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arXiv · 1601.04088

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

Gogi Pantsulaia·Tengiz Kiria
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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$β$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 $(\{αn\})_{n \in {\bf N}}$ with irrational number $α$ 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 $(0,1)$.

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Gogi Pantsulaia, Tengiz Kiria. 2016-01-19. Calculation of Lebesgue Integrals by Using Uniformly Distributed Sequences in $(0,1)$. https://doi.org/10.1016/j.trmi.2016.06.003

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