3) If p is a lifting of L (u), we can choose V uniquely v-almost
everywhere such that p[V ] = V (of. Definition 1.5.5). If,
m a
in addition, there exists a>0 such that p 'lvl, then we can
choose V uniquely such that p(V ) = V .
m a m
4) We can choose V (t) E L(E,F) for every t c T, in each of the
m
following cases:
a) F= Z'
b) There exists a family A covering T such that v has the
direct sum property with respect to A such that for every
A E A x c E, the convex equilibrated cover of the set
ifAJxdm: iR-step function, JAlI IdivI < 1I
relatively compact in F for the topology o(F,Z).
b') The same statement as (b), with A such that V has the
direct sum property with respect to A and with
IJidpj 5 1 instead of AI l@dlvl 5 1. In this case we may
not have p[V ] = V .
m m
c) For every x e E, the convex equilibrated cover of the set
iJfxdm: 4 R-step function, II|Idlvl < 1} is relatively
compact in F for the topology o(F,Z).
c') The same condition as (c) with fIIJId 5 1 instead of
fl 4jdjvj 5 1. In this case we may not have p[V ] = V .
Theorem 1.5.7 gives a "weak density" of a vector measure m with
respect to its variation p, whereas Theorem 1.5.8, more generally,
gives such a density of m with respect to a scalar measure v not
obtained from m. The former is a particular case of the latter, but