Calculation of Improper Integrals by Using Uniformly Distributed Sequences

TL;DR

This paper extends the use of uniformly distributed sequences to compute Lebesgue integrals, proving a modified law of large numbers that encompasses a broader class of sequences.

Contribution

It generalizes previous results by including a maximal set of uniformly distributed sequences for Lebesgue integral calculation, beyond just irrational multiples.

Findings

Established a modified strong law of large numbers for Lebesgue integrals.

Extended the class of sequences used for integral approximation.

Proved that the set of such sequences has full measure in the infinite product space.

Abstract

We present the proof of a certain modified 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 $(0,1)$.