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

TL;DR

This paper introduces a measure-induced pseudometric on measurable functions, exploring its topological properties and characterizing measures that influence the space's connectedness, compactness, and zero-dimensionality.

Contribution

It defines a new pseudometric on measurable functions and thoroughly analyzes its topological properties, linking them to measure-theoretic characteristics.

Findings

The space is connected iff the measure is non-atomic.

The space is zero-dimensional iff the measure is purely atomic.

Conditions are established for the measure to be bounded away from zero, relating to local compactness.

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 $\mathcal{M}_\delta$.