For two functions U1,U2: T L(E,F) we shall write U1 = U if
= p-a.e. for every xEE, zeZ.
Let p be a lifting of L (u).
Definition 1.5.5. Let U: T + L(E,F) be a function.
a) We shall write p(U) = U if for every xEE and zEZ we have
E L (1) and
p() =
b) We shall write p[U] = U if there exists a family A of subsets
of T such that T has the direct sum property with respect to
A, such that for every AcA, xeE, zeZ, we have
1A E L () and
p(1 ) = p(A)
(Note: If p is o-finite (and in particular if p is a probability
measure), the relation in (b) holds for all A measurable.) The
functions U with p(U) = U or p[U] U have the following properties:
1) p(U) = U implies p[U] = U.
2) If p[U] = U, then is u-measurable for every xcE,
ZEZ.
3) If U U2, p(U ) = U, p(U2) = U2, then U1 U2'
4) If U1 U2, p[U ] = U1 and p[U2] = U2, then U1 = U2 p-a.e.
5) If U = U2 p-a.e. and p[U ] = U1, then p[U2] = U2'
We shall also make use of the following: