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Concept
Complex measure
Summary
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 on a measurable space is a complex-valued function
that is sigma-additive. In other words, for any sequence of disjoint sets belonging to , one has
As for any permutation (bijection) , it follows that converges unconditionally (hence absolutely).
One can define the integral of a complex-valued measurable function with respect to a complex measure in the same way as the Lebesgue integral of a real-valued measurable function with respect to a non-negative 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 real-valued function with respect to a non-negative measure. To that end, it is a quick check that the real and imaginary parts μ1 and μ2 of a complex measure μ are finite-valued signed measures. One can apply the Hahn-Jordan decomposition to these measures to split them as
and
where μ1+, μ1−, μ2+, μ2− are finite-valued non-negative measures (which are unique in some sense). Then, for a measurable function f which is real-valued for the moment, one can define
as long as the expression on the right-hand side is defined, that is, all four integrals exist and when adding them up one does not encounter the indeterminate ∞−∞.
Given now a complex-valued measurable function, one can integrate its real and imaginary components separately as illustrated above and define, as expected,
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 (An)n whose union is A.
Official source
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EPFL2015Related concepts (4)
Set function
In mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes its values in the extended real number line which consists of the real numbers and A set function generally aims to subsets in some way. Measures are typical examples of "measuring" set functions. Therefore, the term "set function" is often used for avoiding confusion between the mathematical meaning of "measure" and its common language meaning.
Signed measure
In mathematics, signed measure is a generalization of the concept of (positive) measure by allowing the set function to take negative values. There are two slightly different concepts of a signed measure, depending on whether or not one allows it to take infinite values. Signed measures are usually only allowed to take finite real values, while some textbooks allow them to take infinite values. To avoid confusion, this article will call these two cases "finite signed measures" and "extended signed measures".
Sigma-additive set function
In mathematics, an additive set function is a function mapping sets to numbers, with the property that its value on a union of two disjoint sets equals the sum of its values on these sets, namely, If this additivity property holds for any two sets, then it also holds for any finite number of sets, namely, the function value on the union of k disjoint sets (where k is a finite number) equals the sum of its values on the sets. Therefore, an additive set function is also called a finitely additive set function (the terms are equivalent).