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Quantitative form of the Luzin C-property

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Ukrainian Mathematical Journal Aims and scope

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 EX, μE = 0 , for which

$$ \left| {f(x) - f(y)} \right| \leqslant \left[ {g(x) + g(y)} \right]\eta \left( {d\left( {x,y} \right)} \right),\,x,y \in {{X} \left/ {E} \right.} $$

If fL p (X), p >0, then it is possible to choose g belonging to L p (X).

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 62, No. 3, pp. 387–395, March, 2010.

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Krotov, V.G. Quantitative form of the Luzin C-property. Ukr Math J 62, 441–451 (2010). https://doi.org/10.1007/s11253-010-0363-1

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  • DOI: https://doi.org/10.1007/s11253-010-0363-1

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