I am talking about the Measure Theoretic definition of Conditional Expectation. We say that for two random variables $X$ and $Y$, the conditional expectation $E(X|Y)$ is a $\sigma(Y)$-measurable function of $Y$ such that, for all $A\in\sigma(Y)$ we have,$$E[E(X|Y)1_A]=E(X1_A)$$.
Then, how does one "define" things like $E(X|Y\in B)$ where $B\in\sigma(Y)$? Does it mean that $$E[E(X|Y)1_A]=E(X1_A)$$ for every $A\in B\cap\sigma(Y)$?
In that case, how does one define conditional probabilities like $P(X\in T|Y\in B)$? For simplicity, we can assume $Y$ to be a discrete and absolutely continuous r.v. only.