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I know the Sigma Algebra generated by a random variable $X: (\Omega, \mathcal{F}) \to (I, \mathcal{A})$, can be defined as $\{ Y \in \mathcal{F} \mid \exists B \in \mathcal{A}(Y = X^{-1}(B)) \}$. However, I think my professor has somewhat implicitly mentioned that when I is a countable space, and $\mathcal{A} = P(I)$, the generated Sigma Algebra is $\sigma(\cup_{i \in \mathbb{N}} \{ X =i \})$, where $\{ X = i\}$ is all the s in $\Omega$ s.t $X(s) = i$

This is what I recalled of how he defined it as he never really write down the exact definition, and I'm not sure if it's even true or not. If it is, I cannot really see why would these two definition coincide, as the latter is not really considering all event. I guess it might be true in the case of I is a countable space, but I doubt it would be true in general. So are these definitions equal in this case, and in general?

patchouli
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1 Answers1

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I was a bit thrown by your use of $\Sigma$, so I'm using $\Upsilon$ (and the lower case $\upsilon$) instead.

Let $\Upsilon$ be a countable set, and let $\mathcal A = \mathcal P(\Upsilon)$ be the power set $\sigma$-algebra. If $S = \left\{ \{ \upsilon \} \colon \upsilon \in \Upsilon\right\} \subset \mathcal{P}(\Upsilon)$, then $\sigma(S) = \mathcal{A}$. In particular, $\mathcal{A}$ can be generated from $S$ via countable unions.

As you know, $\sigma(X) = \left\{ X^{-1}(A) \colon A \in \mathcal{A} \right\}$. Using certain properties of inverse images, we have

\begin{align*} X^{-1}(A) &= X^{-1}\left( \bigcup_{\upsilon \in A} \{ \upsilon \} \right) = \bigcup_{\upsilon \in A} X^{-1}\left( \{ \upsilon \} \right) \end{align*}

This implies that $\sigma(X) = \sigma\left(\left\{ X^{-1}(\upsilon) \colon \upsilon \in \Upsilon \right\}\right)$.

Novice
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  • Thank you for your answer, I have upvoted it. I understand the bulk, but before accepting it, would you mind helping me clarify a few details? (1) What is {v}? I am new to this, and I saw many places where people just surrounds event with curly braces like X = v to {X = v} what's the point of doing so? (2) Does this apply in general when it's not countable, and the event space is not the power set? – patchouli Feb 10 '23 at 16:32
  • (1) ${ \upsilon }$ (lower case upsilon; just a Greek letter I chose) is the singleton set containing upsilon. It's just a set containing one element. Strictly speaking, events are sets, so something like $P(X > 1)$ is more properly written $P({X > 1})$, but the former is just shorthand. (2) This does not work in general. If we work with the identity function $X \colon \mathbb R \to \mathbb R$, using the Borel sigma algebra on both sides, then $\sigma(X)$ is the Borel sigma algebra, but the sigma algebra generated by preimages of singletons is the countable-cocountable sigma algebra. – Novice Feb 10 '23 at 17:00
  • @wsz_fantasy I ran out of room to tag you in my reply above. – Novice Feb 10 '23 at 17:02
  • Thank you. I'll accept it right now. One quick thing do you mean $X^{-1}({v})$ in the last line? – patchouli Feb 10 '23 at 17:09
  • @wsz_fantasy Yes, I think I goofed there – Novice Feb 10 '23 at 17:10
  • I see. Btw, I was originally wondering about the Sigma Algebra generated by multiple random variables. I assume it is the sigma field generated by the union of the sigma field generated by each random variable. Can it also be simplified into something in this case? – patchouli Feb 10 '23 at 17:22
  • I am not quite sure what you mean, and I think site guidelines would encourage you to ask a separate question, if you have one. – Novice Feb 10 '23 at 19:42
  • I have asked the new question here – patchouli Feb 12 '23 at 02:21