-1
$\begingroup$

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; $X (\omega)=(A\cap B)$ ?

  • 2
    What is $\mathcal R$? $\mathcal B(\mathcal R)$?2012-09-18
  • 0
    Oh thanks for the comments. I edited my post. I meant real line and borel field.2012-09-18
  • 0
    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.2012-09-18
  • 0
    I clarified more my question. Hope this helps!2012-09-18

1 Answers 1

0

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.

  • 0
    thanks for the clarification...but does this takes into account the intersection of A and B?2012-09-19
  • 0
    The *intersection* of A and B?? B is a subset of R and A is a subset of Omega, how could they intersect?2012-09-19
  • 0
    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!2012-09-19