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.

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

that is sigma-additive. In other words, for any sequence ${\displaystyle (A_{n})_{n\in \mathbb {N} }}$ of disjoint sets belonging to ${\displaystyle \Sigma }$, one has

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