Quantitative form of the Luzin $C$-property

Abstract

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 $η ∈ Ω\; \left(η(+0) = 0\right.$ and $η(t)t^{−a}$ decreases for a certain $\left. a > 0\right)$, a nonnegative function $g$ measurable on $X$, and $a$ set $E ⊂ X, μE = 0$, for which $$|f(x)−f(y)| ⩽ [g(x)+g(y)]η(d(x,y)),\;x,y ∈ X \setminus E.$$ If $f ∈ L^p(X),\; p >0$, then it is possible to choose $g$ belonging to $L^p (X)$.