Dilated sections and extremal preimage measures
type: Item
instance of: scholarly article
title: Dilated sections and extremal preimage measures
zbMATH Open document ID: 0538.28002
DOI: 10.1007/BF01193852
author: Wolfgang Hackenbroch
published in: Archiv der Mathematik
publication date: 1984-01-01T00:00:00Z
review text: Let a measurable mapping \(\tau: (Y,{\mathfrak B})\to(X,{\mathfrak A})\) between two measurable spaces be fixed. Define a dilation of \(\tau\) as a triplet \(((S;{\mathfrak S}),\rho,\pi)\) consisting of a measurable space \((S;{\mathfrak S})\), a measurable imbedding \(\rho: Y\to S\) (i.e. \(\rho\) injective and \(\rho\) (\({\mathfrak B})=\rho(Y)\cap {\mathfrak S})\) and a measurable ma\(p\pi: (S;{\mathfrak S})\to(X;{\mathfrak A})\) such that \(\pi {\mathbb{O}}\rho =\tau.\) Given then a fixed measure \(\mu\) on \({\mathfrak A}\) with nonempty set \(M=\{\nu:\nu \quad measure\quad on\quad {\mathfrak B}\quad with\quad \nu {\mathbb{O}}\tau^{-1}=\mu \}\) of preimage measures, it is well-known [cf. \textit{G. A. Edgar}, Ill. J. Math. 20, 630-646 (1976; Zbl 0327.28001)] that image measures \(\nu =\mu {\mathbb{O}}\sigma^{-1}\) to measurable sections \(\sigma\) of \(\tau\) areextremal elements of M. But measurable sections of \(\tau\) exist only under additional hypotheses. The main result of the present paper states however that there is always a dilation \(((S;{\mathfrak S}),\rho,\pi)\) of \(\tau\) characterizing \(ex M\) via sections, i.e. \(\nu\in M\) is extremal in M iff \(\nu {\mathbb{O}}\rho^{-1}=\mu^*{\mathbb{O}}\sigma^{- 1} (=\mu {\mathbb{O}}\sigma^{-1}\), iff \(\tau\) is onto) for some measurable section \(\sigma\) of \(\pi\).
Mathematics Subject Classification ID: 28A12
Mathematics Subject Classification ID: 28D05
zbMATH DE Number: 3855418
zbMATH Keywords: dilated sections
zbMATH Keywords: extremal preimage measures
zbMATH Keywords: dilation
zbMATH Keywords: measurable imbedding
zbMATH Keywords: measurable sections
zbMATH Keywords: extremal elements
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3855418