Suppose that X and Y are independent random variables. $X,Y : \Omega \rightarrow \mathbb{R}$. Show that $X^+$ and $Y^+$ are independent random variables.
$X^+= \text{max}(X,0)$, $\ Y^+ = \text{max}(Y,0)$
My Attempt: I was trying to prove this by using their sigma algebras, since random variables are independent if their sigma algebras are independent. X and Y are independent so $\sigma (X)$ and $\sigma (Y)$ are independent. If I can show that $X^+ \in \sigma (X)$ and $Y^+ \in \sigma (Y)$ then I think it should prove it.
The pre-image of $X^+$ is a subset of the pre-image of $X$. So it is an element of the set $\{X^{-1}(A):A \in \mathcal{B}\}$, which is $\sigma (X)$. Similarly for $Y$, $Y^+ \in \sigma (Y)$. Since $X$ and $Y$ are independent, so are their $\sigma$-algebra, which means $X^+$ and $Y^+$ are independent.
Could you tell me if/where this is wrong?