# Definition:Random Variable/Definition 3

## Definition

Let $\EE$ be an experiment with a probability space $\struct {\Omega, \Sigma, \Pr}$.

A random variable on $\struct {\Omega, \Sigma, \Pr}$ is a mapping $X: \Omega \to \R$ such that:

$\forall x \in \R: X^{-1} \sqbrk {\hointl {-\infty} x} \in \Sigma$

where:

$\hointl {-\infty} x$ denotes the unbounded closed interval $\set {y \in \R: y \le x}$
$X^{-1} \sqbrk {\hointl {-\infty} x}$ denotes the preimage of $\hointl {-\infty} x$ under $X$.