A pseudometric on $\mathcal{M}(X,\mathscr{A})$ induced by a measure

Abstract

For a probability measure space $(X,\mathscr{A},\mu)$, we define a pseudometric $\delta$ on the ring $\mathcal{M}(X,\mathscr{A})$ of real-valued measurable functions on $X$ as $\delta(f,g)=\mu(X\setminus Z(f-g))$ and denote the topological space induced by $\delta$ as $\mathcal{M}_\delta$. We examine several topological properties, such as connectedness, compactness, Lindel\"{o}fness, separability and second countability of this pseudometric space. We realise that the space is connected if and only if $\mu$ is a non-atomic measure and we explicitly describe the components in $\mathcal{M}_\delta$, for any choice of measure. We also deduce that $\mathcal{M}_\delta$ is zero-dimensional if and only if $\mu$ is purely atomic. We define $\mu$ to be bounded away from zero, if every non-zero measurable set has measure greater than some constant. We establish several conditions equivalent to $\mu$ being bounded away from zero. For instance, $\mu$ is bounded away from zero if and only if $\mathcal{M}_\delta$ is a locally compact space. We conclude this article by describing the structure of compact sets and Lindel\"{o}f sets in $\mathcal{M}_\delta$.