2 ZxA)
Denote the rectangle [O,z] by R ; for z E R2 set m(A) = m(R xA)
z + 2
for A E F. We verify that mz: F L(E,F) is a o-additive measure
with finite variation 1mZ and that mz is absolutely P-continuous:
i) mz Is o-additive: First of all, mz is additive. Let
A,B E F, disjoint. Then R xA and R xB are disjoint, so we have
z z
m (AL B) = m'(R x(A JB)) = m((R xA) J(R xB)) = m(R xA) + m(R xB)
SZ
= mZ(A) + mZ(B). Now, let (A ) E F, A + 0. Then (R xA )(R x 0)
n n z n z
= 0, so lim mz(A ) = 11m m(R xA ) = m(0) = 0, so mz is indeed
n z n
n n
o-additive.
1i) mz has finite variation: we show in particular that
ImzJ Iliz, where Im1z(A) = Im (RzxA). Let A E F, and let (A ),
n
i 1,...,n be disjoint sets from F with AJAiC A. We have, since
i=1
n n n n
Rzx( ( Ai) = (RxA ) C RzxA, Z m z(A )I = Jm(RzxAi )
=1 1i=1 i=1
n
E Iml(Rz xA) ml (R xA) = ImlZ(A). Taking supremum, we obtain
i=1
ImZl(A) 5 Im z(A).
iii) mz << P: In fact, we have Imz(A)l = Im(RzxA)1
Sml(RzxA) = E(f 21 xAdVu) = E(1AVz); hence Imlz << P, so
R z
m << P as well.
Applying the Extended Radon Nikodym Theorem (1.5.8), we get, for
2
each z E R, a function B : n + L(E,Z') satisfying:
1) IBz is P-integrable, and for Ec L1 (ImZ), we have
fIdlm l = I 0 B dP.