Are there any concrete examples of $\sigma$-algebra generated by a random variable?

Question

I have searched around the internet for any concrete example of $\sigma$ algebra generated by a random variable $X$ but failed to find any nontrivial, concrete examples.

For example,

https://stats.stackexchange.com/questions/312474/what-is-it-meant-with-the-sigma-algebra-generated-by-a-random-variable

Consider a single tosses of a fair coin,

Then the sample space is $\Omega = \{\{H\}, \{T\}\}$, with $\sigma$-algebra on $\Omega$ equal to$ \mathcal{F} = \{\{\varnothing\}, \{H\}, \{T\}, \{H,T\}\}$

We can define a random variable $X$, where $X(H) = 1$, $X(T) = -1$

Now I ask, what is the $\sigma$-algebra generated by $X$?

By definition,

$\sigma$-algebra generated by $X$, denoted $\sigma(X)$ is the collection of sets $\sigma(X) = \{\{\omega \in \Omega, X(\omega) \in B\}: B \in \mathcal{B}\}$, where $\mathcal{B}$ is the Borel set.

(Is my definition correct?)

Let's pick some sets,

$B = \{1\}, \omega = \{H\} \in \Omega$

$B = \{-1\}, \omega = \{T\} \in \Omega$

For $B$ excluding both of these singletons, $\omega = \varnothing \in \Omega$, and for $B$ including both of these singletons, $\omega = \Omega \in \Omega$

So the $\sigma$-algebra generated by $X$ is the same as generated by the subsets of $\Omega$.

Am I right in my reasoning?

What is a concrete example where $\sigma$-algebra generated by $X$ is not the same as that generated by the subsets of $\Omega$?

Answer 1

For example, you might try $\Omega = \{-2, -1, 0, 1, 2\}$ and $X(\omega) = \omega^2$. Note that $X$ takes the same value on $\omega$ and $-\omega$, so any member of the $\sigma$-algebra it generates contains either both $\omega$ and $-\omega$ or neither of them.

Answer 2

For any $A \subset \Omega$, $\{\emptyset, A,A^{c}, \Omega\}$ is a sigma algebra and it is generated by $X=I_A$.