Journal of Convex Analysis 20 (2013), No. 2, 339--353
Copyright Heldermann Verlag 2013
Bartle--Dunford--Schwartz Integral versus Bochner, Pettis and Dunford Integrals
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
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 mf . The Dunford (or Pettis) integrability of mf 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 mf , a new function space appears. It is defined through the Choquet integrability of f with respect to the semivariation ||m|| of the measure m . We also study this space and present its main properties.
Keywords: Bartle-Dunford-Schwartz integral, Dunford integral, Pettis integral, Bochner integral, Choquet integral, vector measures.
MSC: 46G10, 28B05, 46E30, 28B15
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