A characterization of almost everywhere continuous functions
| Autor: | Fernando Mazzone |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 1995 |
| Předmět: |
Discrete mathematics
Dominated convergence theorem Uniform convergence 28A60 Mathematics::Spectral Theory almost everywhere continuous functions Separable space Sobolev space Random measure Physics::Atomic and Molecular Clusters Almost surely weak convergence Geometry and Topology Borel set Metric differential Analysis Mathematics |
| Zdroj: | Real Anal. Exchange 21, no. 1 (1995), 317-319 |
| Popis: | 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 \(\mu\). |
| Databáze: | OpenAIRE |
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