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