Quantitative form of the Luzin $C$-property
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...
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| Datum: | 2010 |
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| Sprache: | Russisch Englisch |
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Institute of Mathematics, NAS of Ukraine
2010
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508864250642432 |
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| author | Krotov, V. G. Кротов, В. Г. Кротов, В. Г. |
| author_facet | Krotov, V. G. Кротов, В. Г. Кротов, В. Г. |
| author_sort | Krotov, V. G. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:39:19Z |
| description | 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 $L^p (X)$. |
| first_indexed | 2026-03-24T02:31:59Z |
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| id | umjimathkievua-article-2874 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
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| language | rus English |
| last_indexed | 2026-03-24T02:31:59Z |
| publishDate | 2010 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/26/2f8a2cc6d098a732d32503ba15ff7d26.pdf |
| spelling | umjimathkievua-article-28742020-03-18T19:39:19Z Quantitative form of the Luzin $C$-property Количественная форма $C$-свойства Лузина Krotov, V. G. Кротов, В. Г. Кротов, В. Г. 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 $L^p (X)$. Доведено наступне твердження, яке є кількісною формою теореми Лузіна про $C$-властивість. Нехай $(X, d, μ)$—обмежений метричний простір із метрикою $d$ і регулярною борелевого мірою $μ$, що пов'язані умовою подвоєння. Тоді для будь-якої вимірної на $X$ функції $f$ існують додатна зростаюча функція $η ∈ Ω \;\left(η(+0) = 0\right.$ і $η(t)t^{−a}$ спадає при деякому $\left. a > 0\right)$, вимірна на $X$ невід'ємна функція $g$ та множина $E ⊂ X, μE = 0$, для яких $$|f(x)−f(y)| ⩽ [g(x)+g(y)]η(d(x,y)),\;x,y ∈ X\setminus E.$$ Якщо $f ∈ L^p(X),\; p >0$, то можна вибрати $g \in L^p (X)$. Institute of Mathematics, NAS of Ukraine 2010-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2874 Ukrains’kyi Matematychnyi Zhurnal; Vol. 62 No. 3 (2010); 387–395 Український математичний журнал; Том 62 № 3 (2010); 387–395 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/2874/2499 https://umj.imath.kiev.ua/index.php/umj/article/view/2874/2500 Copyright (c) 2010 Krotov V. G. |
| spellingShingle | Krotov, V. G. Кротов, В. Г. Кротов, В. Г. Quantitative form of the Luzin $C$-property |
| title | Quantitative form of the Luzin $C$-property |
| title_alt | Количественная форма $C$-свойства Лузина |
| title_full | Quantitative form of the Luzin $C$-property |
| title_fullStr | Quantitative form of the Luzin $C$-property |
| title_full_unstemmed | Quantitative form of the Luzin $C$-property |
| title_short | Quantitative form of the Luzin $C$-property |
| title_sort | quantitative form of the luzin $c$-property |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2874 |
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