Recent progress in Rings and Subrings of Real Valued Measurable Functions

Abstract: Two separated realcompact measurable spaces $(X,\mathcal{A})$ and $(Y,\mathcal{B})$ are shown to be isomorphic if and only if the rings $\mathcal{M}(X,\mathcal{A})$ and $\mathcal{M}(Y,\mathcal{B})$ of all real valued measurable functions over these two spaces are isomorphic. It is furthermore shown that any such ring $\mathcal{M}(X,\mathcal{A})$, even without the realcompactness hypothesis on $X$, can be embedded monomorphically in a ring of the form $C(K)$, where $K$ is a zero dimensional Hausdorff topological space. It is also shown that given a measure $\mu$ on $(X,\mathcal{A})$, the $m_\mu$-topology on $\mathcal{M}(X,\mathcal{A})$ is 1st countable if and only if it is connected and this happens when and only when $\mathcal{M}(X,\mathcal{A})$ becomes identical to the subring $L^\infty(\mu)$ of all $\mu$-essentially bounded measurable functions on $(X,\mathcal{A})$. Additionally, we investigate the ideal structures in subrings of $\mathcal{M}(X,\mathcal{A})$ that consist of functions vanishing at all but finitely many points and functions 'vanishing at infinity' respectively. In particular, we show that the former subring equals the intersection of all free ideals in $\mathcal{M}(X,\mathcal{A})$ when $(X,\mathcal{A})$ is separated and $\mathcal{A}$ is infinite. Assuming $(X,\mathcal{A})$ is locally finite, we also determine a pair of necessary and sufficient conditions for the later subring to be an ideal of $\mathcal{M}(X,\mathcal{A})$.