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A Note on Regular Measures

Published online by Cambridge University Press:  20 November 2018

Elias Zakon
Elias Zakon*
Affiliation:
University of Windsor
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As is well known, every Borel measure in a metric space S is regular, provided that S is the union of a sequence of open sets of finite measure. It seems, however, not yet to have been noticed that this theorem can be easily extended to all spaces with Urysohn' s f "F-property", i.e., spaces in which every closed set is a countable intersection of open sets (we call such spaces "F-spaces"). Indeed, various theorems are unnecessarily restricted to metric spaces, while weaker assertions are made about F-spaces. This seems to justify the publication of the following simple proof which extends the theorem stated above to F-spaces.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 7 , Issue 1 , March 1964 , pp. 41 - 44
Copyright
Copyright © Canadian Mathematical Society 1964

References

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2. Saks, S., Theory of the Integral. Monografie Matem., Warsaw, 1937.Google Scholar
3. Schaerf, H. M., Regular measures (abstract). Bull. Am. Math. Soc., 54(1948), 660.Google Scholar
4. Schaerf, H. M., On the continuity of measurable functions in neighborhood spaces. Portug. Mathem. (1947) 33–34 and (1948) 91-92.Google Scholar