Normal numbers from Steinhaus viewpoint

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].