Conditional Expectation and Ordering

Abstract

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