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 \((0,1)\).