Let $(\Omega,\mathscr{F},P)$ be a probability space, let $(\mathscr{X},\mathscr{B})$ be a measurable space, and let $X:\Omega\to\mathscr{X}$ be a measurable function, which means that $X^{-1}(B)=\{\omega:X(\omega)\in B\}\in\mathscr{F}$ for every $B\in\mathscr{B}$. The distribution of $X$ is the probability measure $\mu_X$ over $(\mathscr{X},\mathscr{B})$ defined by $\mu_X(B)=P(X\in B)$. When $\mathscr{X}=\mathbb{R}$ and $\mathscr{B}$ is the Borel sigma-field, we say that the function $X$ is a random "variable".

Let $(\Omega,\mathscr{F},P)$ be a probability space, let $(\mathscr{X},\mathscr{B})$ be a measurable space, and let $X:\Omega\to\mathscr{X}$ be a measurable function, which means that $X^{-1}(B)=\{\omega:X(\omega)\in B\}\in\mathscr{F}$ for every $B\in\mathscr{B}$. The distribution of $X$ is the probability measure $\mu_X$ over $(\mathscr{X},\mathscr{B})$ defined by $\mu_X(B)=P(X\in B)$. When $\mathscr{X}=\mathbb{R}$ and $\mathscr{B}$ is the Borel sigma-field, we refer to the function $X$ as a random "variable".