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

Theorem

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

Let $f: \Omega \to \R$ be a function such that $\Img f$ is countable.

Then $f$ is a discrete random variable on $\struct {\Omega, \Sigma, \Pr}$.

Proof

By definition, $\map {f^{-1} } x \subseteq \Omega$.

But then $\map {f^{-1} } x \in \powerset \Omega$.

Hence the result.

$\blacksquare$

Sources

• 1986: Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction ... (previous) ... (next): $\S 2.1$: Probability mass functions: Exercise $2$