Kurzweil-Henstock absolute integrable means McShane integrable (Q1890640)
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scientific article; zbMATH DE number 756682
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Kurzweil-Henstock absolute integrable means McShane integrable |
scientific article; zbMATH DE number 756682 |
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Kurzweil-Henstock absolute integrable means McShane integrable (English)
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4 September 1995
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In this excellent paper, the author proves that if \(f\) and \(|f|\) are both Kurzweil-Henstock integrable then \(f\) is McShane integrable. This is of course known but the paper's originality is that the proof lies within the theory of these two integrals. The only facts used beyong the two definitions are the following: (i) Fatou's lemma for the McShane integral; (ii) if \(\forall \varepsilon> 0\) \(\exists\) McShane integrable \(H\), \(h\) with \(h\leq f\leq H\) and with \(\int H- \int h< \varepsilon\) then \(f\) is McShane integrable; (iii) if \(f\) is Kurzweil-Henstock integrable then \(M_\delta= \sup_D \{\sum_D f\); \(D\) a \(\delta\)-fine partition\}, and \(m_\delta= \inf_D \{\sum_D f\); \(D\) a \(\delta\)- fine partition\} are major and minor functions of \(f\) in the sense of Ward.
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Kurzweil-Henstock integral
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Kurzweil-Henstock absolute integrable means
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Fatou's lemma
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McShane integral
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