# Sigma Algebra on a finite set and set of partitions on the same set X

I ran across this statement I am trying to prove, but I am sure if I am in the right direction.

Statement: Show there is a natural correspondence between the set of sigma algebras on a finite set $X$ and the set of partitions of $X$.

Proof: To me, it seems that I must show two things. Given a set of sigma algebras, how do each of these sigma algebras relate to the set of partitions of $X$, and given a set of partitions of $X$, how do these relate to the sigma algebras.

(Partition $\rightarrow$ sigma algebra) For each partition, it generates a given sigma algebra by taking the power set of the cells in the given partition of $X$. The element in this set guarantees that I am closed under the operation of complements and union (finite). Also $X$ lie sin the given sigma-algebra.

(Sigma Algebra $\rightarrow$ Partition) For a given sigma algebra, I want to find its generating set. The set where I take its powerset will give me the sigma algebra. If we examine the set of all partitions of $X$ and superimpose each partition of $X$ on each other, the resulting partition is the desired partitition which will give us the sigma algebra.

At least, I believe, this is one way I see it.

The first part looks fine: you’re letting the cells of the partition be the atoms of the Boolean algebra. I can’t quite tell what you’re trying to do in the other direction. You have a Boolean algebra $B$ on $X$; try reversing what you did for the first half. That is, try showing that the atoms (minimal non-zero elements) of $B$ form a partition of $X$. – Brian M. Scott Feb 10 '12 at 6:41