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

Abstract

For a probability measure space $(X,\mathcal{A},\mu )$, the topology ${\mathcal{M}}_{\mu }$, is defined on the ring $\mathcal{M}(X,\mathcal{A},\mu )$ of real-valued measurable functions on $(X,\mathcal{A},\mu )$ 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 $f,g\in \mathcal{M}(X,\mathcal{A},\mu )$. 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 Lindelöf, 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.