1') IH| is ix i-measurable and |H| S 1. In fact, Theorem 1.5.8
says that |H| is iXi-integrable and that for p E LI (lI)
we have fIdlIwxJ = fIHdixli Taking -= 1A, A e M, we
obtain IpXI(A) = JH| 1Ady li x IxI(A); hence |H| 1 on
A, so IH| 1 except on a upix-negligible set (on which we
modify H appropriately, say by setting H = 0).
2') is u XI-integrable for every z E Z, and we have
= fMdI xi for every M E M. In fact, taking
f 1 in Theorem 1.5.8 (2), we get IXI-integrable for
all z. Also, for z E Z and M E M, we have, taking f = 1M,
= fdplXI, i.e., = fIMdp XI
(since = 1M) = fdpi X.
3') IuJ1(M) = fM IHIdjx for M E M. We showed this in proving
(1').
Now, taking M = [0,u] x A, A E F, in (2'), we deduce that
(4.2.3) E( A ) E(1Af O,u]dllw )
In fact, on the left hand side of (2'), we get <(X(M),z> =
dP M E),z = = dP = E(1A ), which is the left hand side of (4.2.3). As