Error bounds for quasi-Monte Carlo integration for L∞ with uniform point sets

Niederreiter (Niederreiter in J Comput Appl Math 150: 283-292, 2003) established new bounds for quasi-Monte Carlo integration for nodes sets with a special kind of uniformity property. Let (X, A, mu) be an arbitrary probability space, i. e., X is an arbitrary nonempty set, A a sigma-algebra of subsets of X, and mu a probability measure defined on A. The functions considered in Niederreiter (J Comput Appl Math 150: 283-292, 2003) are bounded mu-integrable functions on X. In this note, we extend some of his results for bounded mu-integrable functions to essentially bounded A-measurable functions. So Niederreiter's bounds can be used in a more general setting.