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

AbstractWe 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