> [!NOTE] Theorem > Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a [[Probability Space|probability space]]. Let $X$ be a [[Random Variables|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 $X.$ Then $\mathbb{P}_{X}$ is a [[Probability Measure|probability measure]] on $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$: that is, $(\mathbb{R}, \mathcal{B}(\mathbb{R}),\mathbb{P}_{X})$ is a probability space. > **Proof**: We check $\mathbb{P}_{X}$ satisfies Kolmogorov's axioms: (1) $\mathbb{P}_{X}(B)=\mathbb{P}(\{\omega: X(\omega)\in B \})\in [0,1].$ (2) $\mathbb{P}_{X}(\mathbb{R})=\mathbb{P}(\{ \omega\in \Omega : X(\omega) \in \mathbb{R} \}) = \mathbb{P}(\Omega)=1.$ (3) Let $B_{1},B_{2},\dots \in \mathcal{B}(\mathbb{R})$ be pairwise disjoint. Then $\begin{aligned} \mathbb{P}_X(\cup_{n=1}^{\infty}B_n)& =\mathbb{P}(\{\omega:X(\omega)\in\cup_{n=1}^{\infty}B_n\}) \\ &=\mathbb{P}(\cup_{n=1}^\infty\{\omega:X(\omega)\in B_n\}) \\ &=\sum_{n=1}^{\infty}\mathbb{P}(\{\omega:X(\omega)\in B_{n}\}) \\ &=\sum_{n=1}^\infty\mathbb{P}_X(B_n). \end{aligned}$