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?

• Can it not help if you define $\lambda_{+}\left(E\right):=\int f_{+}d\mu$, $\lambda_{-}\left(E\right):=\int f_{-}d\mu$ and finally $\lambda\left(E\right):=\lambda_{+}\left(E\right)-\lambda_{-}\left(E\right)$? Then $\lambda_{+}$and $\lambda_{-}$ are both measures and at least one of them is finite. Commented Mar 17, 2014 at 18:40
• Then how can I prove that these series cannot oscillate? Commented Mar 17, 2014 at 19:22
• Are the sets $E_i$ disjoint? Commented Mar 17, 2014 at 19:28

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.
• All clear except for the last passage. In which way the fact that $\ f$ is integrable does imply that the series cannot oscillate? Commented Mar 17, 2014 at 19:53
• $f$ integrable means that at least one of $\int f_{+}d\mu$ and $\int f_{-}d\mu$ is finite. Then $\lambda_{+}\left(E\right)\leq\lambda_{+}\left(X\right)=\int f_{+}d\mu<\infty$ or $\lambda_{+}\left(E\right)\leq\lambda_{-}\left(X\right)=\int f_{-}d\mu<\infty$. Commented Mar 17, 2014 at 19:57
• Why $\ \sum_{i=1}^{+\infty}\int_{E_i}f_+ d\mu$ and $\ \sum_{i=1}^{+\infty}\int_{E_i}f_- d\mu$ have to exist? Commented Mar 17, 2014 at 20:00
• Both are summations of nonnegative terms and both are allowed to diverge to $+\infty$. Such 'things' always exist. The only thing is that they cannot both diverge to $+\infty$ (that would contradict the integrability of $f$). Commented Mar 17, 2014 at 20:06