 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.
Contents
Definition
Formally, a complex measure μ on a measurable space (X,Σ) is a function
defined on Σ and taking complex values, which is sigmaadditive; that is, for any sequence (A_{n})_{n} of disjoint sets in Σ one has
provided that the sum on the right converges absolutely or diverges properly, by analogy with the realvalued signed measures.
Integration in respect to a complex measure
One can define the integral of a complexvalued measurable function with respect to a complex measure in the same way as the Lebesgue integral of a realvalued measurable function with respect to a nonnegative measure, by approximating a measurable function with simple functions. Just as in the case of ordinary integration, this more general integral might fail to exist, or its value might be infinite (the complex infinity).
Another approach is to not develop a theory of integration from scratch, but rather use the already available concept of integral of a realvalued function with respect to a nonnegative measure. To that end, it is a quick check that the real and imaginary parts μ_{1} and μ_{2} of a complex measure μ are finitevalued signed measures. One can apply the HahnJordan decomposition to these measures to split them as
and
where μ_{1}^{+}, μ_{1}^{}, μ_{2}^{+}, μ_{2}^{} are finitevalued nonnegative measures (unique in some sense). Then, for a measurable function f which is realvalued for the moment, one can define
as long as the expression on the righthand side is defined, that is, all four integrals exist and when adding them up one does not encounter the indeterminate ∞−∞.
Given now a complexvalued measurable function, one can integrate its real and imaginary components separately as illustrated above and define, as expected,
Variation of a complex measure and polar decomposition
For a complex measure μ, one defines its variation, or absolute value, μ by the formula
where A is in Σ and the supremum runs over all sequences of disjoint sets (A_{n})_{n} whose union is A. Taking only finite partitions of the set A into measurable subsets, one obtains an equivalent definition.
It turns out that μ is a nonnegative finite measure. In the same way as a complex number can be represented in a polar form, one has a polar decomposition for a complex measure: There exists a measurable function θ with real values such that
meaning
for any absolutely integrable measurable function f, i.e., f satisfying
One can use the Radon–Nikodym theorem to prove that the variation is a measure and the existence of the polar decomposition.
The space of complex measures
The sum of two complex measures is a complex measure, as is the product of a complex measure by a complex number. That is to say, the set of all complex measures on a measure space (X, Σ) forms a vector space. Moreover, the total variation μ defined as
is a norm in respect to which the space of complex measures is a Banach space.
See also
External links
Categories: Measures (measure theory)
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