A Perron-type integral of order 2 for Riesz spaces: A Perron-type integral of order \(2\) for Riesz spaces
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| Názov: | A Perron-type integral of order 2 for Riesz spaces: A Perron-type integral of order \(2\) for Riesz spaces |
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| Autori: | BOCCUTO, Antonio |
| Informácie o vydavateľovi: | Slovak Academy of Sciences, Mathematical Institute, Bratislava; De Gruyter, Warsaw, 2001. |
| Rok vydania: | 2001 |
| Predmety: | Riesz space, Perron integral of order 2, trigonometric series, Set functions, measures and integrals with values in ordered spaces, Vector-valued set functions, measures and integrals, Denjoy and Perron integrals, other special integrals, 0101 mathematics, Vector-valued measures and integration, Henstock-Kurzweil integral, 01 natural sciences |
| Popis: | A Perron-type integral of order 2 is introduced. Integrable functions are defined on \([a, b]\) (\(a\) and \(b\) are real numbers) and the integral takes values in a Dedekind complete Riesz space \(R\). After proving some assertions about the integral, the author compares the introduced integral with the Perron integral of order 1 which is equivalent to the Henstock-Kurzweil integral introduced in [\textit{B. Riečan}, ``On the Kurzweil integral for functions with values in ordered spaces. I'', Acta Math. Univ. Comenianae 56/57, 75-83 (1990; Zbl 0735.28008)]. The usefulness of the introduced integral is shown on the following nice application when the classical approach is not working since there are Riesz spaces \(R\) and Lipschitz functions \(f\:[a, b] \to R\) which are not differentiable in any point of \(]a, b[\). Let \(f\) be a function defined on \([a, b]\) and with values in \(L^0(X, B, \mu)\), where \((X, B, \mu)\) is a measure space with the positive, \(\sigma \)-additive and \(\sigma \)-finite measure; let \(X\) be the time space; and \(f\) can be represented as a sum of a trigonometric series, convergent pointwise with respect to the space variable and almost everywhere with respect to the time variable. Then \(f\) is Perron integrable of order 2. |
| Druh dokumentu: | Article |
| Popis súboru: | application/xml |
| Prístupová URL adresa: | https://zbmath.org/1737278 https://hdl.handle.net/11391/156809 |
| Prístupové číslo: | edsair.dedup.wf.002..97c01c48fb9f9af88fc16c63c6960a7a |
| Databáza: | OpenAIRE |
Nájsť tento článok vo Web of Science | Abstrakt: | A Perron-type integral of order 2 is introduced. Integrable functions are defined on \([a, b]\) (\(a\) and \(b\) are real numbers) and the integral takes values in a Dedekind complete Riesz space \(R\). After proving some assertions about the integral, the author compares the introduced integral with the Perron integral of order 1 which is equivalent to the Henstock-Kurzweil integral introduced in [\textit{B. Riečan}, ``On the Kurzweil integral for functions with values in ordered spaces. I'', Acta Math. Univ. Comenianae 56/57, 75-83 (1990; Zbl 0735.28008)]. The usefulness of the introduced integral is shown on the following nice application when the classical approach is not working since there are Riesz spaces \(R\) and Lipschitz functions \(f\:[a, b] \to R\) which are not differentiable in any point of \(]a, b[\). Let \(f\) be a function defined on \([a, b]\) and with values in \(L^0(X, B, \mu)\), where \((X, B, \mu)\) is a measure space with the positive, \(\sigma \)-additive and \(\sigma \)-finite measure; let \(X\) be the time space; and \(f\) can be represented as a sum of a trigonometric series, convergent pointwise with respect to the space variable and almost everywhere with respect to the time variable. Then \(f\) is Perron integrable of order 2. |
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