Function of Discrete Random Variable

Theorem

Let $X$ be a discrete random variable on the probability space $\struct {\Omega, \Sigma, \Pr}$.

Let $g: \R \to \R$ be any real function.

Then $Y = g \sqbrk X$, defined as:

$\forall \omega \in \Omega: \map Y \omega = g \sqbrk {\map X \omega}$

As $\Img X$ is countable, then so is $\Img {g \sqbrk X}$.

Now consider $g^{-1} \sqbrk Y$.

We have that:

$\forall x \in \R: \map {X^{-1} } x \in \Sigma$

We also have that:

$\ds \forall y \in \R: \map {g^{-1} } y = \bigcup_{x: \map g x = y} \set x$

But $\Sigma$ is a sigma-algebra and therefore closed for unions.

Thus:

$\forall y \in \R: \map {X^{-1} } {\map {g^{-1} } y} \in \Sigma$

Hence the result.

$\blacksquare$