Given the probability space $(\Omega, \mathcal{F}, P)$ any random variable $X\colon \Omega \to \mathbb{R}$ is $ \mathcal{F}$ measurable, 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 <</SPAN>#56#>$\sigma$ algebra generated by $X$ .