Conditional Expectation, is this true?

Question

I'd like to know if this statement is true:

$$ \mathbb{E}[f(X)|X\in A] = \int f(X) \mathbb{I}_{(X\in A)} dP $$

for any $f(X)$ and $X$ random variables and $A\subset\mathbb{R}$.

Answer 1

It's not true; just take $A$ to be an event with probability zero.

For example consider $X \sim U[0, 1]$, $f = \text{id}_\mathbb{R}$ and $A = \{\frac{1}{2}\}$. Then the left-hand side is $\frac{1}{2}$ and the right-hand side is zero.