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Complex Measure


A measure which takes values in the complex numbers. The set of complex measures on a measure space X forms a vector space. Note that this is not the case for the more common positive measures. Also, the space of finite measures (|mu(X)|<infty) has a norm given by the total variation measure ||mu||=|mu|(X)|, which makes it a Banach space.

Using the polar representation of mu, it is possible to define the Lebesgue integral using a complex measure,

 intfdmu=inte^(itheta)fd|mu|.

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 C, which take values in the bounded linear maps from a Hilbert space to itself, represent the operator spectrum of an operator.


See also

Banach Space, Lebesgue Integral, Measure, Measure Space, Polar Representation, Spectral Theory

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "Complex Measure." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/ComplexMeasure.html

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