# Definition:Random Variable/Real-Valued

## Definition

### Definition 1

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 $X$.

### Definition 2

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

Let $\map \BB \R$ be the Borel $\sigma$-algebra on $\R$.

Let $X$ be a random variable on $\struct {\Omega, \Sigma, \Pr}$ taking values in $\struct {\R, \map \BB \R}$.

Then we say that $X$ is a real-valued random variable on $\struct {\Omega, \Sigma, \Pr}$.

## Notation

As an abuse of notation, we may write:

$\set {\omega \in \Omega : \map X \omega \le x}$ as $\set {X \le x}$
$\set {\omega \in \Omega : \map X \omega \ge x}$ as $\set {X \ge x}$
$\set {\omega \in \Omega : \map X \omega < x}$ as $\set {X < x}$
$\set {\omega \in \Omega : \map X \omega > x}$ as $\set {X > x}$
$\set {\omega \in \Omega : \map X \omega = x}$ as $\set {X = x}$
$\set {\omega \in \Omega : \map X \omega \in A}$ as $\set {X \in A}$

Generally, we write:

$\set {\omega \in \Omega : \map P {\map X \omega} }$ as $\set {\map P X}$

for any propositional function of $\map X \omega$ such that:

$\set {\omega \in \Omega : \map P {\map X \omega} }$ is $\Sigma$-measurable.

We may therefore write, for example:

$\map \Pr {\set {\omega \in \Omega : \map X \omega = x} }$

as:

$\map \Pr {\set {X = x} }$

Usually the curly brackets are dropped and we write:

$\map \Pr {\set {\omega \in \Omega : \map X \omega = x} } = \map \Pr {X = x}$