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:
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(i)
\(\lim \nolimits _{n\rightarrow \infty }\int _{\Omega _{n}}\left\| \varphi _{n}\left( \omega _{n}\right) -a\right\| _{F}d\mu _{n}\left( \omega _{n}\right) =0\).
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(ii)
For each uniformly continuous and bounded function \(f:X\rightarrow E\), the following equality holds
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We would like to thank the referee of our paper for carefully reading the manuscript and for such constructive comments, remarks and suggestions which helped improving the first version of the paper.
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Popa, D. Limits of Sequences of Bochner Integrable Functions Over Sequences of Probability Measures Spaces. Results Math 73, 134 (2018). https://doi.org/10.1007/s00025-018-0899-1
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DOI: https://doi.org/10.1007/s00025-018-0899-1