Now, if uSv, then A S A (since for B E F, A (B) =
i(Bx[0,u]) S p(Bx[0,v]) = A (B)); hence a S a a.s. We set, for w
v u v
running through the rationals in R2 (i.e., points with rational
1 1
coordinates), a = sup a a is increasing* in both senses of
z w
wSz
Definition 2.1.3, as we now show.
The order sense is no problem, as we are taking supremum over a
bigger set for bigger z. For the other, let (s,t) < (s',t'), denote
R ((s,t),(s',t')]. First of all, for any such rectangle R, we have
A R(a) 2 0 a.s. In fact, for B E F, E(lBAR(a)) = E(1Bas't ) -
E(Bas't) E(IBast,) + E(Bast) st') A't() As'( +
Ast(B) = p(Bx[(O,O),(s',t')]) u(Bx[(O,O),(s',t)]) -
p(Bx[(O,O),(s,t')]) + p(Bx[(0,0),(s,t)]) = p(Bx((s,t),(s',t')]) 2 0.
Since the four functions that make up AR(a) are F-measurable, we have
AR(a) Z 0 a.s. (i.e., for w e N) for R with rational coordinates.
Now, we show that for w e N (see note preceding page), we have
AR(a (w)) 2 0 for all R = ((s,t),(s',t')] C R2. Let w e N and suppose
* Outside an evanescent set: for uSv, u,v rationals, a a a.s.,
u v
so for each pair u,v there is a negligible set N outside of
U,V
which a (w) s a (w). We put these together into a common
u v
negligible set N outside of which a (w) S a (w) for any u,v
2 u v
rational in R .
+