Definition:Random Variable/Real-Valued/Definition 1

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Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

A real-valued random variable on $\struct {\Omega, \Sigma, \Pr}$ is a $\Sigma$-measurable function $X : \Omega \to \R$.

That is, a function $X : \Omega \to \R$ is a real-valued random variable if and only if:

$X^{-1} \sqbrk {\hointl {-\infty} x} = \set {\omega \in \Omega : \map X \omega \le x} \in \Sigma$

for each $x \in \R$, 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 $X$.