Error bounds for quasi-Monte Carlo integration for \mathscr{L}^{\infty} with uniform point sets

Niederreiter [H.Niederreiter, Error bounds for quasi-Monte Carlo integration with uniform point sets, Journal of computational and applied mathematics 150 (2003), 283-292] established new bounds for quasi-Monte Carlo integration for nodes sets with a special kind of uniformity property. Let (X,\mathscr{A},\mu) be an arbitrary probability space, i.e., X is an arbitrary nonempty set, \mathscr{A} a \sigma-algebra of subsets of X, and \mu a probability measure defined on \mathscr{A}. The functions considered in Niederreiter's paper are bounded \mu-integrable functions on X. In this note, we extend some of his results for bounded \mu-integrable functions to essentially bounded \mathscr{A}-measurable functions. So Niederreiter's bounds can be used in a more general setting.