$X^+$ and $Y^+$ are independent random variables if $X$ and $Y$ are independent random variables

Question

I am aiming to prove that $X^+$ and $Y^+$ are independent random variables if $X$ and $Y$ are independent random variables.

I know that $$P(X=x \text{ or } X=−x)=P(X=x\text{ or } X=−x \mid Y=y \texttt{ or } Y=−y)$$ by independence of $X$ and $Y$, and thus $$P(X^2=x^2)=P(X^2=x^2 \mid Y^2=y^2)$$ demonstrating $X^2$ is independent of $Y^2$.

However, this only proves the case for $X^2$ and $Y^2$. Is there a way I can generalize the proof for $X^+$ and $Y^+$?

Answer 1

Hint: Use the theorem that states: If $X$ and $Y$ are independent, then $g(X)$ and $h(Y)$ are independent for any measurable $g$, $h$.