Koksma's Inequality

Page 18 of Niederreiter, his Theorem 2.9 states Koksma's inequality

Theorem 3 (Koksma's Inequality)   If h has bounded (total) variation on the closed interval [0,1], then for any points $x_1,...,x_N$, in the interval [0,1], we have

$$\vert\frac{1}{N} \sum_{i=1}^{N} h(x_i) - \int_{0}^1 h(u) du \le V(h) D_{N}^{*}(x_1,...,x_N)$$ (2.12)

Koksma's inequality applies to where the integral is over one dimension. This applies for example to the Boness [#!kn:BonessJPE1964!#] formula for the European Put, or special cases of it like the Black Scholes formula.