Measure with complex values
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.
Definition.
Formally, a "complex measure" formula_1 on a measurable space formula_2 is a complex-valued function
formula_3
that is sigma-additive. In other words, for any sequence formula_4 of disjoint sets belonging to formula_5, one has
formula_6
As formula_7 for any permutation (bijection) formula_8, it follows that formula_9 converges unconditionally (hence, since formula_10 is finite dimensional, formula_11 converges absolutely).
Integration with respect to a complex measure.
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
formula_12
and
formula_13
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
formula_14
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,
formula_15
Variation of a complex measure and polar decomposition.
For a complex measure μ, one defines its "variation", or "absolute value", |μ| by the formula
formula_16
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 non-negative 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
formula_17
meaning
formula_18
for any absolutely integrable measurable function "f", i.e., "f" satisfying
formula_19
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 over the complex numbers. Moreover, the "total variation" formula_20 defined as
formula_21
is a norm, with respect to which the space of complex measures is a Banach space.
References.
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