Let $X$ and $Y$ be independent random variables. Prove that $X^2$ and $Y$ are also independent.
I saw this question was asked but never explicitly solved so was wondering if this was the correct strategy: Since $X$ and $Y$ are independent there joint distribution is $F(X, Y)$ and can be written as $$F(X, Y) = F_X(x)F_Y(y) \implies P(X \leq x, Y \leq y) = P(X \leq x)P(Y \leq y)$$ Now, let $g(x) = X^2 \text{ and } h(y) = y$ Then: $$F_{g(x), g(y)}(x, y) = P(g(x) \leq x, h(y) \leq y)$$ $$= P(x \in g^{-1}(0, x], y \in h^{-1}(0, h])$$ $$= P(x \in g^{-1}(0, x])P(y \in h^{-1}(0, h])$$ $$= P(g(x) \leq x)P(h(y) \leq y)$$ $$= F_{g(x)}(x)F_{h(y)}(y) = F_{X^2}F_{Y}(y) $$
Let $A$ and $B$ be Borel sets in $\mathbb R$. Let $C=\{x: x^{2} \in A\}$. $P(X^{2} \in A, Y \in B)=P(X \in C,Y \in B)=P(X \in C) P(Y \in B)=P(X^{2} \in A)P( Y \in B)$. [$C$is a Borel set becaushe the map $x \to x^{2}$ is continuous and hence Borel measurable].