# On measure spaces where Egoroff's Theorem holds

- UNCP Author/Contributor (non-UNCP co-authors, if there are any, appear on document)
- Dr. Laszlo Zsilinszky, Professor (Creator)
- Institution
- The University of North Carolina at Pembroke (UNCP )
- Web Site: http://www.uncp.edu/academics/library

**Abstract:** A measure space (X, S, µ) is called almost f inite if X is a union of a setof finite measure and finite many atoms of infinite measure. It is shown that Egoroff’sTheorem for sequences of measurable functions holds if and only if the underlyingmeasure space is almost finite. As a consequence we obtain several theorems on theinteraction between convergences almost everywhere, almost uniform and in measure,respectively, with no preliminary conditions on the measure space (X, S, , µ), thusextending results from [2],[4],[6],[10]. It is proved further that if (X, S, µ) is almostfinite (is not almost finite) then f : R ? R preserves almost uniform convergenceand convergence in measure, respectively, if and only if f is continuous (is uniformlycontinuous), thus augmenting a result of [3].

On measure spaces where Egoroff's Theorem holds

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Created on 10/31/2017

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## Additional Information

- Publication
- Real Anal. Exchange Vol. 20
- Language: English
- Date: 1994
- Keywords
- measure spaces, convergence, Egoroff's theorem, almost finite

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