Functions of Independent Random Variables are Independent

Theorem
Let $X$ and $Y$ be independent random variables on a probability space $\left({\Omega, \Sigma, \Pr}\right)$.

Let $g$ and $h$ be real-valued functions defined on the codomains of $X$ and $Y$ respectively.

Then $g \left({X}\right)$ and $h \left({Y}\right)$ are independent random variables.

Proof
Let $A$ and $B$ be subsets of the real numbers $\R$.

Let $g^{-1} \left[{A}\right]$ and $h^{-1} \left[{B}\right]$ denote the preimages of $A$ and $B$ under $g$ and $h$ respectively.

Applying the definition of independent random variables:

Hence, $g \left({X}\right)$ and $h \left({Y}\right)$ are independent random variables.

Note that we can expand this proof by replacing the word real with the word complex.