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The following is a remark from my textbook:

One might think that $\sigma(\mathcal{G})$ (a $\sigma$-algebra generated by $\mathcal{G}$) can be explicitly constructed for any given $\mathcal{G}$ (where $\mathcal{G} \subset \mathcal{P}(X)$ is a system of sets and $\mathcal{P}(X)$ is the power set) by adding to the family $\mathcal{G}$ all possible countable unions of its members and their complements: $$\mathcal{G}_{\sigma c} = \bigg\{ \bigcup_{n \in N} G_n, \bigg(\bigcap_{n \in N} G_n\bigg)^c : G_n \in \mathcal{G} \bigg\}. $$ But $\mathcal{G}_{\sigma c}$ is not necessarily a $\sigma$-algebra.

My thinking is if I let $\mathcal{G} = \{A,B\}$ then $A^c$ is not in $\mathcal{G}_{\sigma c}$ as only $A \cup B$, $(A\cap B)^c$ are in $\mathcal{G}_{\sigma c}$. Am I correct? or am i missing something?

HMPtwo
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    If $\mathcal G$ is countable, then the proposed family would also be countable, but there are no countable $\sigma$ algebras. And no, your example is not valid. $A^c$ is indeed in $\mathcal G_{\sigma c}$ In fact, if you take $\mathcal G$ to be finite, you won't be able to find a counterexample. – Andrew Jun 26 '23 at 20:15
  • I was not aware of this (there is no countable $\sigma$-algebra) – HMPtwo Jun 26 '23 at 20:19
  • First, I do not see why $A^c$ is in $\mathcal{G}$. Secondly, I can't find the result that says that there are no countable σ-algebras. @AndrewZhang – HMPtwo Jun 26 '23 at 21:20
  • In the notation of $\mathcal G_{\sigma c}$, take $n = 1$. And https://math.stackexchange.com/questions/320035/if-s-is-an-infinite-sigma-algebra-on-x-then-s-is-not-countable – Andrew Jun 26 '23 at 21:49
  • On a more practical note, the first thing just about everyone tries when they first start working with $\sigma$-fields is trying to explicitly write out the $\sigma$-field. After trying this approach for many hours, you will realize it simply does not work. Except for a few special cases such as the cylindrical $\sigma$-field, trying to explicitly write out a $\sigma$-field is a hopeless endeavor. – Andrew Jun 26 '23 at 21:55

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