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.References
- Arlen Brown and Carl Pearcy, Introduction to operator theory. I, Graduate Texts in Mathematics, No. 55, Springer-Verlag, New York-Heidelberg, 1977. Elements of functional analysis. MR 0511596
- H. D. Brunk, On an extension of the concept conditional expectation, Proc. Amer. Math. Soc. 14 (1963), 298–304. MR 148090, DOI 10.1090/S0002-9939-1963-0148090-5
- E. J. McShane, Families of measures and representations of algebras of operators, Trans. Amer. Math. Soc. 102 (1962), 328–345. MR 137002, DOI 10.1090/S0002-9947-1962-0137002-X
- M. M. Rao, Measure theory and integration, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1987. A Wiley-Interscience Publication. MR 891879
- Adriaan Cornelis Zaanen, Integration, North-Holland Publishing Co., Amsterdam; Interscience Publishers John Wiley & Sons, Inc., New York, 1967. Completely revised edition of An introduction to the theory of integration. MR 0222234
- A. C. Zaanen, The Radon-Nikodym theorem. I, II, Nederl. Akad. Wetensch. Proc. Ser. A 64 = Indag. Math. 23 (1961), 157–170, 171–187. MR 0146340
Additional Information
- © Copyright 1995 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 123 (1995), 3681-3685
- MSC: Primary 28A99; Secondary 60A10
- DOI: https://doi.org/10.1090/S0002-9939-1995-1277115-X
- MathSciNet review: 1277115