Normal numbers from Steinhaus viewpoint

Details Normal numbers from Steinhaus viewpoint Normal numbers from Steinhaus viewpoint

2008-06-22

arxiv.org/abs/0806.3559v1

Mathematics

Number Theory

11 pages

Scientific paper

In this paper we recall a non-standard construction of the Borel sigma-algebra B in [0,1] and construct a family of measures (in particular, Lebesgue measure) in B by a completely non-topological method. This approach, that goes back to Steinhaus, in 1923, is now used to introduce natural generalizations of the concept of normal numbers and explore their intrinsic probabilistic properties. We show that, in virtually all the cases, almost all real number in [0,1] is normal (with respect to this generalized concept). This procedure highlights some apparently hidden but interesting features of the Borel sigma-algebra and Lebesgue measure in [0,1].

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