Conditional probability and indicator function Can someone give me a formal rigorous proof of the following equation? 
$$\frac{E\{X \cdot I(T=1) \}}{\Pr(T=1)}= E(X|T=1)$$
Many thanks!

\begin{align}
\mathsf{E}(X \cdot \mathbb{I}(T = 1)) &= \sum_t \mathsf{E}(X \cdot \mathbb{I}(T = 1) \mid T = t)\cdot\Pr(T = t) \\
&= \mathsf{E}(X \cdot \mathbb{I}(T = 1) \mid T = 1) \cdot \Pr(T = 1) \\
&= \mathsf{E}(X \mid T = 1) \cdot \Pr(T = 1)
\end{align}
where the first equality is by the law of total expectation.
