Normal numbers from Steinhaus viewpoint

TL;DR

This paper revisits Steinhaus's non-topological approach to constructing measures on [0,1], generalizes the concept of normal numbers, and demonstrates that almost all real numbers are normal under these new definitions.

Contribution

It introduces a novel non-topological method for constructing measures and generalizes the concept of normal numbers, revealing new probabilistic properties.

Findings

Almost all real numbers are normal under the generalized concept.

The approach uncovers hidden features of Borel sigma-algebra and Lebesgue measure.

The method provides a new perspective on measure construction in [0,1].

Abstract

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].