On almost periodicity defined via non-absolutely convergent integrals
We investigate some properties of the normed space of almost periodic functions which are defined via the Denjoy-Perron (or equivalently, Henstock-Kurzweil) integral. In particular, we prove that this space is barrelled while it is not complete. We also prove that a linear differential equation with the non-homogenous term being an almost periodic function of such type, possesses a solution in the class under consideration.