NDLI: Naturalness of the class of Lebesgue-Borel-Hausdorff measurable functions

Content Provider	Springer Nature Link
Author	Zakharov, V. K. Rodiov, T. V.
Copyright Year	2014
Abstract	It is well known that the family of all continuous functions on a topological space contains all constant functions and is closed with respect to the usual pointwise operations (addition, multiplication, finite supremum and infimum, division) and uniform convergence. The complete description of such function families (normal families) was given by Borel, Lebesgue, and Hausdorff. The normal families turned out to be exactly the families of all functions measurable with respect to multiplicative σ-additive families of sets. In 1914, Hausdorff also described the normal envelope of an arbitrary family of functions. If uniform convergence is replaced by pointwise convergence, then the notions of completely normal family and completely normal envelope arise. In 1977, Regoli described all completely normal families. They turned out to be precisely the families of all functions measurable with respect to σ-algebras of sets. Moreover, Regoli described the completely normal envelope of a specific family of functions. The present paper gives descriptive and some constructive characterizations of the completely normal envelope of an arbitrary family of functions.
Starting Page	500
Ending Page	508
Page Count	9
File Format	PDF
ISSN	00014346
Journal	Mathematical Notes
Volume Number	95
Issue Number	3-4
e-ISSN	15738876
Language	English
Publisher	Pleiades Publishing
Publisher Date	2014-04-30
Publisher Place	Moscow
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	normal family of functions completely normal family normal envelope completely normal envelope pointwise convergence measurable function Mathematics
Content Type	Text
Resource Type	Article

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