# 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