A complex measure is a generalized concept in measure theory that extends the notion of a measure to allow for complex-valued measures. While a traditional measure assigns a non-negative real number to a set (such as its "size" or "volume"), a complex measure can assign a complex number to a set.
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A complex measure is a countably additive complex-valued set function on a sigma-algebra. Its total variation norm of a measure is obtained by taking the supremum of sums over finite measurable partitions. For finite variation and absolute continuity of measures with respect to a sigma-finite positive measure, the Radon-Nikodym theorem supplies an integrable complex density.