Quantitative form of the Luzin C-property - Krotov, V. G. (Ukrains’kyi Matematychnyi Zhurnal, 2010)
We prove the following statement, which is a quantitative form of the Luzin theorem on C-property: Let (X, d, μ) be a bounded metric space with metric d and regular Borel measure μ that are related to one another by the doubling condition. Then, for any function f measurable on X, there exist a positive increasing function η ∈ Ω (η(+0) = 0. and η(t)t^−a decreases for a certain . a > 0), a nonnegative function g measurable on X, and a set E ⊂ X, μE = 0, for which |f(x)−f(y)| ⩽ [g(x)+g(y)]η(d(x,y)), x,y ∈ X E. If f ∈ L^p(X), p >0, then it is possible to choose g belonging to L^p (X).