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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.

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

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  • $\begingroup$ Thank you, but could you explain it a little more precisely? I just started studying probability theory and it's very difficult subject for me. $\endgroup$ – user62136 Feb 18 '13 at 20:39
  • $\begingroup$ OK. First, by definition of sigma-algebra, the empty set is in it, and the complement of empty set(E) is in it, and there are some other subsets. The goal is to partition E into small sets so that every other sets can be expressed as a union of them. So we start with E, and split it by the other sets. And then split the splitted sets by some other sets and so on. Finally, we will end up with the smallest (no more splittable) set that we can have. Then, every set can be written in term of them. $\endgroup$ – NECing Feb 18 '13 at 20:52
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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 $x\in S_y$.

  • Show that it's an equivalence relation.
  • What about the equivalence classes?
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