Ideals in Rings and Intermediate Rings of Measurable Functions

TL;DR

This paper characterizes the maximal ideals of the ring of real measurable functions on a space, showing their homeomorphism to ultrafilters and describing the structure spaces of intermediate subrings.

Contribution

It provides a topological description of maximal ideals and structure spaces of subrings of measurable functions, linking algebraic and topological properties.

Findings

Maximal ideals correspond to ultrafilters of measurable sets.

Structure spaces of intermediate subrings are compact, Hausdorff, zero-dimensional.

When X is a P-space, measurable functions coincide with continuous functions.

Abstract

The set of all maximal ideals of the ring $\mathcal{M}(X,\mathcal{A})$ of real valued measurable functions on a measurable space $(X,\mathcal{A})$ equipped with the hull-kernel topology is shown to be homeomorphic to the set $\hat{X}$ of all ultrafilters of measurable sets on $X$ with the Stone-topology. This yields a complete description of the maximal ideals of $\mathcal{M}(X,\mathcal{A})$ in terms of the points of $\hat{X}$. It is further shown that the structure spaces of all the intermediate subrings of $\mathcal{M}(X,\mathcal{A})$ containing the bounded measurable functions are one and the same and are compact Hausdorff zero-dimensional spaces. It is observed that when $X$ is a $P$-space, then $C(X) = \mathcal{M}(X,\mathcal{A})$ where $\mathcal{A}$ is the $\sigma$-algebra consisting of the zero-sets of $X$.