More on rings of functions which are discontinuous on a set of measure zero (Q2099380)

The quadruplet \((X, \tau, \mathcal{A}, \mu)\) where \((X, \tau) \) is a topological space, \(\mathcal{A}\) is a \(\sigma\)-\textit{algebra} containing \(\tau\) and \(\mu\) is a measure defined on \(\mathcal{A}\), is called a \(\tau\mathcal{A}\mu\)-space. The concepts of \(\tau\mathcal{A}\mu\)-\textit{real compact} and \(\tau\mathcal{A}\mu\)-\textit{pseudocompact} are defined using the set \(\mathcal{M}_0(X, \mu)\) of all real-valued functions that are discontinuous on a set of measure zero. The paper shows that \((X, \tau, \mathcal{A}, \mu)\) is \(\tau\mathcal{A}\mu\)-\textit{compact} if and only if it is \(\tau\mathcal{A}\mu\)-\textit{real compact} and \(\tau\mathcal{A}\mu\)-\textit{pseudocompact}. For a topological space \(X\), the ring \(C(X)_F\) of all real-values functions with at most a finite number of points of discontinuity introduced in [\textit{Z. Gharebaghi} et al., Houston J. Math. 44, No. 2, 721--739 (2018; Zbl 1403.54012)] and studied in [\textit{S. Bag} et al., Positivity 26, No. 1, Paper No. 12, 15 p. (2022; Zbl 1495.54012)] is revisited here: In \((X, \tau, \mathcal{A}, \mu)\) with restriction on \(\mu\) having measure zero for any finite set on \(X\), we have \(C(X)_F \subseteq\mathcal{M}_0(X, \mu)\). The authors show that there is no \(T_1\) space \(Y\) such that \(\mathcal{M}_0(X, \mu)\) is isomorphic to \(C(Y)_F\).