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

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∈X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a\in X$$\end{document} and E a non-null Banach space. Let Ωn,Σn,μnn∈N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( \Omega _{n},\Sigma _{n},\mu _{n}\right) _{n\in {\mathbb {N}}}$$\end{document} be a sequence of probability measure spaces, φn:Ωn→X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi _{n}:\Omega _{n}\rightarrow X$$\end{document} a sequence of μn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _{n}$$\end{document}-Bochner integrable functions. Then the following assertions are equivalent:limn→∞∫Ωnφnωn-aFdμnωn=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lim \nolimits _{n\rightarrow \infty }\int _{\Omega _{n}}\left\| \varphi _{n}\left( \omega _{n}\right) -a\right\| _{F}d\mu _{n}\left( \omega _{n}\right) =0$$\end{document}.For each uniformly continuous and bounded function f:X→E\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:X\rightarrow E$$\end{document}, the following equality holdslimn→∞∫Ωnfφnωndμn(wn)=fain norm ofE.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim \limits _{n\rightarrow \infty }\int _{\Omega _{n}}f\left( \varphi _{n}\left( \omega _{n}\right) \right) d \mu _{n} (w_n)=f\left( a\right) \text { in norm of }E. \end{aligned}$$\end{document}