Let $A \subset B$ be a subalgebra. This means that $E(X|A)$ is also $B$ measurable. Now, if I can show that $E(X|B)$ is also $A$ measurable, does this imply that the conditional expectations are the same?
I know that conditional expectations are uniquely defined. But I don't see where
$\int_{H} E(X|A) = \int_{H} X $ for all $H \in B$ follows from.