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Some properties of algebras of real-valued measurable functions

Ali Akbar Estaji, Ahmad Mahmoudi Darghadam (2023)

Archivum Mathematicum

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 ∈ 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...

Some properties of g- and P-spaces

Kalamidas, N. (1999)

Serdica Mathematical Journal

A γ-space with a strictly positive measure is separable. An example of a non-separable γ−space with c.c.c. is given. A P−space with c.c.c. is countable and discrete.

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