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 $T = E_\mu(\cdot \mid \mathscr{A}_0)$.
Citation
Ep de Jonge. "Conditional Expectation and Ordering." Ann. Probab. 7 (1) 179 - 183, February, 1979. https://doi.org/10.1214/aop/1176995162
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