Some properties of algebras of real-valued measurable functions

Abstract

. Let M ( X, A ) ( M ∗ ( X, A )) be the f -ring of all (bounded) real-mea-surable functions on a T -measurable space ( X, A ), let M K ( X, A ) be the family of all f ∈ M ( X, A ) such that coz( f ) is compact, and let M ∞ ( X, A ) be all f ∈ M ( X, A ) that { x ∈ X : | f ( x ) | ≥ 1 n } is compact for any n ∈ 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 ϕ : M ( X, A ) → R , we prove that there is an element x 0 ∈ X such that ϕ ( f ) = f ( x 0 ) for every f ∈ 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 M K ( X, A ) is equal to the intersection of all free ideals of M ( X, A ) (or M ∗ ( X, A )). We show that M ∞ ( X, A ) and M K ( X, A ) do not have free ideal.