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SUPERPOSITION AL MEASURABILITY IN THE PRESENCE OF BOTH ε -SEMICONTINUITIES OF X-SECTIONS

SUPERPOSITION AL MEASURABILITY IN THE PRESENCE OF BOTH ε -SEMICONTINUITIES OF X-SECTIONS DEMONSTRATIO MATHEMATICAVol. XXXIIINo 22000Wlodzimierz A. SlçzakSUPERPOSITION AL MEASURABILITY IN THE PRESENCEOF BOTH 5-SEMICONTINUITIES OF X-SECTIONSThe purpose of the present article is to give an answer to two problemsconcerning superpositional measurability in the Carathéodory ([4]) sense ofreal functions, published by Z. Grande [12, problème 8, p. 20], improvingat the same time earlier results of [17]. In order to formulate and solvethese problems we need some preliminary notions. Let X, Y be two separable metric spaces. We intend to deal simultaneously with the notion ofsupmeasurability [1, 4, 9, 10, 11, 16, 17] and 23-supmeasinability [6], so itis reasonable to introduce the notion of a measurable space with negligibles{X, M.{X), J"{X)), which is adequate as a common generalization of thetwo principal examples:(X, M.{X),J\f(X)), where X is a measure space and A/"(X) is the cr-idealof measure zero subsets of X, and(X, B*(X),X(X)), where B*(X) is the cr-algebra of subsets of X with theBaire property and X(X) is the cr-ideal of meager subsets of X. The symbol&(X) without asterisk we reserve for the cr-algebra of Borel subsets of X.A measurable space with negligibles is a triple (X,A4(X), ¿TX)), whereAi(X) is a cr-algebra of subsets of X and 3"{X) C "P{X) is a cr-ideal generated http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Demonstratio Mathematica de Gruyter

SUPERPOSITION AL MEASURABILITY IN THE PRESENCE OF BOTH ε -SEMICONTINUITIES OF X-SECTIONS

Ślęzak, Włodzimierz A.
Demonstratio Mathematica , Volume 33 (2): 12 – Apr 1, 2000
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Publisher
de Gruyter
Copyright
© by Włodzimierz A. Ślęzak
ISSN
0420-1213
eISSN
2391-4661
DOI
10.1515/dema-2000-0208
Publisher site
See Article on Publisher Site

Abstract

DEMONSTRATIO MATHEMATICAVol. XXXIIINo 22000Wlodzimierz A. SlçzakSUPERPOSITION AL MEASURABILITY IN THE PRESENCEOF BOTH 5-SEMICONTINUITIES OF X-SECTIONSThe purpose of the present article is to give an answer to two problemsconcerning superpositional measurability in the Carathéodory ([4]) sense ofreal functions, published by Z. Grande [12, problème 8, p. 20], improvingat the same time earlier results of [17]. In order to formulate and solvethese problems we need some preliminary notions. Let X, Y be two separable metric spaces. We intend to deal simultaneously with the notion ofsupmeasurability [1, 4, 9, 10, 11, 16, 17] and 23-supmeasinability [6], so itis reasonable to introduce the notion of a measurable space with negligibles{X, M.{X), J"{X)), which is adequate as a common generalization of thetwo principal examples:(X, M.{X),J\f(X)), where X is a measure space and A/"(X) is the cr-idealof measure zero subsets of X, and(X, B*(X),X(X)), where B*(X) is the cr-algebra of subsets of X with theBaire property and X(X) is the cr-ideal of meager subsets of X. The symbol&(X) without asterisk we reserve for the cr-algebra of Borel subsets of X.A measurable space with negligibles is a triple (X,A4(X), ¿TX)), whereAi(X) is a cr-algebra of subsets of X and 3"{X) C "P{X) is a cr-ideal generated

Journal

Demonstratio Mathematica – de Gruyter

Published: Apr 1, 2000

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