  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].
