# Importance of $|\mu(A)|<\infty$ in complex measure?

In the book 'An Introductory Course in Functional Analysis' by Bowers and Kalton, they give the definition of positive measure at page $209:$

Let $(X,\mathcal{A})$ be a measurable space, A set function $\mu:\mathcal{A} \rightarrow [0,\infty]$ is said to be countably additive if $$\mu\left( \bigcup_{j=1}^\infty A_j \right) = \sum_{j=1}^\infty \mu(A_j)$$ whenever $(A_j)_{j=1}^\infty$ is a sequence of pairwise disjoint measurable sets. A countably additive set function $\mu:\mathcal{A} \to [0,\infty]$ such that $\mu({\emptyset})=0$ is called a positive measure.

The authos give definition of complex measure at page $213:$

Let $(X,\mathcal{A})$ be a measurable space, A countably additive set function $\mu:\mathcal{A} \rightarrow \mathbb{C}$ is called a complex measure. When we say $\mu$ is countably additive, we mean
$$\mu\left( \bigcup_{j=1}^\infty A_j \right) = \sum_{j=1}^\infty \mu(A_j)$$ whenever $(A_j)_{j=1}^\infty$ is a sequence of pairwise disjoint measurable sets in $\mathcal{A},$ where the series is absolutely convergent.

At the same page, the authors mentioned the following:

There is a significant difference between positive measures and complex measures. In the definition of complex measure, we require that a complex measure $\mu$ be finite; that is, $|\mu(A)|<\infty$ for all $a\in\mathcal{A}.$ This was not a requirement for a positive measure.

Question: Why do we need to assume that $|\mu(A)| < \infty$ for complex measure while we do not need to assume it in positive measure? I strongly believe that purpose of inequality is not to ensure the series converges absolutely only, but it has other purpose.

• To avoid having things like $\infty-\infty$. – Lord Shark the Unknown Sep 9 '17 at 17:38
• Can you elaborate more on your statement? Can you give an example where we encounter the mentioned situation? – Idonknow Sep 10 '17 at 0:28

In the case of positive measures we had that $$\mu(A) \leq \mu(B)$$ whenever $$A\subseteq B$$, and we would like to recover a similar if yet weaker property for complex measures.