A Koksma–Hlawka inequality for general discrepancy systems

Motivated by recent ideas of Harman (Unif. Distrib. Theory, 2010) we develop a new concept of variation of multivariate functions on a compact Hausdorff space with respect to a collection DD of subsets. We prove a general version of the Koksma–Hlawka theorem that holds for this notion of variation and discrepancy with respect to DD. As special cases, we obtain Koksma–Hlawka inequalities for classical notions, such as extreme or isotropic discrepancy. For extreme discrepancy, our result coincides with the usual Koksma–Hlawka theorem. We show that the space of functions of bounded DD-variation contains important discontinuous functions and is closed under natural algebraic operations. Finally, we illustrate the results on concrete integration problems from integral geometry and stereology.