A Generalization of m -topology and U -topology on rings of measurable functions

2022/07/11 by Soumyadip Acharyya, Rakesh Bharati, Acharyya, Soumyadip +5
Mathematics · #Rings, Modules, and Algebras #Advanced Topology and Set Theory #Mathematical and Theoretical Analysis

paper · pdf · doi:10.48550/arxiv.2207.05550

Abstract

For a measurable space (X,A), let M(X,A) be the corresponding ring of all real valued measurable functions and let μ be a measure on (X,A). In this paper, we generalize the so-called m_μ and U_μ topologies on M(X,A) via an ideal I in the ring M(X,A). The generalized versions will be referred to as the m_μ_I and U_μ_I topology, respectively, throughout the paper. L_I^∞ (μ) stands for the subring of M(X,A) consisting of all functions that are essentially I-bounded (over the measure space (X,A, μ)). Also let I_μ (X,A) = \ f ∈ M(X,A) : for every g ∈ M(X,A), fg is essentially I-bounded \. Then I_μ (X,A) is an ideal in M(X,A) containing I and contained in L_I^∞ (μ). It is also shown that I_μ (X,A) and L_I^∞ (μ) are the components of 0 in the spaces m_μ_I and U_μ_I, respectively. Additionally, we obtain a chain of necessary and sufficient conditions as to when these two topologies coincide.

Related

• U-topology and m-topology on the ring of Measurable Functions, generalized and revisited
• Measure finite topology on the ring of measurable functions
• Structure spaces and allied problems on a class of rings of measurable functions
• Recent progress in Rings and Subrings of Real Valued Measurable\n Functions
• Ideals in Rings and Intermediate Rings of Measurable Functions