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We have $(\Omega, \mathcal F, P)$, some probability space. Let $\mathcal F_1$ - some sub-algebra of $\mathcal F$ and $\forall n$ define $\mathcal F_{n+1}$ as class of sets, which received by countable intersecting or countable union from $\mathcal F_n$. How to prove, that $\cup_{n \in \mathbb N} \mathcal F_n$ mustn't be even $\sigma$-algebra? Does it exists some counterexample?

user157895564
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Ok so I can give you an example to this. Assuming that you already aware of this question which says that there exists a Borel set which is not obtained by finitely many applications of intersections/unions of closed or open sets.

Let $X$ be such a space. We let $F_1$ be the set of open and closed sets of $X$. This is an algebra.

Let $F$ be the union of all $F_n$, if by contradiction $F$ is an $\sigma$-algebra it must contain all Borel sets (because the Borel is the minimal $\sigma$-algebra which contains all open sets).

But if you believe to the answer in the linked question, there exists a Borel measurable set that is not a obtained from an open or closed sets by finitely many applications of intersection or union. Hence for this $X$ we have a counter-example.

Yanko
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