Functions of bounded variation, signed measures, and a general Koksma-Hlawka inequality

TL;DR

This paper establishes a link between functions of bounded variation and signed measures, leading to a generalized Koksma-Hlawka inequality applicable to non-uniform measures, with applications in Quasi-Monte Carlo methods.

Contribution

It introduces a correspondence principle connecting bounded variation functions and signed measures, simplifying proofs of generalized inequalities and analyzing measure transformation limitations.

Findings

Proves a correspondence principle between bounded variation functions and signed measures.

Derives a generalized Koksma-Hlawka inequality for non-uniform measures.

Shows limitations of Chelson's method for transforming low-discrepancy sequences.

Abstract

In this paper we prove a correspondence principle between multivariate functions of bounded variation in the sense of Hardy and Krause and signed measures of finite total variation, which allows us to obtain a simple proof of a generalized Koksma--Hlawka inequality for non-uniform measures. Applications of this inequality to importance sampling in Quasi-Monte Carlo integration and tractability theory are given. Furthermore, we discuss the problem of transforming a low-discrepancy sequence with respect to the uniform measure into a sequence with low discrepancy with respect to a general measure $\mu$, and show the limitations of a method suggested by Chelson.