Convergence of conditional expectation

Question

I have the following questions. For random variables $X$ and $Y$ (suppose for simplicty that the are $\mathbb{R}^n$ valued). Now, we have some abstract way of constructing $P(Y\in B|X=x)$ for some Borel set $B$. Now my first question is, under which conditions does: $$ \lim_{\epsilon \to 0 }P(Y\in B|X\in B_\epsilon (x))=P(Y\in B|X=x)\, . $$ We write $B_\epsilon(x)$ for the ball around $x$ with radius $\epsilon$.Of course, if the decay is of the same order as $\epsilon\to 0$ this is trivial, but are there some more general conditions such that this holds or counterexamples? And of course we are interested in the non-trivial case, i.e. $P(X=x)=0$.