The Ward property for a P-adic basis and the P-adic integral

An Henstock–Kurzweil type integral with respect to a P-adic basis is considered. It is shown that a P-adic basis possesses the Ward property if and only if the sequence by which it is defined is bounded. As a consequence, some full descriptive characterizations of the P-adic integral in the bounded case are obtained. Moreover, an example of an exact P-adic primitive which is not a VBG function and does not satisfy the Lusin condition (N) is constructed.  2003 Elsevier Inc. All rights reserved