Normal numbers from Steinhaus's viewpoint (Q1945731)

In this expository paper, the author, by recalling a non-standard construction of the Borel sigma-algebra \({\mathfrak B}\) in \([0,1]\), managed to characterize the Borel sigma-algebra in \([0,1]\) as a product sigma-algebra and to construct a family of measures (in particular, Lebesgue measure) in \([0,1]\) using a non-topological method. Moreover, by applying the previous results in order to study natural generalizations of the concept of normal numbers and by exploring their probabilistic behaviour, the author managed to provide a different approach through modern notation and terminology of Steinhaus' ideas concerning normal numbers in the beginning of the conception of modern probability theory.