On the convergence of sequences of integrally quasicontinuous functions.

Abstract

A function $ f: \mathbb{R}^n \to \mathbb{R}$ satisfies condition $(Q_i(x))$ (resp. $(Q_s(x))$, [$Q_o(x)$]) at a point $x$ if for each real $r > 0$ and for each set $U \ni x$ belonging to the Euclidean topology in $\mathbb{R} ^n$ (resp.~to the strong density topology [to the ordinary density topology]) there is an open set $I$ such that $I \cap U \neq \emptyset $, $f$ is Lebesgue integrable on $I\cap U$ and $$\left|\frac{1}{\mu (U\cap I)}\int_{U \cap I} f(t)dt - f(x) \right| < r.$$ These notions are modifications of quasicontinuity or approximate quasicontinuity. In this article we investigate the limits of sequences of such functions.