Abstract

We study some relationships between the Bartle-Dunford-Schwartz integral of a scalar valued function f, with respect to a vector measure m, and the Dunford, Pettis or Bochner integrals of its (vector valued) distribution function mfm_f. The Dunford (or Pettis) integrability of mfm_f is strongly related to the weak integrability (or the integrability) of f in the sense of Bartle-Dunford-Schwartz. In the case of the Bochner integrability of mfm_f, a new function space appears. It is defined through the Choquet integrability of f with respect to the semivariation ∥m∥\|m\| of the measure m. We also study this space and present its main properties.

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Antonio Fernández

Dpto. de Matemática Aplicada II, E.T.S. de Ingeniería, Universidad de Sevilla, Camino de los Descubrimientos s/n, 41092 Sevilla, Spain

afcarrion@etsi.us.es

Fernando Mayoral

Dpto. Matemática Aplicada II, E.T.S. de Ingeniería, Universidad de Sevilla, Camino de los Descubrimientos s/n, 41092 Sevilla, Spain

mayoral@us.es

Francisco Naranjo

Dpto. Matemática Aplicada II, E.T.S. de Ingeniería, Universidad de Sevilla, Camino de los Descubrimientos s/n, 41092 Sevilla, Spain

naranjo@us.es

Keywords

  • Bartle-Dunford-Schwartz integral
  • Dunford integral
  • Pettis integral
  • Bochner integral
  • Choquet integral
  • vector measures

Mathematics Subject Classification

46G10, 28B05, 46E30, 28B15

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Suggested citation

A. Fernández, F. Mayoral, F. Naranjo. “Bartle–Dunford–Schwartz Integral versus Bochner, Pettis and Dunford Integrals.” Journal of Convex Analysis 20 (2013), No. 2, 339–353. https://doi.org/10.68381/jca20022

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Published by Heldermann Verlag, 2013. Rights now held by Banach Press LLC.