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

Abstract

Consider $(X,\tau ,{\mathcal {A}},\mu )$, where $(X,\tau )$ is a topological space, ${\mathcal {A}}$ is a $\sigma$-algebra containing $\tau$ and $\mu$ is a measure defined on ${\mathcal {A}}$. We call the quadruplet $(X,\tau ,{\mathcal {A}},\mu )$ a $\tau {\mathcal {A}}\mu$-space. Let ${\mathcal {M}}_\circ (X,\mu )$ be the set of all real valued functions which are discontinuous on a set of measure zero. We define and study $\tau {\mathcal {A}}\mu$-compact, $\tau {\mathcal {A}}\mu$-real compact and $\tau {\mathcal {A}}\mu$-pseudocompact spaces. Main aim of this article is to investigate the relations among these concepts. We show that $(X,\tau ,{\mathcal {A}},\mu )$ is $\tau {\mathcal {A}}\mu$-compact if and only if it is $\tau {\mathcal {A}}\mu$-pseudocompact and $\tau {\mathcal {A}}\mu$-real compact. Also we establish Stone Weierstrass like theorem for a $\tau {\mathcal {A}}\mu$-compact space. We define ${\mathcal {Z}}$-filter on X and characterize $\tau {\mathcal {A}}\mu$-compact, $\tau {\mathcal {A}}\mu$-real compact spaces via ${\mathcal {Z}}$-filters on X and ideals of ${\mathcal {M}}_\circ (X,\mu )$. Finally, we answer a question raised by S. Bag et al. [Rings of functions which are discontinuous on a set of measure zero, Positivity, 16(12)(2022),1-15].