Variations on the Koksma-Hlawka inequality

The Koksma-Hlawka inequality gives a bound for the error inapproximating the multidimensional integral of a function by the average of the function over a discrete set of points. The error is essentially a product of a measure of the function's variation and the discrepancy of the set of points. For the standard form of the inequality this discrepancy is with respect to boxes aligned to the axes and the measure of variation is then dependent on this choice of axes. This rules out various natural choices of function (or even integration domains) and we here explore a new choice for the measure of variation that enables a wider class of functions to be investigated, expanding comments we stated without proof in the author monograph Metric Number Theory, Oxford, 1998, p. 161-2.