This comes from page 2 of General Theory of Markov Processes by Michael Sharpe, with some changes in notation.

Suppose $$P(A_1\cap A_2\,|\, {\cal F}_{=t} )= P(A_1 \,|\, {\cal F}_{=t} )P( A_2\,|\, {\cal F}_{=t} )$$ for all $A_1\in {\cal F}_{\leq t}$ and $A_2\in {\cal F}_{\geq t}$. Using well known properties of conditional expectations, \begin{eqnarray*} P(A_1\cap A_2) &=&P(P(A_1\cap A_2\ |\ {\cal F}_{=t}))\cr &=&P\left( P(A_1\ |\ {\cal F}_{=t})\ P(A_2 \ | \ {\cal F}_{=t}) \right)\cr &=&P(P(A_2\ |\ {\cal F}_{=t}) ; A_1). \end{eqnarray*}

My question is by what property of conditional expectations, we can have $$P\left( P(A_1\ |\ {\cal F}_{=t})\ P(A_2 \ | \ {\cal F}_{=t}) \right) = P(P(A_2\ |\ {\cal F}_{=t}) ; A_1)?$$

What is the general form for the property of conditional expectation?

Any reference?

-
@Byron: I still don't understand your reply there. –  Ethan May 4 '11 at 5:12

Apply this to G your sigma-algebra $\mathcal{F}_{=t}$, A the event $A_1$ and Y the conditional probability of $A_2$ conditionally on G.