we shall make good use of it in its own right, and so we take the time
to state it separately here.
The final result we shall need is a "converse" of these theorems.
Theorem 1.5.9. Let v be a scalar measure on R and U: T L(E,F)
a function such that JU' is locally v-integrable and the function
is v-measurable for every x E E and z E Z.
Then the function is v-integrable for f E L1 (lUJDvi)
and z E Z and there exists a measure m: R L(E,Z') such that
= f<(t)f(t),z>dv for f E L1(Iu vl) and z E Z,
and Jljld J IUI ldlvl for L e Lll(uIV ).
The measure m has values in L(E,F) in each of the following cases:
a) F= Z'
b) For every x E E there exists a family A such that v has
direct sum property with respect to A such that for every
A E A the convex equilibrated cover of the set (U(t)x; t E A}
is relatively compact in F for the topology o(F,Z).
c) For every x E E, the convex equilibrated cover of the set
(U(t)x: t E T} Is relatively compact in F for the topology
o(F,Z).
d) The function t U(t)x is v-measurable for each x E E; in
particular if F is separable.
If v has the direct sum property, we have the equality
S= IIIv!, hence