# $\sigma$-algebra generated by a random variable

Given the probability space $(\Omega,\mathcal{F},P)$, any random variable $X\colon\Omega\to\mathbb{R}$ is $\mathcal{F}$- measurable (http://planetmath.org/MeasurableFunctions),  in the following sense:

 $X^{-1}(U)=\{\omega\in\Omega\colon X(\omega)\in U\}\in\mathcal{F}$

for any open sets $U\subseteq\mathbb{R}$, or equivalently any Borel sets $U\subset\mathbb{R}$.

We now define $\mathcal{F}_{X}$ as follows:

 $\mathcal{F}_{X}=X^{-1}(\mathcal{B}):=\{X^{-1}(B)\colon B\in\mathcal{B}\},$

where $\mathcal{B}$ is the Borel $\sigma$-algebra on $\mathbb{R}$. $\mathcal{F}_{X}$ is sometimes denoted as $\sigma(X)$. $\mathcal{F}_{X}$ is a sigma algebra since it satisfies the following:

• $\varnothing=X^{-1}(\varnothing)\in\mathcal{F}_{X}$,

• $\Omega-X^{-1}(B)=X^{-1}(\mathbb{R}-B)\in\mathcal{F}_{X}$, and

• $\bigcup X^{-1}(B_{i})=X^{-1}(\bigcup B_{i})\in\mathcal{F}_{X}$.

It is also clear that $\mathcal{F}_{X}$ is the smallest $\sigma$-algebra containing all sets of the form $X^{-1}(B)$, $B\in\mathcal{B}$. $\mathcal{F}_{X}$ as defined above is called the $\sigma$-algebra $X$.

Title $\sigma$-algebra generated by a random variable sigmaalgebraGeneratedByARandomVariable 2013-03-22 15:48:19 2013-03-22 15:48:19 PrimeFan (13766) PrimeFan (13766) 19 PrimeFan (13766) Definition msc 60A99 msc 60A10 SigmaAlgebra