Let $(X,\mathcal{M},\mu)$ be a measure space and let $f:X \to [-\infty,\infty]$ be an extended $\mu$-integrable function (i.e. $f$ is measurable and at least one of $\int f^{+} \,d\mu$ and $\int f^{-} \,d\mu$ is finite). In chapter 3 of Folland's Real Analysis, it is claimed that the set function $\nu: \mathcal{M} \to [-\infty,\infty]$ defined by$$ \nu(E) := \int_E f \,d\mu$$is a signed measure, which Folland defines as follows:
Definition. Let $(X,\mathcal{M})$ be a measurable space. A signed measure on $(X,\mathcal{M})$ is a function $\nu: \mathcal{M} \to [-\infty,\infty]$ such that:
- $\nu(\emptyset) = 0$;
- $\nu$ assumes at most one of the values $\pm \infty$;
- If $\{E_j\}_{j=1}^{\infty}$ is a sequence of disjoint sets in $\mathcal{M}$, then $\nu \left( \bigcup_{j=1}^{\infty} E_j \right) = \sum_{j=1}^{\infty} \nu(E_j)$, where the latter sum converges absolutely if $\nu \left( \bigcup_{j=1}^{\infty} E_j \right)$ is finite.
I was able to verify (1), (2), and the first part of (3), but I am having trouble with the absolute convergence criterion. Namely, if $\{E_j\}_{j=1}^{\infty} \subset \mathcal{M}$ are disjoint and $\nu \left( \bigcup_{j=1}^{\infty} E_j \right) < \infty$, how can I show that $ \sum_{j=1}^{\infty} |\nu(E_j)| < \infty$? It seems to me that this criterion could fail: for example, if $f$ is a nonnegative function such that $\int_{\cup_{j=1}^{\infty} E_j} f \,d\mu = + \infty$, then we would have\begin{align*} \sum_{j=1}^{\infty} |\nu(E_j)| = \sum_{j=1}^{\infty} \left| \int_{E_j} f \,d\mu \right| = \sum_{j=1}^{\infty} \int_{E_j} f \,d\mu = \int_{\cup_{j=1}^{\infty} E_j} f \,d\mu = \infty. \end{align*}
Related questions: