A pseudometric on $\mathcal{M}(X,\mathscr{A})$ induced by a measure
Amrita Dey
TL;DR
The paper introduces a measure-induced pseudometric $\delta(f,g)=\mu(X\setminus Z(f-g))$ on the real-valued measurable function ring $\mathcal{M}(X,\mathscr{A})$, forming the topological space $\mathcal{M}_\delta$ and linking its topological structure to the underlying measure $\mu$. By examining atomicity and the notion of being bounded away from zero, it characterizes when $\mathcal{M}_\delta$ is connected (non-atomic $\Rightarrow$ connected) and when it is zero-dimensional (purely atomic $\Rightarrow$ zero-dimensional), among other dualities such as local compactness and extremal disconnectedness. The work establishes a complete description of components, path components, and quasicomponents, and provides a detailed analysis of compactness and Lindelöfness, including necessary and sufficient conditions tied to the atomic decomposition of $\mu$. Overall, it reveals deep connections between measure-theoretic properties and the induced topological structure on function spaces, contrasting the $\mathcal{M}_\delta$ topology with existing $u_\mu$- and $m_\mu$-topologies.
Abstract
For a probability measure space $(X,\mathscr{A},μ)$, we define a pseudometric $δ$ on the ring $\mathcal{M}(X,\mathscr{A})$ of real-valued measurable functions on $X$ as $δ(f,g)=μ(X\setminus Z(f-g))$ and denote the topological space induced by $δ$ as $\mathcal{M}_δ$. We examine several topological properties, such as connectedness, compactness, Lindelöfness, separability and second countability of this pseudometric space. We realise that the space is connected if and only if $μ$ is a non-atomic measure and we explicitly describe the components in $\mathcal{M}_δ$, for any choice of measure. We also deduce that $\mathcal{M}_δ$ is zero-dimensional if and only if $μ$ is purely atomic. We define $μ$ to be bounded away from zero, if every non-zero measurable set has measure greater than some constant. We establish several conditions equivalent to $μ$ being bounded away from zero. For instance, $μ$ is bounded away from zero if and only if $\mathcal{M}_δ$ is a locally compact space. We conclude this article by describing the structure of compact sets and Lindelöf sets in $\mathcal{M}_δ$.