I'm trying to solve this Hann decomposition problem, it is as follows:
Let $\mu$ a finite measure,such that $\lambda \ll \mu$ and let $P_n, N_n$ Hann decomposition for $\lambda - n\mu$. Let $P = \bigcap_{n}P_n$, $N = \bigcup_{n}N_n $. Show that N is $\sigma$ - finite for $\lambda$ and that if $E \subset P$. $E \in \mathcal{A}$ then $\lambda(E) = 0$ or $\lambda(E)= \infty $
STEPS:
Well firs I try to prove that N is $\sigma$ - finite for $\lambda$, since N is equal to $\bigcup_{n}N_n$ then it would be enough to show that $\lambda(N_n) < \infty $, but each $N_n$ it's a Hann decomposition for $\lambda - n\mu$ then $\lambda(N_n) - n\mu(N_n) \leq 0$ so $\lambda(N_n) \leq n\mu(N_n)$ also $\mu$ a finite measure then $\lambda(N_n) < \infty$ this prove that N is $\sigma$ - finite for $\lambda$, on the other hand now suppose $E \in \mathcal{A}$ and $E \subset P$, since P is equal to $\bigcap_{n}P_n$ then for each "n", E $\subset P_n$ then $\lambda(E) - n\mu(E) \geq 0$ for each "n" then if $\mu(E) = 0$ since $\lambda \ll \mu$ then $\lambda(E) = 0$ on the other hand if $0 <\mu(E) < \infty$ then as it is for every $n$ natural $\lambda(E) = + \infty $ but I don't know how to conclude if it is negative, Any suggestions and would my idea be fine or am I wrong about something? thanks for your attention