so A CB E H. We get A B H by writing 1AU B 1 1 B
and Ac by 1 = 1 1A (E(fldX ) = E(X )).
A
H is closed under monotone convergence. Let A be a sequence of sets
n
from H with An increasing to A. For almost all w, 2A (w)dX (w)
R n
S21A(w)dXz(w) by Lebesgue (since 1 A 1 E L (IX(w))). For each n,
R n
the map w + jlA (w)dXz(w) is bounded by IXJl(w) E L (P), so these
n
converge to w + 1A(w)dXz(w) a.s. and in L (p). In particular,
E(f1A(w)dXz(w)) exists. The proof for decreasing sequences is the
same.
H contains a semiring generating M. Let S be the semiring of half-
open rectangles in the plane from before, and denote P = S R2.
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 BdXz) defined, but we can compute it
explicitly. There are four types of sets in P (cf. Theorem 3.2.2):
1) A = ((s,t),(s',t')]: then E(i AxFdXz) = E(IFfIAdX )
E(1F (A (X))).
2) A = (s,s'] x [O,t']: then we get E(J AxFdX =
E(IF(X(s,t') X(s,t') )
3) A = [O,s'] x (t,t']: we get E(f1AxFdX) =
E(1F(X(s ,t') (s',t)))