In this talk we consider the possibility to approximate (with respect to some topology induced by the Alexiewicz norm) non absolutely integrable functions defined on the unit interval by step functions. In particular we show that any Henstock (respectively Henstock-Kurzweil-Pettis, Denjoy-Khintchine-Pettis) integrable functions can be scalarly approximate in the Alexiewicz norm by a sequence of step functions. Moreover the approximation may be done in the Alexiewicz norm if and only if the range of the integral is relatively norm compact (property which is automatically satisfied by the Henstock integrable functions). We also provide an example to show that, unlike the Pettis case, Henstock-Kurzweil-Pettis integrable functions may not have relatively compact range.
Di Piazza, L. (2008). Approximation of Banach space valued Riemann type integrable functions by step functions.