On the equivalence of McShane and Pettis integrability in non-separable Banach spaces ✩

We show that McShane and Pettis integrability coincide for functions f : [0, 1] → L1(μ), where μ is any finite measure. On the other hand, assuming the Continuum Hypothesis, we prove that there exist a weakly Lindelöf determined Banach space X, a scalarly null (hence Pettis integrable) function h: [0, 1]→X and an absolutely summing operator u from X to another Banach space Y such that the composition u ◦ h: [0, 1]→Y is not Bochner integrable; in particular, h is not McShane integrable. © 2007 Elsevier Inc. All rights reserved