How to prove that functions of independent variables are independent?

Question

How do I prove the rather obvious result that if random variables $X$ and $Z$ are independent, then $f(X)$ and $g(Z)$ are independent.

Answer 1

Notice that for random variables $X,~Z:\Omega \rightarrow \mathbb{R}^k$ and Borel functions $f,~g:\mathbb{R}^k \rightarrow \mathbb{R}^m$, we have:

$$P\big[ f(X)^{-1}(A)\cap g(Z)^{-1}(B)\big]=P\big[ X^{-1}(f^{-1}(A))\cap Z^{-1}(g^{-1}(B))\big]=P\big[X^{-1}(f^{-1}(A))\big]~ P\big[Z^{-1}(g^{-1}(B))\big]~(since ~X,Z~ are ~independent)= P\big[ f(X)^{-1}(A)\big]~P\big[g(Z)^{-1}(B)\big]$$

for all $A,~B \subset \mathbb{R}^m$ Borel sets.