### 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.

#### Full Text:

PDFDOI: https://doi.org/10.2478/tatra.v49i0.124