Calculation of Lebesgue integrals by using uniformly distributed sequences

A certain modified version of Kolmogorov’s strong law of large numbers is used for an extension of the result of C. Baxa and J. Schoißengeier (2002) to a maximal set of uniformly distributed sequences in (0,1)(0,1) which strictly contains the set of all sequences having the form ({αn})n∈N for some irrational number αα and having the full ℓ1∞-measure, where ℓ1∞ denotes the infinite power of the linear Lebesgue measure ℓ1ℓ1 in (0,1)(0,1).