Real Analysis Exchange

A characterization of almost everywhere continuous functions

Fernando Mazzone

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Abstract

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 \(\mu\).

Article information

Source
Real Anal. Exchange, Volume 21, Number 1 (1995), 317-319.

Dates
First available in Project Euclid: 3 July 2012

Permanent link to this document
https://projecteuclid.org/euclid.rae/1341343248

Mathematical Reviews number (MathSciNet)
MR1377542

Zentralblatt MATH identifier
0866.28003

Subjects
Primary: 28A60: Measures on Boolean rings, measure algebras [See also 54H10]

Keywords
weak convergence almost everywhere continuous functions

Citation

Mazzone, Fernando. A characterization of almost everywhere continuous functions. Real Anal. Exchange 21 (1995), no. 1, 317--319. https://projecteuclid.org/euclid.rae/1341343248


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