Generalized Conditional Expectations
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.
Proceedings of the American Mathematical Society
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