Pointwise Limits of Sequences of Right-Continuous Functions and Measurability of Functions of Two Variables

Abstract

In this article I prove that the pointwise limit $f:\mathbb{R}\to \mathbb{R}$ of a sequence of right-continuous functions has some special property (G) and that bounded functions of two variables $g:{\mathbb{R}}^2\to \mathbb{R}$ whose vertical sections $g_x$, $x\in \mathbb{R}$, are derivatives and horizontal sections $g^y$, $y\in \mathbb{R}$, are pointwise limits of sequences of right-continuous functions, are measurable and sup-measurable in the sense of Lebesgue.