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 $(\{\alpha n\}{)}_{n\in \mathbf{N}}$ 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 $(0,1)$.

12 pages

References in corpus (1)

• Infinite-Dimensional Monte-Carlo Integration