On a. c. limits of decreasing sequences of continuous or right continuous functions (Q1396767)

A function \(f:X\to \mathbb R\) belongs to the class \(B_1^*\) if there is a sequence of continuous functions \(f_n:X\to \mathbb R\) such that \(f= \text{a.c.}\lim_{n\to \infty } f_n\), i.e.\ for each point \(x\in X\) there is a positive integer \(k\) such that \(f_n(x)=f(x)\) for any \(n>k\). The a.c. limits were introduced by Á. Császár and M. Laczkovich. In the first part of the paper, the a.c. limits of decreasing sequences of continuous functions are investigated. Let \(X\) be a perfectly normal \(\sigma \)-compact Hausdorff topological space. The following result is obtained: An upper semicontinuous function \(f:X\to \mathbb R\) belongs to the class \(B_1^*\) iff there is a decreasing sequence of continuous functions \(f_n:X\to \mathbb R\) such that \(f= \text{a.c.}\lim _{n\to \infty } f_n\). In the second part, sequences, resp.\ decreasing sequences of right continuous functions on \(X=[a,b)\) are discussed. If \(f\) is upper semicontinuous from the right, a sufficient condition for the existence of a decreasing sequence of right continuous functions \(f_n\) such that \(f= \text{a.c.}\lim _{n\to \infty } f_n\) is given.