Signed measure defined by an integral Let $ (X,\mathcal{M},\mu)$ be a measure space and let $ f:X\to[-\infty,+\infty]$ be an integrable function (i.e. at least one of $ f_+ $ and $ f_-$ is integrable).
I want to prove that $ \lambda:\mathcal{M}\to[-\infty,+\infty]$ defined as
$$ \lambda(E)=\int_{E}f d\mu $$
is a signed measure.
When I prove the countable additivity, how can I prove that the series $$ \sum_{i=1}^{+\infty}\int_{E_i}f d\mu$$ may only converge absolutely or diverge?
 A: Define for every $E\in\mathcal M$:
$\lambda_{+}\left(E\right):=\int_{E}f_{+}d\mu$, $\quad\lambda_{-}\left(E\right):=\int_{E}f_{-}d\mu$
and finally $\lambda\left(E\right):=\lambda_{+}\left(E\right)-\lambda_{-}\left(E\right)$.
Let $E=\cup_{i}E_{i}$ where the $E_{i}$ are measurable and disjoint.
Then $\sum_{i}\left|\lambda\left(E_{i}\right)\right|=\sum_{i}\left[\lambda_{+}\left(E_{i}\right)+\lambda_{-}\left(E_{i}\right)\right]=\lambda_{+}\left(E\right)+\lambda_{-}\left(E\right)$.
If $\lambda_{+}\left(E\right)+\lambda_{-}\left(E\right)<\infty$ then
the sequence converges absolutely. If $\lambda_{+}\left(E\right)+\lambda_{-}\left(E\right)=\infty$
then $\lambda_{+}\left(E\right)=\infty\wedge\lambda_{-}\left(E\right)<\infty$
or $\lambda_{+}\left(E\right)<\infty\wedge\lambda_{-}\left(E\right)=\infty$.
In both cases $\sum_{i}\lambda\left(E_{i}\right)$ diverges. In the
first case to $+\infty$ and in the second to $-\infty$. The option
$\lambda_{+}\left(E\right)=\infty=\lambda_{-}\left(E\right)$  is not
there since it contradicts that $f$ is integrable. 
