this and still get a R-N derivative of y. Hence we can take F separable. We can then choose Z separable in F', norming for F.
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1411 2 Let B: R + xft * L(E,Z') be the stochastic function associated with m for this choice of Z. We have, then, for z in a countable dense subset Z C Z, and for A elf E(1,**** = E(<1 B' ,z >) u A x,z A x,z = ^Vx.zW ' = E(1 A ****); hence **** = **** a,S ' for each z e Z ' Now, since Z Q is countable, there is a common negligible set; since Z Q is dense in Z we have B' = B x outside this negligible set. There is then a common negligible set such that 3' = B x x,z z for all z rational, x in a countable dense set of E. Then, by right continuity of B and closure of L(E,F) in L(E.Z'), we have B = B' e L(E,F) outside this negligible set, i.e., up to evanescence. By modifying B on this evanescent set, we obtain B with values in L(E,F). Moreover, since B x = B' , B x is intestable (in z x, z z particular Bx is measurable and separably valued by right continuity) and so, by 4.2.3(2), for X e L^m) , we have m(X) = E( f X dB ) t ' ; 2 v v K + (m is the measure associated with B via 4.2.3), and in particular, m(M)x = m(y) = E(/l M xdB ), which completes (c). d) Assume the range of m is contained in GCL(E.F) with G e RNP. We write G = L(R,G) and apply (c): R is separable, G e RNP. Then B has values in L(R,G) = GCL(E.F), and 8a is measurable for a e R; hence B is measurable. Also, for e L 1 (m) we have by (c) m(<{>) = Etj^dB ), which is (d), and completes the proof of this theorem. m Remark . In the last part, once we have B measurable, we can also get the equality by applying Theorem 4.2.1.
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145 4.4 On the Equality J m j = y. , In Theorem 4.2.3, we began with a stochastic function B with integrable variation, and associated a measure m with finite variation, and we proved that |m| < y. .. We now consider some cases where this is in fact an equality. Theorem 4.4.1 Let E,F be two Banach spaces and Z C F' a subspace norming for F. Let B: R^xP. -> L(E,F) be a right continuous stochastic function satisfying conditions (i) and (ii) of Theorem 4.2.3, and let ra: M » L(E,Z') be the corresponding measure with finite variation |m| satisfying = E() V V V for any E-valued measurable process X e L^y, ,). We have the E |B| equality ]m| = y, . , i.e., IH(| X I> = E(J 2 lX v |d|3J v ) for X e Lg(|a|) R + in each of the following cases: 1) There is a lifting p of P such that p[B ] = B for z z z c R . + 2) E is separable and there is a countable subset SCz norminj for F. 3) E is separable and B x is integrable for every x e E and 2 z e R . + 4) B is measurable and B is integrable for z e r 2 .
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146 Proof.' Ca se (4) has been dealt with in Theorem 4.2.1; we include it here for completeness. We have a measure m: M -> L(E.Z') with finite variation. By Theorem 4.3.1, there exists a stochastic function B' with values also in L(E,Z') (by 3(a)) satisfying (i) and (ii) of Theorem 4.3.1, such tnat p[B'] = B' for z g R and such that for every X e L 1 (m) we have z z + jr (X),z > E«/ 2 X v dB;, Z() >) for z Q e Z (not_e: here we consider Z embedded in Z" as a norming subspace in order to apply the theorem), and I»I<|X|> = E(J |X |d|B'| ). R Now, for X = i M x with M e M, x e E, and for z e Z , we have E(/l M d****) = E(/l M d****). In fact, E( Jl M d**** ) = E ^) (by 11J ' 3) = E(< / X v dB v» Z >) = z > = E() = E() = E(/l M d****). Taking M = [0,z]xA with A eFwe obtain '^[O.zlxA^v*^ = ^V^'V 5 ' and ^Ko.zJxA^V'V) = E(1 A ****): hence E(1 A < B ;x,z » E(1 A ****) for A ef so (1.1) **** = **** a.s. for x e E, z e Z. We shall prove that in cases (1)-(3), B and B' are
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147 indistinguishable; hence |B'| = |b| up to evanescence, so IB'I = l m l= y 1) Assume p[B z ] = B^ for all z e R^. We have, from above, p[B^] = p[B z ] for z e R + as well. Then from (1.1) we conclude that B z = B z a ' 3, for each z ( P r °P ert y ^ following Defn. 1.5.5). Since both are right continuous, they are indistinguishable. 2) Assume E is separable, let E Q C E be a countable dense set, SCZ a countable subset norming for F. We have **** = z **** a,S * for ail x G E o' z e S " There is then a common negligible set N such that the equality is valid for all z e R 2 + rational, x e E , z Q z S. By right continuity, then, this holds 2 outside N for all z e R . Since S is norming, we have B'x = B x + z z 2 outside N for all z e R + , x e E Q . Since E Q is dense in E, we have , 2 B x = B x for all z e R x t E (still outside N 1 ) ; hence 3' = B c z + z z outside the evanescent set R + xN, i.e., B and B' are indistinguishable. 3) Assume now that E is separable and 3 x is integrable for z 2 every x c E, z e R + . Then B z x is almost separably valued; by right continuity Bx is separably valued outside an evanescent set for x z E. Since E is separable there is (as before) a common evanescent set A outside of which Bx is separably valued for all x e E. We modify B on A by setting it equal to zero on A and get a process B" indistinguishable from B, with B" taking values in a separable space F Q C F for all x e E. men E(/l M d****) = E(/l M d****) = ;
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118 hence m is the measure associated with the stochastic function 2 B": R + xn * L(E,F ). Since F is separable, there is a countable subset SCZ norming for F . By (2), we have |m|(M) = E( fl d|B"| ). U ' ' J M ' ' v Since B = B" outside an evanescent set, |b| = |B"| outside an evanescent set, so J 1 d | B I = / _1 u dlB"| a.s. for any M e M; ; 2 M ' ' v ; D 2 M ' ' v R + R + hence |m|(M) = E(fl d|B"| ) = E ( f 1 d I B I ) for M e M, i.e., ' M ' ' v 'M' l v | m | = ui R |, and this completes the proof. I Remark . If we start with a stochastic measure and associate a function, we always have |m| = u,_i, but if we start with a stochastic l B l function, we do not get equality not even if the measure has values in L(E,F). Equality (1.1) seems to be as close as we can come in general; in order to get everywhere from there, it seems we need for E and Z not to be "too large."
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CHAPTER V CONCLUSION We have seen that the usual definition of the variation on a rectangle of a function of two variables is insufficient to yield all the properties necessary to extend the theory of Stieltjes measures to functions of finite variation on the plane. We have given some additional conditions sufficient to establish a proper definition of the variation of a function, and although these were not shown to be minimal, it would seem to be difficult to weaken them further. We have shown that, starting with a two-parameter stochastic function X with values in L(E,F), we can associate a measure u with x values in L(E,Z') and that under certain conditions u has values in x L(E,F) as well. We have also established a similar correspondence, starting with a measure and obtaining a stochastic function. We have also shown that, if the spaces E and F are not "too large," we have the equality Ki = M |x|We hope that this lays the groundwork for exploring the question of existence of optional and predictable projections of vector-valued multiparameter processes. 1 49
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BIBLIOGRAPHY 1) Bakry, D., Limites "Quadrantales" des Martingales , in Processus Ale"atoires a Deux Indices , Colloque ENST-CNET, Lecture Notes in Mathematics no. 863, Springer-Verlag, New York, 1980, pp. 40-49. 2) Cairoli, R., and Walsh, J.B., Stochastic Integrals in the Plane , Acta Math., 134(1975), pp. 1111 83~~ 3) Chevalier, L., Martingales Continue a Deux Parametres , Bull. Sc. Math., 106(1982) , pp. 19-62. 4) Dellacherie, C, and Meyer, P. A., Probabilities and Potential B , chaps. I-IV, North-Holland, New York, 1978. 5) Dellacherie, C, and Meyer, P. A., Probabilities and Potential , chaps. V-VIII, North-Holland, New York, 1982. 6) Dinculeanu, N., Vector Measures , Pergamon Press, New York, 1967. 7) Dinculeanu, N., Vector-Valued Stochastic P r ocesses I: Vector Measures and Vector-Valued Stochastic Processes with Finite Variation , Journal of Theoretical Probability (to appearl. 8) Dunford, N., and Schwartz, J.T., Linear Operators, Part I: General Theory , Pure and Applied Math., no. 7, WileyInterscience, New York, 1958. 9) Fouque, J. P., The Past of a Stopping Point and Stopping for TwoParameter Processes , Journal of Multivariate Analysis. 13(1983), pp. 561-577. 10) Kussmaul, A.V., Stochastic Integration and Generalized Martingales , Pitman, London, 1977. 11) Me'tivier, M. , Semimartingales , Walter de Gruyter, Berlin, 1982. 12) Meyer, P. A., The'orie Ele'mentaire de Processus a Deux Indices in Processus Altjatoires a Deux Indices , Colloque ENST-CNET. Lecture Notes in Mathematics no. 863, Springer-Verlag, New York, 1980, pp. 1-39. 13) Millet, A., and Sucheston, L., On Regularity of Multiparameter Amarts and Martingales , Z. Wahr . , 56(1981), pp. 21-45. 1 50
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151 11) Neveu, J., Discrete Parameter Martingales , North-Holland, New York, 1975. 15) Nualart, D., and Sanz, M., The Conditional Independence Property in nitrations Associated to Stopping Lines in Processus Aldatoires a Deux Indices , Colloque ENST-CNET, Lecture Notes in Mathematics no. 863, Springer-Verlag, New York, 1980, pp. 202-210. 16) Radu, E., Mesures Stieltjes Vectorielles sur R n , Bull. Math, de la Soc. Sci. Math, de la R.S. de Roumanie, 9(1965), pp. 1 29-1 36. 17) Rao, K.M., On Decomposition Theorems of Meyer , Math. Scand., 21(1969), pp. 66-78. 18) Walsh, J.B., Optional Increasing Paths in Processus Ale~atoires a Deux Indices , Colloque ENST-CNET, Lecture Notes in Mathematics no. 863, Springer-Verlag, New York, 1980, pp. 172-201.
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BIOGRAPHICAL SKETCH Charles Lindsey was born in Lexington, Kentucky, on April 7, 1962, and lived there until 1969 when his family moved to Merritt Island, Florida (where his parents still live). He graduated from Merritt Island High School in June, 1979. He began his undergraduate career in fall 1979 at the California Institute of Technology and stayed there until December, 1980, when he transferred to Auburn University. He attended Auburn the first six months of 1981, then transferred to the University of Florida and has been there until the present. He received his B.S. degree from the University of Florida in April, 1983; his M.S. in December, 1984; and expects to receive his Ph.D. in April, 1988. 152
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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. ( / / .' U i L Nicolae Dinculeanu, Chair Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. James Brooks Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. wi SCUlJ, cuccr\/^ Louis 31ock Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Joseph Glover Professor of Mathematics
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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. James Keesling Prjofessor of Mathematics 1 I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. ftjdL^ £&i£AJ> William Dolbier Professor of Chemistry This dissertation was submitted to the Graduate Faculty of the Department of Mathematics in the College of Liberal Arts and Sciences and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. Dean, Graduate April, 1988 M~*s**>U£*r^*^ > <^J^^U^l~*-^C~ School
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