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Say that we observe random variable $X$, can I prove whether an event in $\sigma(X)$ (sigma field constructed from $X$) has occured or not. That is for each $x \in \mathcal{R}$ (real line) and each $\omega \in \Omega$ such that $X(\omega) = x,$ we can tell whether $\omega \in E$ or $\omega$ is not in $E$ for all $E \in \sigma(X)$. I must take for granted that for any $x \in \mathcal{R}, \{x\} \in \mathcal{B}$ $(\mathcal{R})$ (Borel field) and for any function $f$ and $g$ if $f^{-1} (A\cap B)$?

Do I have to state that an outcome $\omega \in \Omega$ where $\Omega \in \sigma(x)$ must have a combination of two singleton? For example $ (G;E;GE;...)\in \Omega$ and measurable function $X(\omega)$ for $\omega$ = GE; $X (\omega)=(A\cap B)$ ?

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What is $\mathcal R$? $\mathcal B(\mathcal R)$? –  Davide Giraudo Sep 18 '12 at 8:05
    
Oh thanks for the comments. I edited my post. I meant real line and borel field. –  Charles M Sep 18 '12 at 19:42
    
I don't understand: $x$ seems to be a real number at the end of the first line, but in the second it's a Borel set. –  Davide Giraudo Sep 18 '12 at 19:43
    
I clarified more my question. Hope this helps! –  Charles M Sep 18 '12 at 20:39

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Events in $\sigma(X)$ are exactly sets $A=X^{-1}(B)$ for $B$ in the Borel sigma-field $\mathcal B(\mathbb R)$. To determine whether $\omega\in A$ or not, check whether $X(\omega)\in B$ or not.

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thanks for the clarification...but does this takes into account the intersection of A and B? –  Charles M Sep 19 '12 at 2:48
    
The intersection of A and B?? B is a subset of R and A is a subset of Omega, how could they intersect? –  Did Sep 19 '12 at 5:16
    
yes but in my question I must take for granted the following fact: $f^{-1}(A \cap B)$ that's what confuses me. Thanks for all the help! –  Charles M Sep 19 '12 at 5:32

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