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Generalizations of Cyclostationary Signal Processing: Spectral Analysis and Applications by Antonio Napolitano

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6.2 Relative Measurability

6.2.1 Relative Measure of Sets

Given a set img, img being the σ-field of the Borel subsets and μ the Lebesgue measure on the real line img, the relative measure of A is defined as (Kac and Steinhaus 1938), (Leimgkow and Napolitano 2006)

(6.2) equation

provided that the limit exists. If the limit in (6.2) exists, it is independent of t0 and the set A is said to be relatively measurable. From definition (6.2) it follows that the relative measure of a set is the Lebesgue measure of the set normalized to that of the whole real line. Thus, sets with finite Lebesgue measure have zero relative measure, and only sets with infinite Lebesgue measure can have nonzero relative measure. In (Leimgkow and Napolitano 2006), the following properties of the relative measure are proved: The class of RM sets is not closed under union and intersection; The relative measure μR is additive, but not σ-additive; ...

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