Let $(X_i, Y_i)_{i=1}^{\infty}$ be iid continuous random vectors with continuous joint density, where $X_1$ have support $\mathcal{X}$. Let $B_n\subset \mathcal{X}\subset\mathbb{R}$ be decreasing subsets (open intervals) such that $\cap B_n= x_0\in\mathcal{X}$.
Let $S = \{i\leq n: X_i\in B_n\}$. I want to show that $$ \frac{1}{|S|}\sum_{i\in S}Y_i \overset{P}{\to} \mathbb{E}[Y_1\mid X_1=x_0], \,\,as\,\,n\to\infty. $$ I assume that the necessary condition for this convergence is $|S|\to\infty$ or that $nP(X_i\in B_n)\to\infty$. Is it sufficient? Is there some theory that describes this?