Error Bounds for quasi-Monte Carlo integration with nets

We analyze the error introduced by approximately calculating the -dimensional Lebesgue measure of a Jordan-measurable subset of . We give an upper bound for the error of a method using a -net, which is a set with a very regular distribution behavior. When the subset of is defined by some function of bounded variation on , the error is estimated by means of the variation of the function and the discrepancy of the point set which is used. A sharper error bound is established when a -net is used. Finally a lower bound of the error is given, for a method using a -net. The special case of the 2-dimensional Hammersley point set is discussed.