Countable Function on Power Set of Sample Space is Discrete Random Variable

Theorem

Let $\left({\Omega, \Sigma, \Pr}\right)$ be a probability space such that $\Sigma$ is the power set of $\Omega$.

Let $f: \Omega \to \R$ be a function such that $\operatorname{Im} \left({f}\right)$ is countable.

Then $f$ is a discrete random variable on $\left({\Omega, \Sigma, \Pr}\right)$.

Proof

By definition, $f^{-1} \left({x}\right) \subseteq \Omega$.

But then $f^{-1} \left({x}\right) \in \mathcal P \left({\Omega}\right)$.

Hence the result.

$\blacksquare$

Sources

• Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction (1986): $\S 2.1$: Exercise $2$