Relationships between continuity and abstract measurability of functions.

Abstract

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}).