Function of Discrete Random Variable

Theorem
Let $$X$$ be a discrete random variable on the probability space $$\left({\Omega, \Sigma, \Pr}\right)$$.

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

Then $$Y = g \left({X}\right)$$, defined as:
 * $$\forall \omega \in \Omega: Y \left({\omega}\right) = g \left({X \left({\omega}\right)}\right)$$

is also a discrete random variable.

Proof
As $$\operatorname{Im} \left({X}\right)$$ is countable, then so is $$\operatorname{Im} \left({g \left({X}\right)}\right)$$.

Now consider $$g^{-1} \left({Y}\right)$$.

We have that $$\forall x \in \R: X^{-1} \left({x}\right) \in \Sigma$$.

We also have that $$\forall y \in \R: g^{-1} \left({y}\right) = \bigcup_{x: g \left({x}\right) = y} \left\{{x}\right\}$$.

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

Thus $$\forall y \in \R: X^{-1} \left({g^{-1} \left({y}\right)}\right) \in \Sigma$$.

Hence the result.