A Characterization of a Conditional Expectation with Respect to a $\Sigma$- Lattice

Abstract

Several authors [1], [2], $\cdots$, [6], have derived characterizations of a conditional expectation operator. That is, if $T$ is a transformation which maps a particular set of functions into the same set, then necessary and sufficient conditions are specified so that $T$ is a conditional expectation operator. It is shown in the present paper that a similar sort of characterization can be found in the more general case when $T$ is a conditional expectation with respect to a $\sigma$-lattice operator even though $T$ need not be linear.