$\sigma$-algebra generated by a random variableMathcalFMeasurableFunction2006-03-24 15:02:182009-02-06 18:47:32Definition% this is the default PlanetMath preamble. as your knowledge
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% define commands hereGiven the probability space $(\Omega, \mathcal{F}, P)$, any random variable $X\colon \Omega \to \mathbb{R}$ is \emph{$ \mathcal{F}$-} \PMlinkname{measurable}{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:
\begin{itemize}
\item $\varnothing = X^{-1}(\varnothing)\in \mathcal{F}_{X}$,
\item $\Omega-X^{-1}(B) = X^{-1}(\mathbb{R} - B)\in \mathcal{F}_{X}$, and
\item $\bigcup X^{-1}(B_i) = X^{-1}(\bigcup B_i)\in \mathcal{F}_{X}$.
\end{itemize}
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 \emph{$\sigma$-algebra \PMlinkescapetext{generated by} $X$}.