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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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