Pointwise convergence on the rings of functions which are discontinuous on a set of measure zero

Dhananjoy Mandal; Achintya Singha; Sagarmoy Bag

doi:10.4995/agt.2024.18884

Authors

Dhananjoy Mandal University of Calcutta
Achintya Singha Bangabasi Evening College
Sagarmoy Bag Bangabasi Evening College

DOI:

https://doi.org/10.4995/agt.2024.18884

Keywords:

measure spaces, complete measure, real maximal ideal, B1-separated, ZB -filters, B1(X,μ)-compact space

Abstract

Consider the ring ℳ_∘ ( X , μ ) of functions which are discontinuous on a set of measure zero which is introduced and studied extensively in [2]. In this paper, we have introduced a ring B₁ ( X , μ ) of functions which are pointwise limits of sequences of functions in ℳ_∘ ( X , μ ) . We have studied various properties of zero sets, B₁ ( X , μ ) -separated and B₁ ( X , μ ) -embedded subsets of B₁ ( X , μ ) and also established an analogous version of Urysohn's extension theorem. We have investigated a connection between ideals of B₁ ( X , μ ) and Z_B -filters on X. We have studied an analogue of Gelfand-Kolmogoroff theorem in our setting. We have defined real maximal ideals of B₁ ( X , μ ) and established the result | ℛ M a x ( ℳ_∘ ( X , μ ) ) | = | ℛ M a x ( B₁ ( X , μ ) ) | , where ℛ M a x ( ℳ_∘ ( X , μ ) ) and ℛ M a x ( B₁ ( X , μ ) ) are the sets of all real maximal ideals of ℳ_∘ ( X , μ ) and B₁ ( X , μ ) respectively.

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Author Biographies

Dhananjoy Mandal, University of Calcutta

Associate Professor, Department of Pure Mathematics, University of Calcutta

Achintya Singha, Bangabasi Evening College

Department of Mathematics

Sagarmoy Bag, Bangabasi Evening College

Department of Mathematics

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