function of independent random variables

Question

I have following question: If $X$ and $Y$ are independent, then are $g(X)$ and $g(Y)$ independent as well, for any real function $g$?

Answer 1

If $g$ is measurable, the answer is yes (note that otherwise, we do not even know whether $g(X)$ and $g(Y)$ and random variables). Note that, for every pairs of Borel sets $A, B \subseteq \mathbf R$, we have \begin{align*} \def\P{\mathbf P}\P\bigl(g(X)\in A, g(Y)\in B\bigr) &= \P(X \in g^{-1}[A], Y \in g^{-1}[B])\\ &= \P(X \in g^{-1}[A])\P(Y \in g^{-1}[B])\\ &= \P\bigl(g(X) \in A\bigr)\P\bigl(g(Y) \in B\bigr). \end{align*} Hence, $g(X)$ and $g(Y)$ are independent.