Properties of the Riemann–Lebesgue integrability in the non-additive case

Abstract

We study Riemann–Lebesgue integrability of a vector function relative to an arbitrary non-negative set function. We obtain some classical integral properties. Results regarding the continuity properties of the integral and relationships among Riemann–Lebesgue, Birkhoff simple and Gould integrabilities are also established.

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Correspondence to Anna Rita Sambucini.

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Dedicated to Mimmo: a Master, a Colleague, a dear Friend and a Gentleman who passed away, but lives in our hearts.

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The first and the last authors have been partially supported by University of Perugia – Department of Mathematics and Computer Sciences – Ricerca di Base 2018 “Metodi di Teoria dell’Approssimazione, Analisi Reale, Analisi Nonlineare e loro applicazioni” and the last author by GNAMPA - INDAM (Italy) “Metodi di Analisi Reale per l’Approssimazione attraverso operatori discreti e applicazioni” (2019).

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Candeloro, D., Croitoru, A., Gavriluţ, A. et al. Properties of the Riemann–Lebesgue integrability in the non-additive case. Rend. Circ. Mat. Palermo, II. Ser 69, 577–589 (2020). https://doi.org/10.1007/s12215-019-00419-y

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Keywords

  • Riemann–Lebesgue integral
  • Birkhoff simple integral
  • Gould integral
  • Non-negative set function
  • Monotone measure

Mathematics Subject Classification

  • 28B20
  • 28C15
  • 49J53