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Version 14 |
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Given 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: $$X^{-1}(U)=\{\omega\in \Omega \colon X(\omega)\in U\} \in \mathcal{F}$$
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Let $(\Omega,\mathcal{F}, P)$ be a probability space. Any random variable $X\colon \Omega \to \mathbb{R}$ is \emph{$ \mathcal{F}$-} \PMlinkname{measurable}{MeasurableFunctions},\, in the following sense: $$X^{-1}(U)=\{\omega\in \Omega \colon X(\omega)\in U\} \in \mathcal{F}$$
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| for any open sets $U \subseteq \mathbb{R}$, or equivalently any Borel sets $U\subset \mathbb{R}$. |
for any open sets $U \subseteq \mathbb{R}$, or equivalently any Borel sets $U\subset \mathbb{R}$. |
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| We now define $\mathcal{F}_{X}$ as follows: |
We now define $\mathcal{F}_{X}$ as follows: |
| $$\mathcal{F}_{X}=X^{-1}(\mathcal{B}):=\{X^{-1}(B)\colon B\in \mathcal{B}\},$$ |
$$\mathcal{F}_{X}=X^{-1}(\mathcal{B}):=\{X^{-1}(B)\colon B\in \mathcal{B}\},$$ |
| 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: |
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} |
\begin{itemize} |
| \item $\varnothing=X^{-1}(\varnothing)\in \mathcal{F}_{X}$, |
\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 $\Omega-X^{-1}(B)=X^{-1}(\mathbb{R} - B)\in \mathcal{F}_{X}$, and |
| \item $\bigcup X^{-1}(B_i)=X^{-1}(\bigcup B_i)\in \mathcal{F}_{X}$. |
\item $\bigcup X^{-1}(B_i)=X^{-1}(\bigcup B_i)\in \mathcal{F}_{X}$. |
| \end{itemize} |
\end{itemize} |
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| 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} $X$}. |
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} $X$}. |
