How to prove that if $A$ is algebra then $\mu(A)=\sigma(A)$?

Question

Prove that if $A$ is algebra, then $\mu(A)=\sigma(A)\qquad$ $(\mu(A)\quad$is monotone class, generateg by A$)$.

Answer 1

One direction is easy. The other direction takes a bit more effort. Hint:

Set $\mathcal G = \{E \in \mathcal M(\mathcal A) |\, \forall F \in \mathcal M(\mathcal A): E\cap F, E\backslash F, F\backslash E \in \mathcal M(\mathcal A) \}$. Your first aim is to show that $\mathcal G$ is a monotone class. Have fun.

$\mathcal M(\mathcal A)$ is the monotone class generated by $\mathcal A$. Don't use $\mu$!!