A DECOMPOSITION OF BOUNDED, WEAKLY MEASURABLE FUNCTIONS

Abstract

Let (X,A, μ) be a complete probability space,

\rho a lifting, and T_\rho

the associated Hausdorff lifting topology on X.

Suppose F : (X, T\rho) → E′′
_\sigma be a bounded continuous mapping. It is
proved that there is an A ∈ A such that F_{\chi A}

has range in a closed
separable subspace of E (so F_{\chi A} : X → E

is strongly measurable)
and for any B ∈ A with μ(B) > 0 and

B ∩ A = \emptyset, F_{\chi,B} cannot
be weakly equivalent to a E-valued strongly measurable function.
Some known results are obtained as corollaries.