2023/06/03 by Pratip Nandi, Nandi, Pratip, A. Deb Ray +3 · 1 citation
Mathematics · #Advanced Topology and Set Theory #FOS: Mathematics #General Topology (math.GN) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2306.03768
openalex publication_date 2023/06/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let M(X,A) be the ring of all real valued measurable functions defined over the measurable space (X,A). Given an ideal I in M(X,A) and a measure μ:A→[0,∞], we introduce the UμI-topology and the mμI-topology on M(X,A) as generalized versions of the topology of uniform convergence or the U-topology and the m-topology on M(X,A) respectively. With I=M(X,A), these two topologies reduce to the Uμ-topology and the mμ-topology on M(X,A) respectively, already considered before. If I is a countably generated ideal in M(X,A), then the UμI-topology and the mμI-topology coincide if and only if X∖ \bigcap Z[I] is a μ-bounded subset of X. The components of 0 in M(X,A) in the UμI-topology and the mμI-topology are realized as I∩ L^∞(X,A,μ) and I∩ Lψ(X,A,μ) respectively. Here L^∞(X,A,μ) is the set of all functions in M(X,A) which are essentially μ-bounded over X and Lψ(X,A,μ)=\f∈ M(X,A): ~∀ g\inM(X,A), f.g∈ L^∞(X,A,μ)\. It is established that an ideal I in M(X,A) is dense in the Uμ-topology if and only if it is dense in the mμ-topology and this happens when and only when there exists Z∈ Z[I] such that μ(Z)=0. Furthermore, it is proved that I is closed in M(X,A) in the mμ-topology if and only if it is a Zμ-ideal in the sense that if f≡ g almost everywhere on X with f∈ I and g\inM(X,A), then g∈ I.