Suppose $X$ and $Y$ are real-valued random variables, and $f$ and $g$ are Borel measurable real-valued functions defined on $\mathbb{R}$.
If $X$ and $Y$ are independent, then I know that $f(X)$ and $g(Y)$ are also independent.
If $X$ and $Y$ are uncorrelated, are $f(X)$ and $g(Y)$ also uncorrelated?
Thanks and regards!