Generalizations of Cyclostationary Signal Processing: Spectral Analysis and Applications

6.2 Relative Measurability

6.2.1 Relative Measure of Sets

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

(6.2)

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 (Lekow 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 μ_R is additive, but not σ-additive; ...