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

TL;DR

This paper extends Niederreiter's error bounds for quasi-Monte Carlo integration from bounded functions to essentially bounded measurable functions, broadening the applicability of these bounds in numerical integration.

Contribution

It generalizes Niederreiter's error bounds to essentially bounded measurable functions, allowing their use in more general probability spaces.

Findings

Bounds now applicable to essentially bounded functions

Extension to arbitrary probability spaces

Maintains accuracy of quasi-Monte Carlo integration

Abstract

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.