I have seen the following statement in $\pi-\lambda$ system form. But not in the monotone class form. I would just like some verification that this is correct.
Problem: $(\Omega, \Sigma)$ be a measurabla space. $\mathcal{A}$ be a Boolean algebra on $\Sigma$, i.e. closed under finite intersection, union, complement, contains $\emptyset$ and $\Omega$. Then if $\mu_1, \mu_2$ are two $\sigma$-finite measures on $\Omega$ that coincide on $\mathcal{A}$, then they also coincide on $\sigma(\mathcal{A})$.
Proof: without loss of generality, suppose both measures are finite. Let $D:= \{ E \in \Sigma : \mu_1(E)=\mu_2(E)\}$. Then $D$ is closed under increasing union and decreasing intersection. Thus, $D$ is a monotone class, contains $\mathcal{A}$. Hence, $\mu_1$ and $\mu_2$ coincide on $\mathcal{A}$.
Is this correct?