Let $(\mu_n)_{n\in \Bbb N}$ and $\nu$ be $\sigma$-finite measures on $(\Omega , \mathcal A)$ and $(a_n)_{n\in \Bbb N}$ such that $\mu := \sum_{n\le 1} a_n \mu_n $ is a signed measure. Further, there exists for each $n \in \Bbb N$ a set $N_n \in \mathcal A$ so that $\mu _n (N_n) = 0$ and $\nu (\overline N_n) = 0$ (in other words $\mu _n \perp \nu$). Now I need to show that there also exists a set $N \in \mathcal A$ so that $\mu (N) = 0$ and $\nu (\overline N) = 0$. ($\mu \perp \nu$)
I tried finding such a $N$ and found $\cap_{n \in \Bbb N} N_n$ (because i can for $\nu(\overline N)$ form $\nu(\cup \overline N_n) \le \sum \nu(\overline N_n)$ with each part of the sum beeing zero and for $\mu(N)$ say that each $\mu_n(\cap N_n) \le \mu_n(N_n) = 0$ )
What troubles me now is that I didn't use the $\sigma$-finite property of the measures and also not that $\mu$ is a signed measure, so I think I did something wrong.
Maybe someone can either see my fault, or tell me where I unknowingly used those properties.
Thanks in advance!