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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Descriptions of conditional expectations induced by non-measure preserving transformations


Authors: Alan Lambert and Barnet M. Weinstock
Journal: Proc. Amer. Math. Soc. 123 (1995), 897-903
MSC: Primary 28D99; Secondary 46N30, 60A10
MathSciNet review: 1231301
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Abstract: Given a measure-preserving transformation T acting on a $ \sigma $-finite measure space $ (X,\mathcal{A},m)$ and a $ \sigma $-finite sigma algebra $ \mathcal{B} \subset \mathcal{A}$, the conditional expectations $ E( \cdot \vert\mathcal{B})$ acting on $ {L^\infty }(\mathcal{A})$ and $ E( \cdot \vert{T^{ - 1}}\mathcal{B})$ acting on $ {L^\infty }({T^{ - 1}}\mathcal{A})$ are known to be related by the formula $ [E(f\vert\mathcal{B})] \circ T = E(f \circ T\vert{T^{ - 1}}\mathcal{B})$ . In this note the conditional expectation $ E( \cdot \vert{T^{ - 1}}\mathcal{B})$ is investigated in the non-measure-preserving case, and those transformations for which the above equation holds are characterized in terms of measurability conditions for $ d(m \circ {T^{ - 1}})/dm$. It is precisely in the non-measure-preserving case that the measurability of $ d(m \circ {T^{ - 1}})/dm$ plays an important role. Relatedly, it is shown that if composition by T intertwines $ E( \cdot \vert\mathcal{B})$ and any mapping $ \Lambda $, then $ \Lambda $ is a conditional expectation induced by a measure equivalent to m. These results were motivated by a result concerning induced conditional expectation operators on $ {C^ \ast }$-algebras, and the paper concludes with a brief description of this $ {C^\ast}$-algebra setting.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1995-1231301-3
PII: S 0002-9939(1995)1231301-3
Article copyright: © Copyright 1995 American Mathematical Society