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') <H,z> is u XI-integrable for every z E Z, and we have

 <ux(M),z> = fM<H,z>dI xi for every M E M. In fact, taking

 f 1 in Theorem 1.5.8 (2), we get <H,z> IXI-integrable for
 all z. Also, for z E Z and M E M, we have, taking f = 1M,
 <flMduX,z> = f<H-1M,z>dplXI, i.e., <uX(u),z> = fIM<H,z>dp XI
 (since <H-1M,z> = 1M<H,z>) = f<H,z>dpi 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 <X ,z>) E(1Af O,u]<Hw,z>dllw )


In fact, on the left hand side of (2'), we get <(X(M),z> =

<E(X >dP M E),z = <E(XAz, w h dX),z> = <E( AXd ),zs = f (AudP,z) =

IA<XUz>dP = E(1A <X ,z>), which is the left hand side of (4.2.3). As