Suppose I have a random variable $X$ from which I sample $n$ times. Then I take the average of only those realizations that fullfill a certain condition $A$.
Will this average converge in probability to the conditional expecation?
$$ \frac{1}{n}\sum_{i=1\\i \neq j}^{n} x_i \overset{P}{\rightarrow} \mathbb{E}[X|A]\quad j =\{i \in n | x_i \notin A \} $$