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Chapter Info

DATE

2015

AUTHORS

Piermarco Cannarsa , Teresa D’Aprile

ABSTRACT

The class of measurable, or Borel, functions \(f:X\rightarrow \mathbb {R}\cup \{\pm \infty \}\) on a measurable space \((X,\fancyscript{E}, \mu )\) can be defined in natural way using the notion of measurable sets. Such a class is stable under linear operations, product, and pointwise convergence. Moreover, if \(X\) is a topological space and \(\fancyscript{E}\) is the Borel \(\sigma \)-algebra, then every continuous function is Borel. In particular, for a Radon measure \(\mu \) on \(\mathbb {R}^N\), all Borel functions \(f:\mathbb {R}^N\rightarrow \mathbb {R}\cup \{\pm \infty \}\) preserve the regularity properties of \(\mu \). A very useful consequence of this is the fact that measurable function can be approximated with continuous functions. More... »

PAGES

37-80

Book

TITLE

Introduction to Measure Theory and Functional Analysis

ISBN

978-3-319-17018-3
978-3-319-17019-0

Author Affiliations

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/978-3-319-17019-0_2

DOI

http://dx.doi.org/10.1007/978-3-319-17019-0_2

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1001786818


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