after Defn. 1.5.5). By the "converse" of the generalized Radon-

Nikodym Theorem (Theorem 1.5.9), there exist measures

mD: F L(E,Z') and mR: F L(E,Z') (the measures have the same
 o
 values as the function B since Z' is a dual (cf. part 3(a) of the

 statement of this theorem) with finite variation ImD and ImRl

 such that:
 o o
 i) <mD(A)x,z0> = E(1A<(B. Bz)x,z0>) for A e F, x E E,

 z E Z (by taking f = x1A in 1.5.9), and <mR(A)x,z0> =
 0
 E(1A<(AR (B ))x,z0>) likewise. Also,
 zz
 ii) mDJ(A) = E(1IBZ, BzI), and imRI(A) = E(LA l (B )I)
 zz
for A c F (we take p = 1A in 1.5.9).
 0 0
 From (i) we have <mD(A)x,z > = E(1A<(B. B )x,z >)

<(mz mZ)(A)x,z0> from earlier. Likewise, <mR(A)x,z0> =

<(s't s't st + mst)(A)x,z >. Both these hold for all A,

x, z0 so we have mD(A)x = (mz mz)(A)x, and mR(A)x =

(m't ms't st' + mst)(A)x for all A, x; hence mD = m m

 S s't' s't st' st
and m = m m mt (and In pa-ticular m mR

have values in L(E,F)).

 By (11), we have Imz mZ(A) = |mD (A) = E(1AIBz, B ),

and similarly I't' 't stm + (A) = E(1AIAR (B )I) for
 zz
A e F. On the other hand, we have I(mz m)(A)I = Im(Dzz,xA)j [

ml (DzzxA); hence Imz mZJ(A) S JIm(Dzz,xA) = lm]z (A) Im1z(A)

= E(1AVz.) E(1AV ) = E(1A(V z-V )); hence E(1AjBz, Bz)