The proofs of the last two are the same as those of the first
two: to prove (3), we use the same method as that of (1) to get
j-1
E |f(si,ti) f(si+1,ti+1) ) IfI(s,t) (instead of Ifl(sl,tl)),
i=1
and to prove (4) we use the same method as that of (2) to get
j-1
E If(siti) f(si+1,ti+1) I Ifl(sl,t) (instead of If (s,t )).
i=1
This completes the first part of the theorem.
Assume, now, that f has bounded variation, i.e., there exists M
such that IfI(s,t) < M for all (s,t) E R. We will prove the
existence of the "limits of infinity" in pretty much the same manner
as the proofs of the other limits: the main difference occurs in
using M instead of a particular value of Ifl to obtain a
contradiction.
Proof of (1') Let (s ,t ) be a sequence with s ++s, s n < n
n n n n+1 n
Q
V = for all n, t t+-. We proceed by denial as above. The proof of
n
this is much the same as that of (4) (and (2)), with a slight
difference: proceeding in the same fashion as in (2), we obtain
j-1 J-1 j-1
Z If(s ,ti) f(si+t f(s+ ) f(si,O)l
i-1 i=1 1 i=1
5 Jf (sl,t )
(see Figure 2-13). However, the right side now depends on j, so we
must further majorize it by M: