Distributional equality and independence

Question

Let $X$ and $Y$ be random variables. Assume that \begin{align} P(X \in A, Y \in B) = P(X \in A) \, P(X \in B). \end{align} Claim: Then, $X$ and $Y$ are equal in distribution and independent of each other.

I agree with this intuitively but would be interested in a rigorous explanation.

Answer 1

Take $A =\mathbb R$ to see that $P(Y \in B)=P(X \in B)$. Hence $X$ and $Y$ have the same distribution. We now get $P(X \in A, Y \in B)=P(X\in A) P(Y \in B)$ so $X$ and $Y$ are independent.