| Version 4 |
Version 3 |
| Let $(\Omega,\mathcal{F}, P)$ be a probability space then a function $X\colon \Omega \to \mathbb{R}$ |
Let $(\Omega,\mathcal{F}, P)$ be a probability space then a function $X\colon \Omega \to \mathbb{R}$ |
| is called \emph{$ \mathcal{F}$- measurable } if |
is called \emph{$ \mathcal{F}$- measurable } if |
| $$X^{-1}(U)=\{\omega\in \Omega \colon X(\omega)\in U\} \in \mathcal{F}$$ |
$$X^{-1}(U)=\{\omega\in \Omega \colon X(\omega)\in U\} \in \mathcal{F}$$ |
| for all open sets $U \in \mathbb{R}$, equivalently for all Borel sets $U\subset \mathbb{R}$. |
for all open sets $U \in \mathbb{R}$, equivalently for all Borel sets $U\subset \mathbb{R}$. |
| \\We can define now the smallest $\sigma$-algebra $\mathcal{F}_{X}$ generated by $X$ as following$\colon$ |
\\We can define now the smallest $\sigma$-algebra $\mathcal{F}_{X}$ generated by $X$ as following$\colon$ |
| $$\mathcal{F}_{X}=\{X^{-1}(B)\colon B\in \mathcal{B}\}$$ |
$$\mathcal{F}_{X}=\{X^{-1}(B)\colon B\in \mathcal{B}\}$$ |
| where $\mathcal{B}$ is the Borel $\sigma$-algebra on $\mathbb{R}$, since we have that $\mathcal{F}_{X}$ satisfies the following: |
where $\mathcal{B}$ is the Borel $\sigma$-algebra on $\mathbb{R}$ |
| \begin{itemize} |
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| \item $\varnothing=X^{-1}(\varnothing)\in \mathcal{F}_{X}$, |
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| \item $\Omega-X^{-1}(B)=X^{-1}(\mathbb{R} - B)\in \mathcal{F}_{X}$, and |
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| \item $\bigcup X^{-1}(B_i)=X^{-1}(\bigcup B_i)\in \mathcal{F}_{X}$. |
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| \end{itemize} |
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| $\mathcal{F}_{X}$ as defined above is called the \emph{sigma algebra generated by $X$}. |
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