Definitions 1.5.3.* Let u be a positive measure.
A mapping p: L(uv) L (p) is said to be a lifting of L()
if it satisfies the following six conditions:
1) p(f) = f p-a.e.
2) f = g U-a.e. implies p(f) = p(g)
3) p(af + Bg) = ap(f) + Bp(g) for a,B E R
4) frO implies p(f)W O
5) f(z) = a implies p(f)(z) = a
6) p(fg) = p(f)p(g).
We say that L (u) has the lifting property if there exists a lifting p
of L(p5). The following theorem affirms that, for probability
measures P in particular, L (P) always has the lifting property.
Theorem 1.5.4. If i has the direct sum property, then L (u) has the
lifting property.
Let, now, E and F be Banach spaces, T a set, and Z a subspace
of F', norming for F, i.e., such that
lYIF = sup i for every ycF.
ZEZ II"F'
For every function U: T L(E,F) (continuous linear maps E F),
x: T 4 E and z: T + Z, we denote by the map t __
and by [JU the map t |U(t)D.
The definitions and theorems in Dinculeanu [6] are given in
somewhat more generality; we restrict ourselves here to statements
involving the measures and o-fields we shall be working with.
__