Let $\Omega$ be a set, $\mathcal B$ is a semi-algebra that contains $\Omega$, and let $\mu \colon \mathcal B\rightarrow [0,\infty]$ be a measure defined on $(\Omega,B)$. Now define the algebra $\mathcal A$ which contains all the finite (disjoint) unions of sets from $\mathcal B$: $\mathcal A=\left\{E=E_1 \cup E_2 \cup\cdots\cup E_n: E_1,E_2,\cdots,E_n\in \mathcal B, E_i\cap E_j=\varnothing \right\}.$ Now define $\nu\colon\mathcal A\rightarrow [0,\infty]$ as $\nu (E)=\sum_{k=1}^n \mu (E_k)$. I shall prove that $\nu$ is a measure on $\mathcal A$.
I'm having problems with showing the $\sigma$-additivity property. How do I show that if $Q_1,Q_2,\ldots\in\mathcal A$ and $\bigcup_{n=1}^\infty Q_n \in \mathcal A$ then $\nu\left(\bigcup_{n=1}^\infty Q_n\right)=\sum_n \nu(Q_n)$?