On the convergence of sequences of integrally quasicontinuous functions (Q2501033)

In this nice paper, the author considers integrally quasicontinuous functions \(f:\mathbb R^n\rightarrow\mathbb R\) (he writes \(f\in Q_i\)) and investigates the limits of sequences of such functions. Two other notions of integral quasicontinuity of a function \(f:\mathbb R^n\rightarrow\mathbb R\) using the strong density topology and the ordinary density topology instead of the usual Euclidean topology in \(\mathbb R^n\) are considered. In these cases, the author uses the notation \(Q_s\) and \(Q_o\), respectively, instead of \(Q_i\). The inclusions \(Q_o\subset Q_s\subset Q_i\) hold and the author proves that \(Q_o\) and \(Q_s\) are uniformly closed while \(Q_i\) is not. However, the author proves that if a sequence in \(Q_i\) converges uniformly to a locally Lebesgue measurable function \(f:\mathbb R^n\rightarrow\mathbb R\), then \(f\) belongs to \(Q_i\). Yet, the author establishes conditions so that \(f:\mathbb R^n\rightarrow\mathbb R\) is the limit of a sequence in \(Q_i\). It is also proved that if the set of all continuity points of a bounded function \(f:\mathbb R^n\rightarrow\mathbb R\) is dense in \(\mathbb R^n\), then \(f\) is the quasi-uniform limit, in Arselà's sense, of a sequence in \(Q_i\) of quasicontinuous functions. An example of a quasicontinuous bounded function which does not belong to \(Q_i\) is given and it is pointed out that \(Q_i\) contains nonmeasurable functions.