How to solve this exercise?
If $A$ is a collection of $\sigma$-algebra. Show that $\bigcap A=\bigcap_{F\in A}F$ is a $\sigma$-algebra.
2026-05-14 08:42:26.1778748146
collection of sigma algebra
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It is clear that $\emptyset \in \bigcap A$ (?).
Given $E\in \bigcap A$, $E\in F$ for all $F\in A$ hence $E^c$ is in each $F$, i.e., $E^c \in \bigcap A$.
Given a sequence $(E_j) \in \bigcap A$, each $E_j$ is in every $F$, so the sequence is contained in all $F$ and hence $\bigcup E_j \in F \; \forall F\in A$, that's equivalent to say that $\bigcup E_i \in \bigcap A$...
So $\bigcap A$ is closed under complement and countable union.