Give an example for an increasing series of $\sigma$ algebras $$ \mathcal{A}_1\subset\mathcal{A}_2\subset\ldots $$ so that $$\bigcup_{i=1}^{\infty}\mathcal{A}_i $$ is no $\sigma$-algebra.
Could you pls give me a hint how to find such an example?
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Hint: If I remember correctly, almost any example works, as long as $\mathcal A_i \subsetneq \mathcal A_{i+1}$, to give a concrete example think on finite $\sigma$-algebras $\mathcal A_i$ on $\mathbb N$, such that $\bigcup \mathcal A_i$ contains all singletons.
martini
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Would $F_n:=\left{\left{1\right},\left{2\right},\ldots,\left{n\right}\right}$, and then $\sigma(F_n)$ be an appropriate example? – Oct 22 '13 at 15:36
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Exactly this was the example i was thinking of. – martini Oct 22 '13 at 17:10