# Elementary Question on Proof of A Sigma Field

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?

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.
• 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? Commented Sep 8, 2021 at 8:05
• Sorry, but what was your intention when writing $\Omega_X$? Commented Sep 8, 2021 at 8:11
• Probably what you called $A$ should be denoted by $\Omega_X$ and write $B(\Bbb R)$ for the Borel sets of the real line. Commented Sep 8, 2021 at 8:14