Calculation of Lebesgue integrals by using uniformly distributed sequences (Q336016)

In this paper, some metric properties of uniformly distributed sequences are investigated. One particular result is the following. Consider an arbitrary Lebesgue measurable function \(f:[0,1]\rightarrow\mathbb R\) and the set S of all uniformly distributed sequences \((x_k)\) in \([0,1]\) such that NEWLINE\[NEWLINE\lim_{N\rightarrow\infty}\frac{1}{N}\sum^N_{k=1}f(x_k)=\int_0^1f(x)d\lambda(x).NEWLINE\]NEWLINE Then \(\lambda_\infty(S)=1\), where \(\lambda_\infty\) is the infinite product measure of the Lebesgue measure \(\lambda\) on \([0,1]\). This extends a well-known metric result and implies generalizations of results of \textit{C. Baxa} and \textit{J. Schoißengeier} [Monatsh. Math. 135, No. 4, 265--277 (2002; Zbl 1009.11054)].