Conditional Expectation and Ordering

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 $T = E_\mu(\cdot \mid \mathscr{A}_0)$.