Maximal ideals in rings of real measurable functions

Abstract

Let $ M (X)$ be the ring of all real measurable functions on a measurable

space $(X, \mathscr{A})$.

In this article, we show that every ideal of $M(X)$ is a $Z^{\circ}$-ideal.

Also, we give several characterizations of maximal ideals of $M(X)$,

mostly in terms of certain lattice-theoretic properties of $\mathscr{A}$.

The notion of $T$-measurable space is introduced.

Next, we show that for every measurable space $(X,\mathscr{A})$ there

exists a $T$-measurable space $(Y,\mathscr{A}^{\prime})$ such that

$M(X)\cong M(Y)$ as rings. The notion of compact measurable space is introduced. Next, we prove that if $(X, \mathscr{A})$ and $(Y, \mathfrak{M^{\prime}})$ are two compact $T$-measurable spaces, then $X\cong Y$ as measurable spaces if and only if $M(X)\cong M (Y)$ as rings.