An elementary proof of the theorem that absolute gauge integrability implies Lebesgue integrability

Timothy Myers

Displaying similar documents to “An elementary proof of the theorem that absolute gauge integrability implies Lebesgue integrability”

The Vitali convergence theorem for the vector-valued McShane integral

Richard Reynolds, Charles W. Swartz (2004)

Mathematica Bohemica

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The classical Vitali convergence theorem gives necessary and sufficient conditions for norm convergence in the space of Lebesgue integrable functions. Although there are versions of the Vitali convergence theorem for the vector valued McShane and Pettis integrals given by Fremlin and Mendoza, these results do not involve norm convergence in the respective spaces. There is a version of the Vitali convergence theorem for scalar valued functions defined on compact intervals in $ℝ^{n}$ given by...

Beppo Levi's theorem for the vector-valued McShane integral and applications.

Swartz, Charles (1997)

Bulletin of the Belgian Mathematical Society - Simon Stevin

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A scalar Volterra derivative for the PoU-integral

V. Marraffa (2005)

Mathematica Bohemica

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A weak form of the Henstock Lemma for the $P o U$ -integrable functions is given. This allows to prove the existence of a scalar Volterra derivative for the $P o U$ -integral. Also the $P o U$ -integrable functions are characterized by means of Pettis integrability and a condition involving finite pseudopartitions.

The McShane, PU and Henstock integrals of Banach valued functions

Luisa Di Piazza, Valeria Marraffa (2002)

Czechoslovak Mathematical Journal

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Some relationships between the vector valued Henstock and McShane integrals are investigated. An integral for vector valued functions, defined by means of partitions of the unity (the PU-integral) is studied. In particular it is shown that a vector valued function is McShane integrable if and only if it is both Pettis and PU-integrable. Convergence theorems for the Henstock variational and the PU integrals are stated. The families of multipliers for the Henstock and the Henstock variational...

Gauge Integral

Roland Coghetto (2017)

Formalized Mathematics

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Some authors have formalized the integral in the Mizar Mathematical Library (MML). The first article in a series on the Darboux/Riemann integral was written by Noboru Endou and Artur Korniłowicz: [6]. The Lebesgue integral was formalized a little later [13] and recently the integral of Riemann-Stieltjes was introduced in the MML by Keiko Narita, Kazuhisa Nakasho and Yasunari Shidama [12]. A presentation of definitions of integrals in other proof assistants or proof checkers (ACL2, COQ,...