## The Annals of Mathematical Statistics

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

Richard L. Dykstra

#### 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.

#### Article information

Source
Ann. Math. Statist., Volume 41, Number 2 (1970), 698-701.

Dates
First available in Project Euclid: 27 April 2007

https://projecteuclid.org/euclid.aoms/1177697117

Digital Object Identifier
doi:10.1214/aoms/1177697117

Mathematical Reviews number (MathSciNet)
MR258083

Zentralblatt MATH identifier
0202.17601

JSTOR

#### Citation

Dykstra, Richard L. A Characterization of a Conditional Expectation with Respect to a $\Sigma$- Lattice. Ann. Math. Statist. 41 (1970), no. 2, 698--701. doi:10.1214/aoms/1177697117. https://projecteuclid.org/euclid.aoms/1177697117