On the topology of convergence in measure, defined on the ring $\mathcal{M}(X,\mathscr{A},\mu)$

Abstract

For a probability measure space $(X,\mathscr{A},\mu)$, the topology $\mathcal{M}_\mu$, is defined on the ring $\mathcal{M}(X,\mathscr{A},\mu)$ of real-valued measurable functions on $(X,\mathscr{A},\mu)$ involving the notion of \textit{convergence in measure}. It turns out that if $f=g$ is assumed to be in the \textit{almost everywhere} sense, $\mathcal{M}_\mu$ is a completely metrizable space and is induced by the metric given by $\delta(f,g)=\mu(X\setminus Z(f-g))$, for $f,g\in \mathcal{M}(X,\mathscr{A},\mu)$. The notion of a measure being bounded away from zero is introduced and it is observed that a measure $\mu$ is bounded away from zero if and only if it is purely atomic and contains at most finitely many pairwise disjoint atoms. Topological properties, such as being a $P$-space, extremal disconnectedness and local connectedness of $\mathcal{M}_\mu$ are found to be equivalent to the underlying measure $\mu$ being bounded away from zero. The space $\mathcal{M}_\mu$ is proven to be never Lindel\"{o}f, and hence cannot be separable, second countable or compact. It is established that $\mathcal{M}_\mu$ is connected (in fact, path-connected) if and only if $\mu$ is non-atomic and $\mathcal{M}_\mu$ is totally disconnected if and only if $\mu$ is purely atomic. The component of a point in $\mathcal{M}_\mu$ (which is found to be equivalent to the path-component and quasicomponent of that point in $\mathcal{M}_\mu$) is computed in a general setting.