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?
Best Answer
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 $\sigma(X) = \sigma\left(\left\{ X^{-1}(\upsilon) \colon \upsilon \in \Upsilon \right\}\right)$.