> [!NOTE] Theorem > Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a [[Probability Space|probability space]]. Let $X$ be a [[Discrete random variables|discrete real-valued random variable]] on $(\Omega, \mathcal{F}, \mathbb{P}).$ Let $\mathbb{P}_{X}$ be the [[Probability Distribution of Real-Valued Random Variable|distribution]] of $X.$ Let $p_{X}$ denote the [[Probability Mass Function of Discrete Real-Valued Random Variable|probability mass function]] of $X.$ Let $D_{X}$ be the [[Discrete Support of Distribution of Discrete Real-Valued Random Variable|support]] of $\mathbb{P}_{X}.$ Then for all $B\in \mathcal{B}(\mathbb{R}),$ $\mathbb{P}_{X}(B)= \sum_{x\in B \cap D_{X}} p_{X}(x) .$ > **Proof**: Let $B\in \mathcal{B}(\mathbb{R}).$ By definition of $X$ being discrete, there exists a [[Countable Set|countable set]] $S \subset \mathbb{R}$ such that $\mathbb{P}_{X}(\mathbb{R}\setminus S)=0.$ Since $\mathbb{P}_{X}$ is a probability measure, it is countably additive and so $\mathbb{P}_{X}(B)= \mathbb{P}_{X}(B \cap S) + \mathbb{P}(B \cap (\mathbb{R}\setminus S)).$The second term in zero since $B \cap (\mathbb{R}\setminus S)\subset \mathbb{R}\setminus S$ and by [[Probability of Subset of Event is Less Than or Equal to Probability of Event]] $0\leq \mathbb{P}_{X}(B \cap (\mathbb{R}\setminus S))\leq \mathbb{P}_{X}(\mathbb{R}\setminus S)=0.$ Hence, $\mathbb{P}_{X}(B)=\mathbb{P}_{X}(B \cap S)=\mathbb{P}_{X} ( \cup_{x\in B \cap S} \{ x \})= \sum_{x\in B \cap S} \mathbb{P}_{X}(\{ x \})$since $B \cap S$ is countable by [[Intersection with Countable Set is Countable]], and again $\mathbb{P}_{X}$ is countably additive. Now $\mathbb{P}_{X}(\mathbb{R}\setminus D_{X})=\sum _{x\in (\mathbb{R}\setminus D_{X})\cap S} \mathbb{P}(\{ x \})=0$since $\mathbb{P}_{X}(\{ x \})=p_{X}(x)=0$ for all $x\in \mathbb{R}\setminus D_{X}.$ Thus by the same argument for $S,$ $\mathbb{P}_{X}(B)=\sum_{x\in B \cap D_{X}} p_{X}(x) .$