set (fA ixdm: H R-step function, JfAl dl 5 1) is
relatively compact in F for the topology o(F,Z).
c) For every x E E, the convex equilibrated cover of the set
{fixdm: l R-step function, IfIidu S 11 is relatively
compact in F for the topology o(F,Z).
Note: If m is defined, on a o-algebra F, which will be the case in
our uses of this theorem, then the condition that T have the direct
sum property is automatically satisfied, and we can replace the family
by F in part (4b) of the statement.
These remarks also hold for the following generalization of the
Radon-Nikodym Theorem.
Theorem 1.5.8 (Extended Radon-Nikodym Theorem). Let v be a scalar
measure on R and m: R L(E,F) a measure with finite variation U. If
v has the direct sum property and if m is absolutely continuous with
respect to v, then there exists a function V : T + L(E,Z') having the
following properties:
1) The function IV m is locally* v-integrable and
fdy = IV mId ]vI for ip E L1 ()
(here uIv denotes the variation of v).
2) For f e Lc () and z E Z, is v-integrable, and we have
= fdv(t).
SAs before, If the measures are defined on an o-algebra, this can be
dropped.