$\sigma$-algebra generated by a random variable

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

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

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

We now define ${ℱ}_X$ as follows:

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

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

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

• $\mathrm{\Omega }-X^{-1}(B)=X^{-1}(ℝ-B)\in {ℱ}_X$, and

• $\bigcup X^{-1}(B_i)=X^{-1}(\bigcup B_i)\in {ℱ}_X$.

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