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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 . 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 ) , 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...

Spherical completeness with infinitesimals.

José Manuel Bayod (1982)

Revista Matemática Hispanoamericana

In the theory of nonarchimedean normed spaces over valued fields other than R or C, the property of spherical completeness is of utmost importance in several contexts, and it appears to play the role conventional completeness does in some topics of classical functional analysis. In this note we give various characterizations of spherical completeness for general ultrametric spaces, related to but different from the notions of pseudo-convergent sequence and pseudo-limit introduced by Ostrowski in...

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