On averaging of rearrangement invariant ideals of the space \(L^{1} (\Omega , \Sigma , \mu )\) by non-atomic \(\sigma \)-subalgebras of \(\Sigma \)
type: Item
instance of: scholarly article
title: On averaging of rearrangement invariant ideals of the space \(L^{1} (\Omega , \Sigma , \mu )\) by non-atomic \(\sigma \)-subalgebras of \(\Sigma \)
zbMATH Open document ID: 1216.46027
author: Alexander A. Mekler
published in: Positivity
publication date: 2010-08-13T00:00:00Z
review text: Consider a probability space \((\Omega,\Sigma,\mu)\) and a rearrangement invariant ideal \(X\) such that \(L^\infty(\mu) \subseteq X \subseteq L^1(\mu)\). An averaging operator is a conditional expectation operator \(E(\cdot| \mathcal A)\), where \(\mathcal A\) denotes a \(\sigma\)-subalgebra of \(\Sigma\). The image \(E(X| \mathcal A)\) of \(X\) by \(E(\cdot| \mathcal A)\) is a vector ideal. Then it is said that \(\mathcal A\) averages \(X\) if the inclusion \(E(X| \mathcal A) \subseteq X\) holds. The following question is the motivation of the paper. Under which condition does a \(\sigma\)-subalgebra average a rearrangement invariant ideal \(X\)? In particular, the paper deals with the case of non-atomic \(\sigma\)-subalgebras of non-atomic probability spaces (the atomic case is already known). A majorant ideal \(X\) is the one that is averaged by every purely atomic \(\sigma\)-subalgebra. A \(\sigma\)-subalgebra is verifying if it averages only majorant ideals among all rearrangement invariant ideals. After some preliminary results concerning equimeasurability, rearrangement invariance and the majorant property, the main result in Section 5 states that a non atomic \(\sigma\)-subalgebra \(\mathcal A\) is verifying if and only if, for each rearrangement invariant ideal \(X\), the consecutive \(\mathcal A\)-averaging of \(L^1(\mu)\)-rearrangement invariant hull of \(E(X| \mathcal A)\) can be extended to the majorant hull of \(X\) (Theorem 5.3). Finally, Section 6 is devoted to the case of Lebesgue probability spaces, and provides some examples comparing averagings with respect to atomic \(\sigma\)-subalgebras and non atomic ones.
reviewed by: E. A. Sánchez-Pérez
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 5770267
zbMATH Keywords: mathematical expectation operator
zbMATH Keywords: non-atomic \(\sigma\)-subalgebra
zbMATH Keywords: rearrangement invariant ideal
MaRDI profile type: Publication
full work available at URL: s11117-010-0058-8
OpenAlex ID: W2065100822
cites work: Q4781239
cites work: Q5515914
cites work: Q4139196
cites work: On rearrangement invariant and majorant hulls of averages of rearrangement invariant and majorant ideals
cites work: Q4870205
cites work: Contractive projections and prediction operators
cites work: Q4340096
cites work: Orbits of L 1 -Functions Under Doubly Stochastic Transformation
DOI: 10.1007/S11117-010-0058-8
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description: scientific article; zbMATH DE number 5770267