Calculation of Lebesgue Integrals by Using Uniformly Distributed Sequences in $(0,1)$

Abstract

We present modified proof of a certain version of Kolmogorov's strong law of large numbers for calculation of Lebesgue Integrals by using uniformly distributed sequences in $(0,1)$. We extend the result of C. Baxa and J. Schoi$\beta$engeier (cf.\cite{BaxSch2002}, Theorem 1, p. 271) to a maximal set of uniformly distributed (in $(0,1)$) sequences $S_f \subset(0,1)^{\infty}$ which strictly contains the set of sequences of the form $(\{\alpha n\})_{n \in {\bf N}}$ with irrational number $\alpha$ and for which $\ell_1^{\infty}(S_f)=1$, where $\ell_1^{\infty}$ denotes the infinite power of the linear Lebesgue measure $\ell_1$ in $(0,1)$.