## The Annals of Probability

### Conditional Expectation and Ordering

Ep de Jonge

#### Abstract

Let $(\Omega, \mathscr{A}, \mu)$ be a probability space and let $L$ be an ideal in $M(\Omega, \mathscr{A}, \mu)$ containing $\chi_\Omega$. A one-one correspondence between the class of "order closed" linear subspaces of $L$ and the sub $\sigma$-algebras of $\mathscr{A}$ is proved. Furthermore, if $T : L \rightarrow M(\Omega, \mathscr{A}, \mu)$ is a strictly positive order continuous projectionlike linear map then $T$ is shown to be a conditional expectation $E_\nu(\cdot \mid\mathscr{A}_0)$. It follows that if $T: L \rightarrow M(\Omega, \mathscr{A}, \mu)$ is a positive expectation invariant projectionlike linear map, then even $T = E_\mu(\cdot \mid \mathscr{A}_0)$.

#### Article information

Source
Ann. Probab., Volume 7, Number 1 (1979), 179-183.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176995162

Digital Object Identifier
doi:10.1214/aop/1176995162

Mathematical Reviews number (MathSciNet)
MR515827

Zentralblatt MATH identifier
0392.60004

JSTOR