Proposition 1.5.6. Let U: T + L(E,F). If p[U] = U, then the

function t + IU(t) is p-measurable. It is this proposition, along

with property (5) above, that will be used most often later.

1.5.3 Radon-Nikodym Theorems

 We state here some generalizations of the Radon-Nikodym theorem

to vector-valued measures with finite variation. These particular

statements are taken from Dinculeanu [6, pp. 263-274]. Throughout

this section, T denotes a set, R a ring of subsets of T, E and F

Banach spaces, and Z a subspace of F', norming for F.

Theorem 1.5.7. Let m: R 4 L(E,F) be a measure with finite variation

p. If p has the direct sum property, then there exists a function

U : T 4 L(E,Z') having the following properties:
m
 1) IU (t)I = 1 p-a.e.

 2) For all f E L (m) and zeZ, <U f,z> is p-integrable and we have


 <Ifdm,z> f<Um(t)f(t),z>dw.


 3) If p is a lifting of L (u), we can choose U uniquely so

 that p(U ) = Um (cf. Definition 1.5.5).

 4) We can choose U (t) E L(E,F) for every t c T, in each of the

 following cases:

 a) F= Z'

 b) There exists a family A covering T such that u has the

 direct sum property with respect to A such that for

 every A c A x e A, the convex equilibrated cover of the