Functions of Independent Random Variables are Independent

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

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

Then $\map g X$ and $\map h Y$ are independent random variables.

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

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

Applying the definition of independent random variables:

Hence $\map g X$ and $\map h Y$ are independent random variables.