Approximation by step functions of Banach space valued nonabsolute integrals.

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.