[1]
Adams, C. R. and Clarkson, J. A.
Properties of functions f(x, y) of bounded variation. Trans. Amer. Math. Soc.
36 (1934), 711–730.
[2]
Aistleitner, C. and Dick, J.
Functions of bounded variation, signed measures and a general Koksma–Hlawka inequality. Acta Arith
167 (2015), 143–171.
[3]
Antosik, P.
Study of the continuity of a function of many variables (Russian). Prace Mat.
10 (1966), 101–104.
[4]
Blümlinger, M. and Tichy, R. F.
Topological Algebras of Functions of Bounded Variation I. Manuscripta Math.
65 (1989), 245–255.
[5]
Blümlinger, M.
Topological Algebras of Functions of Bounded Variation II. Manuscripta Math.
65 (1989), 377–384.
[6]
Brandolini, L., Colzani, L., Gigante, G. and Travaglini, G.
On the Koksma–Hlawka inequality. J. Complexity
29 (2013), 158–172.
[7]
Brandolini, L., Colzani, L., Gigante, G. and Travaglini, G.
A Koksma–Hlawka inequality for simplices, Trends in Harmonic Analysis. Springer INdAM Ser. 3 (Springer, Milan, 2013), 33–46.
[8]
Clarkson, J. A. and Adams, C. R.
On definitions of bounded variation for functions of two variables. Trans. Amer. Math. Soc.
35 (1933), 824–854.
[9]
Drmota, M. and Tichy, R. F.
Sequences, discrepancies and applications. Lecture Notes in Mathematics (Springer-Verlag, Berlin, 1997), 1651.
[10]
Götz, M.
Discrepancy and the error in integration. Monatsh. Math.
136 (2002), no. 2, 99–121.
[11]
Hardy, G. H.
On double Fourier series, and especially those which represent the double zeta-function with real and incommensurable parameters. Quart. J. Math.
(1)
37 (1906), 53–79.
[12]
Harman, G.
Variations on the Koksma–Hlawka inequality. Unif. Distrib. Theory
5 (2010), 65–78.
[13]
Hlawka, E.
Funktionen von beschränkter Variation in der Theorie der Gleichverteilung. Ann. Math. Pura Appl.
54 (1961), 325–333.
[14]
Idczak, D.
Functions of several variables of finite variation and their differentiability. Ann. Polon. Math.
60 (1994), no. 1, 47–56.
[15]
Koksma, J. F.
A general theorem from the theory of uniform distribution modulo 1. Mathematica B (Zutphen)
11 (1942), 7–11.
[16]
Krause, J. M.
Fouriersche Reihen mit zwei veränderlichen Grössen. Ber. Sächs. Akad. Wiss. Leipzig
55 (1903) 164–197.
[17]
Kuipers, L. and Niederreiter, H.
Uniform Distribution of Sequences (Wiley-Interscience John Wiley & Sons, New York–London–Sydney, 1974).
[18]
Leonov, A. S.
Remarks on the total variation of functions of several variables and on a multidimensional analogue of Helly's choice principle. (in Russian) Mat. Zametki
63 (1998), 69–80.
[19]
Owen, A. B.
Multidimensional variation for quasi-Monte Carlo. Contemporary multivariate analysis and design of experiments. Ser. Biostat.
2 (World Scientific Publishing Co. Pte. Ltd., 2005), 49–74.
[20]
Pausinger, F. and Svane, A. M.
A Koksma–Hlawka inequality for general discrepancy systems. J. Complexity
31 (2015), 773–797.
[21]
Yeh, J.
Real analysis. Theory of measure and integration. Second edition (World Scientific Publishing Co. Pte. Ltd., 2006).
[22]
Young, W. H. and Young, G.
On the discontinuties of monotone functions of several variables. Proc. London Math. Soc.
(2)
22 (1923), 124–142.
${\mathcal D}$
-variation unifies previous concepts of variation of multivariate functions. In this paper, we give an affirmative answer to the open question from [20] whether every function of bounded Hardy–Krause variation is Borel measurable and has bounded
