ss#Probability #MeasureTheory >[!def] Distribution >Let $(\Omega,\mathcal{F},\mathbb{P})$ be a Probability Space. Let $(X_{n})_{n\in\mathbb{N}}$ be a sequence of RVs and $X$ an RV s.t. $X_{n},X:\Omega\to \mathbb{R}$. We define the pushforward measure $X_{\#}P$ as the law or distribution of $X$ i.e. for any $B\in\mathcal{B}(\mathbb{R})$: $$\begin{align*} X_{\#}\mathbb{P}(B):=\mu(B)&=\mathbb{P}(X^{-1}(B))\\ &=\mathbb{P}(\{ \omega \in\Omega:X(\omega)\in B \})\\ &=\mathbb{P}(X \in B) \end{align*}$$Since $\mu$ is the distribution of a rv then $(\mathbb{R},\mathcal{B},\mu)$ is a valid probability space.

This makes some application of Extension Theorem

\begin{proof} Consider the probability space $(\mathbb{R},\mathcal{B},\mu)$ and let $X(\omega)=\omega$. Then $$\text{"Prob}(X\in B)\text{"}=\mu(\omega \in B)=\mu(B)$$ \end{proof}