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

Gogi Pantsulaia, Tengiz Kiria

Published 2016-01-14Version 1

We present the proof of Kolmogorov's strong law of large numbers in particular case and consider its applications for calculations 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 bigger set $D \subset(0,1)^{\infty}$ of uniformly distributed(in $(0,1)$) sequences strictly containing the set of all sequences of real numbers which can be presented in a form $(\{\alpha n\})_{n \in {\bf N}}$ for some irrational numbers $\alpha$ and for which $\ell_1^{\infty}(D)=1$, where $\ell_1^{\infty}$ denotes the infinite power of the linear Lebesgue measure $\ell_1$ in $(0,1)$.