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 -finite measure space and a -finite sigma algebra , the conditional expectations acting on and acting on are known to be related by the formula . In this note the conditional expectation 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 . It is precisely in the non-measure-preserving case that the measurability of plays an important role. Relatedly, it is shown that if composition by *T* intertwines and any mapping , then is a conditional expectation induced by a measure equivalent to *m*. These results were motivated by a result concerning induced conditional expectation operators on -algebras, and the paper concludes with a brief description of this -algebra setting.

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DOI:
https://doi.org/10.1090/S0002-9939-1995-1231301-3

Article copyright:
© Copyright 1995
American Mathematical Society