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 (An)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
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.
Read more about this topic: Complex Measure
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