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, __ is p-integrable and we have
fdw.
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
__