Dealing with Conditinal Expectation

Question

if i have E(X1a|Ft) and 1a is independent of X and Ft.However i dont know if X is independent of Ft. Can i still split the conditional expectation into E(X|Ft)E(1a)=E(X|Ft)P(A)?

cheers

Answer 1

If by "$1_A$ is independent of $X$ and ${\mathcal F}_t$" you mean that then event $A$ is independent of the $\sigma$-algebra ${\mathcal G}$ generated by $X$ and ${\mathcal F}_t$, then the answer is YES. An approach to this would be to use the tower property of conditional expectations. Start with $$ E(X1_A|{\mathcal F}_t) = E(E(X1_A|{\mathcal G})|{\mathcal F}_t) = E(XE(1_A|{\mathcal G})|{\mathcal F}_t), $$ and now use the independence of $A$ and ${\mathcal G}$.