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On coincidence of Pettis and McShane integrability

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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.

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Authors and Affiliations

  1. Mathematical Institute, Czech Academy of Sciences, Žitná 25, 115 67, Praha 1, Czech Republic

    Marián Fabian

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  1. Marián Fabian
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Correspondence to Marián Fabian.

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Dedicated to Jaroslav Kurzweil on the occasion of his 88th birthday and to the memory of Štefan Schwabik

Supported by grant P201/12/0290 and by Institutional Research Plan of the Academy of Sciences of Czech Republic No. RVO: 67985840.

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Fabian, M. On coincidence of Pettis and McShane integrability. Czech Math J 65, 83–106 (2015). https://doi.org/10.1007/s10587-015-0161-x

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  • DOI: https://doi.org/10.1007/s10587-015-0161-x

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