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.
|Numero di pagine||11|
|Rivista||Glasgow Mathematical Journal|
|Stato di pubblicazione||Published - 2008|
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