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Give an example showing that if $\mathcal{F}_1 \subset \mathcal{F}_2 \subset \cdots$ are $\sigma$-algebras then $\cup_i \mathcal{F}_i$ is not necessarily a $\sigma$-algebra.

$\cup_i \mathcal{F}_i$ is an algebra, just not necessarily a $\sigma$-algebra.

This means that any finite union of elements from the $\mathcal{F}_i$ should exist in $\cup_i \mathcal{F}_i$, but a countably infinite union of elements in any $\mathcal{F}_i$ may not necessarily exist in $\cup_i \mathcal{F}_i$. I'm having trouble thinking of such a counter example scenario.

clay
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  • Consider the unit interval $[0,1]$. Define $\mathcal{F}_n$ the algebra (which being finite will be also a $\sigma$-algebra) generated by the intervals ${[(k-1)2^{-n},k2^{-n}):1\leq k<2^n}\cup{[(2^n-1)2^{-n},1]}$. – Mittens Aug 30 '21 at 00:03
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    This is a duplicate but: I think the main obstacle here, besides the fact that infinite-arity operations (countable union in this case) are inherently weird, is that $\sigma$-algebras are really big and complicated objects. So a good first step is to just introduce a simplifying framework. One thing we know about $\sigma$-algebras is that they can be generated: given any collection $\mathcal{C}$ of sets there is a unique "$\sigma$-algebra generated by $\mathcal{C}$," which I'll call "$SA(\mathcal{C})$." (This feature is so common it's easy to overlook but it is super important and useful!) – Noah Schweber Aug 30 '21 at 00:05
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    It's easier to think and write in terms of generators than entire algebras. One key point is that these generated algebras are, by definition, "as small as possible" and so for example we should expect that if we pick a bunch of sets $A_1,A_2,...$ "generically" we'll never have $A_i\in SA({A_j:j\not=i})$. That specific hope is not germane to our current goal, but it's closely related to one that is: that if we pick sets $A_i$ ($i\in\mathbb{N}$) "generically," the infinite union $\bigcup_{i\in \mathbb{N}}A_i$ shouldn't be in $SA({A_i:i<n})$ for any fixed $n$. I think this simplifies things. – Noah Schweber Aug 30 '21 at 00:06

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