Recent progress in rings and subrings of real valued measurable functions

Abstract

Two separated realcompact measurable spaces (X,A) and (Y,B) are shown to be isomorphic if and only if the rings M(X,A) and M(Y,B) of all real valued measurable functions over these two spaces are isomorphic. It is furthermore shown that any such ring M(X,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 µ on (X,A), the m-topology on M(X,A) is 1st countable if and only if it is connected and this happens when and only when M(X,A) becomes identical to the subring L∞(µ) of all µ-essentially bounded measurable functions on (X,A). Additionally, we investigate the ideal structures in subrings of M(X,A) that consist of functions vanishing at all but nitely many points and functions 'vanishing at
innity' respectively. In particular, we show that the former subring equals the intersection of all free ideals in M(X,A) when (X,A) is separated and A is innite. Assuming (X,A) is locally nite, we also determine a pair of necessary and sufficient conditions for the later subring to be an ideal of M(X,A).

Key words: Rings of measurable functions, intermediate rings of measurable functions, separated measurable space, A-lter on X, A-ultralter on X, A-ideal, absolutely convex ideals, hull-kernel topology, Stone-topology, free ideal.