Real Analysis Exchange

Relationships between continuity and abstract measurability of functions.

Artur Bartoszewicz and Elżbieta Kotlicka

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Making use of ideas of Marczewski and Sierpinski we propose a general approach to studies on connections between measurability, continuity and relative continuity of functions. Theorem 2.1 shows that a well-known characterization of $(s)$-measurable Marczewski functions can be extended to the case of functions measurable with respect to a wide class of algebras involved with a topology. Theorem 2.2 gene\-ralizes the Denjoy-Stepanoff theorem and shows that the Denjoy-Stepanoff property stating the continuity of $\mc A$-measurable functions at all points of a co-negligible set is quite common while an algebra $\mc A$ and an ideal $\mc J$ are the results of operations $S$ and $S_0$ on $\tau\setminus\mc I$ for a given topology $\tau$. Also from the obtained results we conclude new theorems concerning the algebras associated with product ideals (Theorems \ref{t320} and \ref{t310}).

Article information

Real Anal. Exchange, Volume 31, Number 1 (2005), 73-96.

First available in Project Euclid: 5 June 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03E15: Descriptive set theory [See also 28A05, 54H05] 03E20: Other classical set theory (including functions, relations, and set algebra) 28A05: Classes of sets (Borel fields, $\sigma$-rings, etc.), measurable sets, Suslin sets, analytic sets [See also 03E15, 26A21, 54H05] 54H05: Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets) [See also 03E15, 26A21, 28A05]

measurability relative continuity MB-representation Denjoy-Stepanoff theorem density-type topologies


Bartoszewicz, Artur; Kotlicka, Elżbieta. Relationships between continuity and abstract measurability of functions. Real Anal. Exchange 31 (2005), no. 1, 73--96.

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