Maximal ideals in rings of real measurable functions

2018/03/16 by Estaji, Ali Akbar, Darghadam, Ahmad Mahmoudi, Yousefpour, Hasan
#06D22 #13A30 #28A20 #54C30 #FOS: Mathematics #Functional Analysis (math.FA) #General Topology (math.GN)

paper · doi:10.48550/arxiv.1803.06271

Abstract

Let M (X) be the ring of all real measurable functions on a measurable space (X, \mathscrA). In this article, we show that every ideal of M(X) is a Z^∘-ideal. Also, we give several characterizations of maximal ideals of M(X), mostly in terms of certain lattice-theoretic properties of \mathscrA. The notion of T-measurable space is introduced. Next, we show that for every measurable space (X,\mathscrA) there exists a T-measurable space (Y,\mathscrA^′) such that M(X)≅ M(Y) as rings. The notion of compact measurable space is introduced. Next, we prove that if (X, \mathscrA) and (Y, \mathfrakM^′) are two compact T-measurable spaces, then X≅ Y as measurable spaces if and only if M(X)≅ M (Y) as rings.

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