A generalization of m-topology and U-topology on rings of measurable functions
Submitted: 2024-05-11
|Accepted:
|Published: 2025-04-01
Copyright (c) 2025 Soumyadip Acharyya, Rakesh Bharati, Atasi Deb Ray, Sudip Kumar Acharyya
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
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Keywords:
component, quasicomponent, atomic and non-atomic measure, non-separable pseudonormed linear space, mμI-topology, UμI-topology, essentially I-bounded measurable functions, μ-stable family
Supporting agencies:
University of South Carolina Sumter 2021 Summer Research Stipend Program
University Grants Commission, New Delhi
Abstract:
In this paper, we generalize mμ and Uμ topologies on M ( X , A ) via an ideal I in the ring M ( X , A ) of all real-valued measurable functions. The collection LI∞ ( μ ) of all essentially I-bounded functions over the measure space (X,A,μ)$ and the ideal Iμ (X,A) ={ f ∈ M (X,A) : for every g ∈ M(X,A), fg is essentially I-bounded } are the components of 0 in the space mμI and UμI respectively. Additionally, we obtain a chain of necessary and sufficient conditions as to when these two topologies coincide. In particular, it is proved that they coincide if and only if the three sets M(X,A), LI∞ ( μ ) , and Iμ (X,A) coincide and this holds when and only when M(X,A), with either topology, is connected, which is further equivalent to each of these two topological spaces being a topological ring as well as a topological vector space. Furthermore, LI∞ ( μ ) is a complete pseudonormed linear space regardless of the choice of I. Finally, we examine when these later spaces are separable.
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