Is it true that $P(X 1_{\{X \in B\}} \in A) = P(X \in A \cap B)$?

Question

Let $X$ be a random variable and $A,B$ measurable sets. Is it true that $P(X 1_{\{X \in B\}} \in A) = P(X \in A \cap B)$?

Let $\omega \in \Omega$. If $X(\omega) \in A \cap B$ then $1_{\{X(\omega) \in B\}}(\omega)=1$ and $X(\omega) \in A$ and hence $X(\omega)1_{\{X(\omega) \in B\}}(\omega)=X(\omega) \in A$.

The other inclusion I have a harder time showing. Or maybe there is a better way?

Answer 1

False. The left side is $P(X\notin B)+P(X \in A\cap B)$ if $0 \in A$.

[If $C$ is the event on the left side use the fact that $P(C)=P(C\cap (X\in B))+P(C\cap (X\notin B))$].