Abstract
DANIEL PELLEGRINOAbstract. In this paper we recall a non-standard construction of theBorelsigma-algebraB in [0,1]andconstructafamily ofmeasures(in par-ticular, Lebesgue measure)in B bya completely non-topologicalmethod.This approach, that goes back to Steinhaus, in 1923, is now used to in-troduce natural generalizations of the concept of normal numbers andexplore their intrinsic probabilistic properties. We show that, in virtu-ally all the cases, almost all real number in [0,1] is normal (with respectto this generalized concept). This procedure highlights some apparentlyhidden but interesting features of the Borel sigma-algebra and Lebesguemeasure in [0,1].
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