On the positive cone of rings of measurable functions

For a measurable space (X, A) , let M⁺(X, 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 M⁺(X, A) and the ideal lattice of its ring of differences M(X, A) . Moreover, we infer that each ideal of M⁺(X, A) is a semiring z -ideal. We investigate the duality between cancellative congruences on M⁺(X, A) and ZA -filters on X . We observe that every σ -algebra is a completely regular σ -frame, so compactness and pseudocompactness coincide in σ -algebras, and we provide a new characterization for compact measurable spaces via algebraic properties of M⁺(X, A) . It is shown that the space of (real) maximal congruences on M⁺(X, A) is homeomorphic to the space of (real) maximal ideals of the M(X, A) . We solve the isomorphism problem for the semirings of the form M⁺(X, A) for compact and realcompact measurable spaces.