  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 $(0,1)$.
