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 $m_f$. The Dunford (or Pettis) integrability of $m_f$ 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 $m_f$, 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.