On the positive cone of rings of measurable functions

Abstract

For a measurable space $(X,\mathcal{A})$, let $\mathcal{M}^+(X,\mathcal{A})$ be the commutative semiring of non-negative real-valued measurable functions with pointwise addition and pointwise multiplication. We show that there is a lattice isomorphism between the ideal lattice of $\mathcal{M}^+(X,\mathcal{A})$ and the ideal lattice of its ring of differences $\mathcal{M}(X,\mathcal{A})$. Moreover, we infer that each ideal of $\mathcal{M}^+(X,\mathcal{A})$ is a semiring $z$-ideal. We investigate the duality between cancellative congruences on $\mathcal{M}^{+}(X,\mathcal{A})$ and $Z_{\mathcal{A}}$-filters on $X$. We observe that every $\sigma$-algebra is a completely regular $\sigma$-frame, so compactness and pseudocompactness coincide in $\sigma$-algebras, and we provide a new characterization for compact measurable spaces via algebraic properties of $\mathcal{M}^+(X,\mathcal{A})$. It is shown that the space of (real) maximal congruences on $\mathcal{M}^+(X,\mathcal{A})$ is homeomorphic to the space of (real) maximal ideals of the $\mathcal{M}(X,\mathcal{A})$. We solve the isomorphism problem for the semirings of the form $\mathcal{M}^+(X,\mathcal{A})$ for compact and realcompact measurable spaces.