On coincidence of Pettis and McShane integrability

Abstract

R.Deville and J.Rodríguez proved that, for every Hilbert generated space X, every Pettis integrable function f: [0, 1] → X is McShane integrable. R.Avilés, G. Plebanek, and J.Rodríguez constructed a weakly compactly generated Banach space X and a scalarly null (hence Pettis integrable) function from [0, 1] into X, which was not McShane integrable. We study here the mechanism behind the McShane integrability of scalarly negligible functions from [0, 1] (mostly) into C(K) spaces. We focus in more detail on the behavior of several concrete Eberlein (Corson) compact spaces K, that are not uniform Eberlein, with respect to the integrability of some natural scalarly negligible functions from [0, 1] into C(K) in McShane sense.