Generalized Conditional Expectations
Document Type
Article
Publication Date
12-1-1995
Department
Mathematics
School
Mathematics and Natural Sciences
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
Recommended Citation
Hornor, W. E.
(1995). Generalized Conditional Expectations. Proceedings of the American Mathematical Society, 123(12), 3681-3685.
Available at: https://aquila.usm.edu/fac_pubs/6005