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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Generalized conditional expectations

Author: William E. Hornor
Journal: Proc. Amer. Math. Soc. 123 (1995), 3681-3685
MSC: Primary 28A99; Secondary 60A10
MathSciNet review: 1277115
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Abstract: In this paper, we state conditions sufficient for the existence of conditional expectations. Given a measure space $ (X,\Sigma ,\mu )$ and a $ \sigma $-subalgebra $ \mathcal{A} \subset \Sigma $, we give conditions on $ \mathcal{A}$ which insure that for every real-valued $ \Sigma $-measurable function f there exists a $ \mathcal{A}$-measurable function $ E(f)$ such that $ \smallint gfd\mu = \smallint gE(f) d\mu $ for every $ \mathcal{A}$-measurable function g for which the left integral exists. These conditions entail a notion of "fineness" of the subalgebra $ \mathcal{A}$ and a "completeness" property of $ (X,\mathcal{A},\mu )$. We then introduce a notion of generalized conditional expectation which requires only the former condition.

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Article copyright: © Copyright 1995 American Mathematical Society

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