Pettis Integrability of Weakly Continuous Functions and Baire Measures

Abstract

We analyse the Pettis integrability of weakly continuous bounded functions defined on a completely regular space S and taking values in a Banach space. We prove that the set of Baire measures with respect to which such functions are universally Pettis integrable is precisely the space Mg(S) of Grothendieck measures introduced by Wheeler. This leads us to prove that Mg(S) is σ(Mg(S), Cb(S))-sequentially complete, and we obtain a characterization in rβ(S) of the measures in Mg(S). We also obtain analogous results for the space of separable measures M∞(S).