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1995/1996 A characterization of almost everywhere continuous functions
Fernando Mazzone
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Real Anal. Exchange 21(1): 317-319 (1995/1996).


Let \((X,d)\) be a separable metric space and \({\mathcal M}(X)\) the set of probability measures on the \(\sigma\)-algebra of Borel sets in \(X\). In this paper we will show that a function \(f\) is almost everywhere continuous with respect to \(\mu\in{\mathcal M}(X)\) if and only if \(\lim_{n\to\infty} \int_{X}f\,d\mu_n=\int_{X}f\,d\mu\), for all sequences \(\{\mu_n\}\) in \({\mathcal M}(X)\) such that \(\mu_n\) converges weakly to \(\mu\).


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Fernando Mazzone. "A characterization of almost everywhere continuous functions." Real Anal. Exchange 21 (1) 317 - 319, 1995/1996.


Published: 1995/1996
First available in Project Euclid: 3 July 2012

zbMATH: 0866.28003
MathSciNet: MR1377542

Primary: 28A60

Keywords: almost everywhere continuous functions , weak convergence

Rights: Copyright © 1995 Michigan State University Press

Vol.21 • No. 1 • 1995/1996
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