Multipliers for generalized Riemann integrals in the real line

Tuo-Yeong Lee

Displaying similar documents to “Multipliers for generalized Riemann integrals in the real line”

A multidimensional integration by parts formula for the Henstock-Kurzweil integral

Tuo-Yeong Lee (2008)

Mathematica Bohemica

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It is shown that if $g$ is of bounded variation in the sense of Hardy-Krause on $\prod_{i = 1}^{m} [a_{i}, b_{i}]$ , then $g χ_{_{\prod_{i = 1}^{m} (a_{i}, b_{i})}}$ is of bounded variation there. As a result, we obtain a simple proof of Kurzweil’s multidimensional integration by parts formula.

Henstock-Kurzweil and McShane product integration; descriptive definitions

Antonín Slavík, Štefan Schwabik (2008)

Czechoslovak Mathematical Journal

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The Henstock-Kurzweil and McShane product integrals generalize the notion of the Riemann product integral. We study properties of the corresponding indefinite integrals (i.e. product integrals considered as functions of the upper bound of integration). It is shown that the indefinite McShane product integral of a matrix-valued function $A$ is absolutely continuous. As a consequence we obtain that the McShane product integral of $A$ over $[a, b]$ exists and is invertible if and only if $A$ is Bochner...

Kurzweil’s PU integral as the Lebesgue integral

Rudolf Výborný (2006)

Mathematica Bohemica

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For a merely continuous partition of unity the PU integral is the Lebesgue integral.

Substitution formulas for the Kurzweil and Henstock vector integrals

Márcia Federson (2002)

Mathematica Bohemica

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Results on integration by parts and integration by substitution for the variational integral of Henstock are well-known. When real-valued functions are considered, such results also hold for the Generalized Riemann Integral defined by Kurzweil since, in this case, the integrals of Kurzweil and Henstock coincide. However, in a Banach-space valued context, the Kurzweil integral properly contains that of Henstock. In the present paper, we consider abstract vector integrals of Kurzweil and...