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

Volume 167 / 2015

Christoph Aistleitner, Josef Dick Acta Arithmetica 167 (2015), 143-171 MSC: 26B30, 65D30, 65C05, 11K38. DOI: 10.4064/aa167-2-4


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. We also 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.


  • Christoph AistleitnerDepartment of Mathematics and Statistics
    Graduate School of Science
    Kobe University
    1-1 Rokkodai, Nada-ku
    Kobe 657-8501, Japan
  • Josef DickSchool of Mathematics and Statistics
    University of New South Wales
    Sydney, NSW 2052, Australia

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