Limits of Sequences of Bochner Integrable Functions Over Sequences of Probability Measures Spaces

Abstract

We prove limits of sequences of Bochner integrable functions over sequences of probability measures spaces. A sample result: Let X be a bounded closed convex set in a Banach space F, $$a\in X$$ and E a non-null Banach space. Let $$\left( \Omega _{n},\Sigma _{n},\mu _{n}\right) _{n\in {\mathbb {N}}}$$ be a sequence of probability measure spaces, $$\varphi _{n}:\Omega _{n}\rightarrow X$$ a sequence of $$\mu _{n}$$ -Bochner integrable functions. Then the following assertions are equivalent: $$\begin{aligned} \lim \limits _{n\rightarrow \infty }\int _{\Omega _{n}}f\left( \varphi _{n}\left( \omega _{n}\right) \right) d \mu _{n} (w_n)=f\left( a\right) \text { in norm of }E. \end{aligned}$$