Generalized conditional expectations

Hornor, William E.

doi:10.1090/S0002-9939-1995-1277115-X

Generalized conditional expectations
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by William E. Hornor PDF

Proc. Amer. Math. Soc. 123 (1995), 3681-3685 Request permission

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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