complex measure

In mathematics, specifically measure theory, a complex measure generalizes the concept of measure by letting it have complex values. In other words, one allows for sets whose size (length, area, volume) is a complex number.

Definition

Formally, a ''complex measure'' $\mu$ on a measurable space $(X,\Sigma)$ is a complex-valued function

:$\mu: \Sigma \to \mathbb$

that is sigma-additive. In other words, for any sequence $(A_)_$ of disjoint sets belonging to $\Sigma$, one has

:$\sum_^ \mu(A_) = \mu \left( \bigcup_^ A_ \right) \in \mathbb.$

As $\displaystyle \bigcup_^ A_ = \bigcup_^ A_$ for any permutation (bijection) $\sigma: \mathbb \to \mathbb$, it follows that $\displaystyle \sum_^ \mu(A_)$ converges unconditionally (hence absolutely).

Integration with respect to a complex measure

One can define the ''integral'' of a complex-valued