A characterization of almost everywhere continuous functions
type: Item
instance of: scholarly article
title: A characterization of almost everywhere continuous functions
zbMATH Open document ID: 0866.28003
published in: Real Analysis Exchange
publication date: 1997-07-07T00:00:00Z
review text: 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, the author proves that a bounded measurable function \(f:X\to\mathbb{R}\) is almost everywhere continuous with respect to \(\mu\in{\mathcal M}(X)\) if and only if \(\lim_{n\to\infty} \int_Xf d\mu_n=\int_Xf d\mu\) for any sequence \(\{\mu_n\}\) in \({\mathcal M}(X)\) weakly convergent to \(\mu\).
Mathematics Subject Classification ID: 28A20
Mathematics Subject Classification ID: 60B05
Mathematics Subject Classification ID: 54C05
zbMATH DE Number: 878227
zbMATH Keywords: almost everywhere continuous functions
zbMATH Keywords: measurable functions
zbMATH Keywords: probability measures
zbMATH Keywords: Borel sets
author: Fernando Mazzone
reviewed by: José M. Ayerbe
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 878227