how to find ALL $\sigma$-algebras of a given sample space?

Question

I have a sample space $\Omega$={$\omega_{1}$,$\omega_{2}$,$\omega_{3}$,$\omega_{4}$} and I need to find ALL $\sigma$-algebra on $\Omega$. I know how to construct some $\sigma$-algebra like $\mathscr{A}$={$\phi$,$\Omega$,$\omega_{1}$,{$\omega_{2}$,$\omega_{3}$,$\omega_{4}$}} but I have no idea about how to find all of them.

Answer 1

Take the collection of all partitions of $\Omega$ and form the $\sigma$-algebra for each. The number of such partitions is called $B_n$, the Bell number (from Wiki, the source of all truth). $B_4 = 15$.

To see that a $\sigma$-algebra ${\cal A}$ corresponds to a partition, choose $\omega \in \Omega$ and let $A_\omega$ be the intersection of all elements $\cal A$ containing $\omega$. Define the relation $\sim$ by $\omega_1 \sim \omega_2$ iff $A_{\omega_1} = A_{\omega_2}$. This is an equivalence relation, hence defines a partition. Then $\cal A$ is the $\sigma$-algebra generated by the partition.