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:

• (i) $\lim \nolimits _{n\rightarrow \infty }\int _{\Omega _{n}}\left\| \varphi _{n}\left( \omega _{n}\right) -a\right\| _{F}d\mu _{n}\left( \omega _{n}\right) =0$.

• (ii) For each uniformly continuous and bounded function $f:X\rightarrow E$, the following equality holds

$$\begin{aligned} \lim \limits _{n\rightarrow \infty }\int _{\Omega _{n}}f\left( \varphi _{n}\left( \omega _{n}\right) \right) d \mu _{n} (w_n)=f\left( a\right) \text { in norm of }E. \end{aligned}$$