Title

Generalized Conditional Expectations

Document Type

Article

Publication Date

12-1-1995

Department

Mathematics

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 A subset of Sigma , we give conditions on A which insure that for every real-valued Sigma-measurable function f there exists a A-measurable function E(f) such that integral gfd mu = integral gE(f) d mu for every A-measurable function g for which the left integral exists. These conditions entail a notion of ''fineness'' of the subalgebra A and a ''completeness'' property of (X, A, mu). We then introduce a notion of generalized conditional expectation which requires only the former condition.

Publication Title

Proceedings of the American Mathematical Society

Volume

123

Issue

12

First Page

3681

Last Page

3685