In many statistics papers, proofs might proceed as follows: Under the event $A$, the random variables $X$ and $Y$ are independent. (Often this means that on $A^C$, they might be dependent). Then some properties of conditional independence might be used, e.g. to calculate $ \mathbb E[ X \mid A, Y] $. I feel quite uncomfortable with the latter type of manipulations.
Therefore, I am wondering two things:
1) What is the proper definition of conditional independence when conditioning on both events and $\sigma$-algebras? My guess would be that we have to check the factorization property only for sets $A \cap B_X$ and $A \cap B_y$, where $B_X \in \sigma(X)$, $B_Y \in \sigma(Y)$ the generated $\sigma$-algebras of the above random variables. (Similarly for the definition of conditional expectation.)
2) Is there any reference on properties of such expectations which are taken conditionally on both $\sigma$-algebras and events? Something akin to the standard properties of conditional expectations (conditioned on $\sigma$-algebras) used to simplify expressions or to calculate them. Or is there any simple trick to generalize the standard results to this setting?