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

TL;DR

This paper extends the use of uniformly distributed sequences in (0,1) for calculating Lebesgue integrals, providing a broader set of sequences that satisfy a version of Kolmogorov's strong law of large numbers.

Contribution

It generalizes previous results by enlarging the class of uniformly distributed sequences applicable for Lebesgue integral calculation, surpassing sequences of the form ({\alpha n}) with irrational lpha.

Findings

Extended the set of sequences for Lebesgue integral calculation.

Proved the measure of the new sequence set is 1.

Provided a modified proof of Kolmogorov's strong law of large numbers.

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 $(0,1)$.