# Show that for every $\sigma$-algebra $\mathcal F$ exist sets…

Show that for every sigma-algebra $\mathcal F$ which consists of finite number of sets exist for some $n$ such sets $A_i \in F$, $i = 1,\dots, n$, that $A_i \cap A_j = \emptyset, i \neq j$, and all the elements of $\mathcal F$ can be presented as unions of sets $A_i, i = 1,\dots, n$.

Thanks for any help.

• Have you tried anything at all, by any chance? – Did Feb 18 '13 at 20:10

Starting with the whole set $E$. Whenever you see a set $F$, split $E$ and $F$ into $F\cap E^C$, $E\cap F$ and $E\cap F^C$. Then, do the same thing for the splitted sets. After running through all sets(as there are finitely many of them), you will get the answer
Let $\Omega$ the set where the $\sigma$-algebra $\mathcal F$ is given. For each $x$, let $S_x$ be the intersection of all the elements of $\mathcal F$ containing $x$. Define the relation $x\sim y$ if and only if $x\in S_y$.