This is the question 8.K of Bartle's book:
Let $(X,\mathcal{X})$ be a measurable space. Let $\mu$ a finite measure, let $\lambda< <\mu,$ and let $P_{n},N_{n}$ be a Hahn's decomposition for $\lambda=n\mu.$ Let $P=\bigcap P_{n},N=\bigcup N_{n}$. Show that $N$ is $\sigma$-finite for $\lambda$, and that if $E\subset P$, $E\in\mathcal{X},$ then either $\lambda(E)=0$ or $\lambda(E)=+\infty.$
The first part was straightforward, but I'm stucked on the second part.
I did:
$\lambda(E)=n\mu(E)\geq\mu(E)\geq 0$. If $\mu(E)=0, \lambda(E)=n\mu(E)=0$ for all $n\in\mathbb{N}$, so $\lambda(E)=0$ is a possibility. Now, I need to prove that, if $\lambda(E)>0$, so $\lambda(E)=+\infty$
Maybe I can use the $\lambda < < \mu$ condition to apply the Radon-Nikodym. So, exists a function $f$ such that $\lambda(F)=\int_{F}f d\mu$ for all $F\in\mathcal{X}$. So,
$$\lambda(E)=\int_{E}fd\mu\leq\int_{P}fd\mu=\int_{\bigcap P_{n}}fd\mu\leq\int_{P_n}fd\mu, $$ but this way leads me no where. What can I do?
You may have copied the wrong question. The exercise says $\lambda-n\mu$ instead of $\lambda=n\mu$.
With such, if $\mu(E)=0$, then $\lambda(E)=0$ because of $\lambda<<\mu$. If $\lambda(E)\ne 0$, then $\mu(E)>0$, but we know that $\lambda-n\mu(E)\geq 0$ for each $n=1,2,...$ and hence $lambda(E)/\mu(E)\geq n$, taking limit $n\rightarrow\infty$ and use the assumption that $\mu(E)<\infty$ to deduce that $\lambda(E)=\infty$.