Prove or disprove that every finite $\sigma$-algebra on $\Omega$ is generated by a finite partition of $\Omega$.
I have a feeling this must be true, but I could not do more than the following:
Being $\Sigma$ a finite $\sigma$-algebra on $\Omega$, I defined a set $S$ as the set of intersections of two arbitrary sets in $\Sigma$, but I don't know if there is a partition $P$ which is a subset of $S$ and if there is, if the $\sigma$-algebra generated by $P$ is $\Sigma$. Any help would be appreciated. Thanks.
And if the fact that every finite $\sigma$-algebra can be generated by a finite partition is false, what conditions do the finite $\sigma$-algebra need to have in order for it to be true?
Best Answer
Intersection of just two sets from the sigma algebra won't suffice. Let $\mathcal A =\{A_1,A_2,\cdots,A_n\}$ be a finite sigma algebra. Verify the following facts: sets of the form $B_1\cap B_2\cap\cdots \cap B_n$ where $B_i$ is either $A_i$ or $A_i^{c}$ for each $i$ are disjoint. Though many of them may be empty, their union is $\Omega$. Also prove that each $A_j$ is a union of those sets of the above form where $B_j=A_j$. Now you can easily verify that the partition we have constructed generates the sigma algebra $\mathcal A$.