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I want to show that the $\sigma$-field generated by a random variable $X$ given by $\{\omega \in \Omega:X(\omega) \in D\}$ is indeed a sigma field. Here, $D \in B(\Omega_X)$, where $B $ denotes a Borel set.

Are the following arguments to show closure under complements and countable unions correct?

Let $\{\omega \in \Omega:X(\omega) \in D\}=X^{-1}(D)$.

Assume $A \in \{X^{-1}(D)|D \in B(\Omega_X)\}$. Then $A=X^{-1}(D)$, where $D \in B(\Omega_X)$. Also, $A^C=X^{-1}(D^C)$, where $D^C \in B(\Omega_X)$. Therefore, $A^C \in \{X^{-1}(D)|D \in B(\Omega_X)\}$.

Assume $A_1,A_2 ,... \in \{X^{-1}(D)|D \in B(\Omega_X)\}$. Then $A_j=X^{-1}(D_j)$, where $D_j \in B(\Omega_X)$ for all $j=1,2,...$. Also, $\cup^{\infty}_{j=1} A_j=X^{-1}(\cup^{\infty}_{j=1} D_j)$, where $\cup^{\infty}_{j=1} D_j \in B(\Omega_X)$. Therefore, $\cup^{\infty}_{j=1} A_j \in \{X^{-1}(D)|D \in B(\Omega_X)\}$.

Also, how can I show that $\Omega$ is contained in the $\sigma$-field?

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It's basically correct, though you might convince yourself that the used properties of the inverse function on subsets of the codomain are indeed valid.

To see $\Omega\in A$, just observe that $X^{-1}(\Omega_X)=\Omega$.

Your notation suggests that $\Omega_X$ is the codomain of the random variable $X:\Omega\to\Omega_X$, but usually $\Omega_X=\Bbb R$ is the default understanding (with its Borel sets as $\sigma$-algebra).

However, this argument works for any measurable space (=set equipped with a $\sigma$-algebra) and any function to it.

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  • $\begingroup$ Just to follow up, is it right to say that the Borel set of $\Omega_X$ is a $\sigma$-field because it contains $\Omega_X$ in addition to the other properties of a sigma field? $\endgroup$ Commented Sep 8, 2021 at 8:05
  • $\begingroup$ Sorry, but what was your intention when writing $\Omega_X$? $\endgroup$
    – Berci
    Commented Sep 8, 2021 at 8:11
  • $\begingroup$ Probably what you called $A$ should be denoted by $\Omega_X$ and write $B(\Bbb R)$ for the Borel sets of the real line. $\endgroup$
    – Berci
    Commented Sep 8, 2021 at 8:14

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