Approximation by step functions of Banach space valued nonabsolute integrals.

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Abstract

The approximation of Banach space valued nonabsolutely integrablefunctions by step functions is studied. It is proved that aHenstock integrable function can be approximated by a sequenceof step functions in the Alexiewicz norm, while aHenstock-Kurzweil-Pettis and a Denjoy-Khintchine-Pettis integrablefunction can be only scalarly approximate in the Alexiewicz normby a sequence of step functions. In case ofHenstock-Kurzweil-Pettis and Denjoy-Khintchine-Pettis integralsthe full approximation can be done if and only if the range ofthe integral is norm relatively compact. It is also proved that ifthe target Banach space X does not contain any isomorphic copyof c_0, then the range of the integral of each X valuedDenjoy-Khintchine-Pettis integrable function is norm relativelycompact.
Lingua originaleEnglish
pagine (da-a)583-593
Numero di pagine11
RivistaGlasgow Mathematical Journal
Volume50
Stato di pubblicazionePublished - 2008

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All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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