A measure which takes values in the complex numbers. The set of complex measures on a measure space forms a vector space. Note that this is not the case for the more common positive measures. Also, the space of finite measures () has a norm given by the total variation measure , which makes it a Banach space.
Using the polar representation of , it is possible to define the Lebesgue integral using a complex measure,
Sometimes, the term "complex measure" is used to indicate an arbitrary measure. The definitions for measure can be extended to measures which take values in any vector space. For instance in spectral theory, measures on , which take values in the bounded linear maps from a Hilbert space to itself, represent the operator spectrum of an operator.