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I am working on a problem to determine the measurability of a random variable.

Define $X : \Omega \rightarrow \mathbb{R}$ with $\Omega = \{\omega_i : i= 1, \ldots, 5\}$.

where $X\{\omega_1\}=0, X\{\omega_2\}=-1, X\{\omega_3\}=1, X\{\omega_4\}=2, X\{\omega_5\}=-2$. The textbook define $\mathcal{F}$ and $\mathcal{G}$ be the smallest $\sigma$-algebra for $X$ and $|X|$.

I need to determine whether X is a $\mathcal{G}$-measurable RV and whether X is a $\mathcal{F}$-measurable RV.

My think: $\mathcal{G}$ contains only information of absolute value of $X(\omega)$, so by giving $X(\omega) = 1$, we can't determine whether it is coming from $\omega_2$ or $\omega_3$. So X is not a $\mathcal{G}$-measurable RV.

In a similar idea, $X$ has more information needed to determine the value of $|X|$, so $|X|$ is $\mathcal{F}$-measurable RV.

I'm not sure if the idea is right or vice versa, and how to write that in mathematical language. From this post, I want to list all the conditions, but it feels too simple.

BlackTea
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    ${\omega_3}=X^{-1}(1)$ does not belong to $\mathcal G$ but it belongs to $\mathcal F$. – geetha290krm Jan 27 '24 at 06:10
  • Is $X:\Omega\to\mathbb R$ the inclusion function? – drhab Jan 27 '24 at 08:23
  • You provide values of the $\omega_i$ but I would rather expect values of the $X(\omega_i)$. – drhab Jan 27 '24 at 08:29
  • @geetha290krm Thank you! How can I prove the existence of $\mathcal{G}$? Should I list all the possible? – BlackTea Jan 27 '24 at 13:59
  • @drhab Yes, that's my mistake. – BlackTea Jan 27 '24 at 14:01
  • $|X|$ induces on $\Omega$ the following partition: ${{\omega_1},{\omega_2,\omega_3},{\omega_4,\omega_5}}$. The elements of $\mathcal G$ are the sets that can be written as a union of sets in this partition (there are $2^3=8$). For $X$ we have similarly the finer partition consisting of all singletons. – drhab Jan 27 '24 at 16:52

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