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 MK(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 x₀∈ X such that φ (f) =f(x₀) 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 MK(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 MK(X, A) do not have free ideal.