# A decomposition of bounded, weakly measurable functions

In: Tatra Mountains Mathematical Publications, vol. 49, no. 2
Surjit Singh Khurana

## Details:

Year, pages: 2011, 67 - 70
Keywords:
liftings, weakly measurable functions, weakly equivalent functions, vector measures with finite variations
Let $(X, \mathcal{A}, μ)$ be a complete probability space, $ρ$ a lifting, $\mathcal{T}ρ$ the associated Hausdorff lifting topology on $X$ and $E$ a Banach space. Suppose $F\colon (X, \mathcal{T}ρ) \to E''σ$ be a bounded continuous mapping. It is proved that there is an $A \in \mathcal{A}$ such that $F χA$ has range in a closed separable subspace of $E$ (so $F χA\colon X \to E$ is strongly measurable) and for any $B \in \mathcal{A}$ with $μ(B) >0$ and $B \cap A = \emptyset$, $F χB$ cannot be weakly equivalent to a $E$-valued strongly measurable function. Some known results are obtained as corollaries.