## Real Analysis Exchange

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

### A characterization of almost everywhere continuous functions

#### 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