Define $ \alpha(E)=\sup\{\mu(F)-\nu(F)\,|\,F\subset E~\text{and}~\nu(F)<\infty\}.$ where $\mu$ and $\nu$ are measures on some $\sigma$-algebra.
How do I show that $\alpha$ is countably additive?
I've already got one direction. That is $\sum \alpha(E_n)\geq \alpha(\cup E_n).$ I am however struggling with the other direction: showing that $\sum\alpha(E_n)\leq \alpha(\cup E_n)$.
thanks.