Calculation of Improper Integrals by Using Uniformly Distributed Sequences

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 $S_f\subset (0,1{)}^{\mathrm{\infty }}$ which strictly contains the set of sequences of the form $({\alpha n})_{n \in {\bf N}}$withirrationalnumber$\alpha$andforwhich$\ell_1^{\infty}(S_f)=1$,where$\ell_1^{\infty}$ denotes the infinite power of the linear Lebesgue measure ${\mathrm{\ell }}_1$ in $(0,1)$.