SOME PROPERTIES OF ALGEBRAS OF REAL-VALUED MEASURABLE FUNCTIONS

Let M(X, .A) (M*(X, A)) be the f-ring of all (bounded) real -measurable functions on a T-measurable space (X, A) , let M-K (X, A) be the family of all f E M(X, A ) such that coz(f) is compact, and let M-infinity(X, A ) be all f E M(X, A) that {x is an element of X : |f (x)| >= 1/n} is compact for any n E N. We introduce realcompact subrings of M(X, A) , we show that M*(X, A) is a realcompact subring of M(X, A ) , and also M(X, A) is a realcompact if and only if (X, A ) is a compact measurable space. For every nonzero real Riesz map cp : M(X, A ) -> R , we prove that there is an element x(0) E X such that phi(f) = f (x(0)) for every f E M(X, A) if (X, A) is a compact measurable space. We confirm that M.(X, A) is equal to the intersection of all free maximal ideals of M*(X, A) , and also MK (X, A) is equal to the intersection of all free ideals of M(X, A ) (or M-K(X,A)). We show that M.(X,A) and M-K(X, A) do not have free ideal.