Bartle–Dunford–Schwartz Integral versus Bochner, Pettis and Dunford Integrals

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.