Measure finite topology on the ring of measurable functions

2025/11/19 by Dey, Soumajit, Acharyya, Sudip Kumar, Mandal, Dhananjoy
#46E30 #54C40 #Commutative Algebra (math.AC) #FOS: Mathematics #General Topology (math.GN)

paper · doi:10.48550/arxiv.2511.15436

Abstract

Let M(X,A,μ) be the ring of all real-valued measurable functions constructed over a measure space (X,A,μ). A topology on M(X,A,μ), called the F_μ-topology weaker than the U_μ-topology is introduced. It is realized that the component, the quasi component and the path component in this F_μ-topology are identical. It turns out that the F_μ-topology on M(X,A,μ) becomes connected if and only if it is path connected if and only if μ is an atomic measure of a special type. It is also proved that the F_μ-topology is first countable when and only when μ is a hemifinite measure. Finally, it is shown that the second countability of the F_μ-topology is equivalent to the hemifiniteness of the measure μ together with the countable chain condition of the F_μ-topology.

Citations

• U-topology and m-topology on the ring of Measurable Functions, generalized and revisited
• Recent progress in Rings and Subrings of Real Valued Measurable\n Functions
• Ideals in Rings and Intermediate Rings of Measurable Functions
• Boundedness properties in function spaces

Related

• A Generalization of m -topology and U -topology on rings of measurable functions
• On the topology of convergence in measure, defined on the ring M(X,\mathscrA,μ)
• U-topology and m-topology on the ring of Measurable Functions, generalized and revisited
• Comparison of topologies on *-algebras of locally measurable operators
• Recent progress in Rings and Subrings of Real Valued Measurable\n Functions