A DECOMPOSITION OF BOUNDED, WEAKLY MEASURABLE FUNCTIONS

Let (X, A, mu) be a complete probability space, rho a lifting, T rho the associated Hausdorff lifting topology on X and E a Banach space. Suppose F: (X, T-rho) -> E-sigma '' be a bounded continuous mapping. It is proved that there is an A is an element of 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 is an element of A with mu(B) > 0 and B boolean AND A = empty set, F chi(B) cannot be weakly equivalent to a E-valued strongly measurable function. Some known results are obtained as corollaries.