 Solution

A discrete random variable on a probability space $(\Omega ,\mathcal{F},\mathbb{P})$ is a real-valued measurable function $X:\Omega \to \mathbb{R}$ whose probability distribution is concentrated on some finite or countable set $S$:Measurability means $\{\omega :X(\omega )\le t\}\in \mathcal{F}$ for every real $t$. Its probability mass function is $p_X(x)=\mathbb{P}(X=x)$, and ${\sum }_{x\in S}{p}_X(x)=1$. Requiring a countable image everywhere, rather than outside a null set, is an equivalent convention after harmless modification on a null set.