Let $\mathcal{F}$ be a finite $\sigma$-algebra.
The problem asks to show there exists a partition $G = \{ G_1,\dots,G_n \}$ of $\Omega$ such that for all $A \in \mathcal{F}$, $A$ is the union of all or some $G_i$:
$$A = \bigcup_{i \in I} G_i$$
The existence of a partition is immediate from the definition of a $\sigma$-algebra, but I'm not sure how to use the fact $\mathcal{F}$ is finite to construct the generating set $G$. Specifically, to show that every member of $\mathcal{F}$ can be generated from a single partition using only union.
Can someone give me a hint?
Not a homework problem, I am working through a textbook for self-study.
Best Answer
Let $\mathcal F =\{B_1,B_2,\cdots ,B_n\}$. Consider all sets of the type $A_1\cap A_2\cap \cdots \cap A_n$ where $A_i$ is either $B_i$ or $B_i^{c}$ for each $i$. You get $2^{n}$ such intersection but many of them be just empty. It is now routine to check that the nonempty sets in this collection form a partition of $\Omega$ and that each $B_i$ is a union of some sets in this partition. For example, you can write $B_1$ as a union of such sets by fixing $A_1$ to be $B_1$ and varying $A_2,A_3,\cdots,A_n$.