Rod Nillsen
Mathematics research
Normal numbers and randomness
Introduction
Many interesting results in mathematical analysis require a knowledge of measure theory, and yet measure theory is often not taught at the undergraduate level, especially with the changing external circumstances in our discipline. The main purpose in writing the paper "Normal numbers without measure theory" ( American Mathematical Monthly , 107 (2000), 639-644), was to make it possible to include Borel's Theorem on normal numbers in the undergraduate syllabus without any prior knowledge of measure theory.
Comments
Each number between 0 and 1 may be expanded to the base 2, and in so doing we represent such a number by an infinite sequence of 0s and 1s. Borel's Theorem says that for "almost all" such numbers, that is for all those numbers outside some set of Lebesgue measure zero, there is an asymptotically equal number of 0s and 1s in the expansion to the base 2. Such numbers are called normal .
We can also think of an infinite sequence of 0s and 1s as the outcome of a coin-tossing "mind experiment", where an unbiased coin is tossed consecutively an infinite number of times, obtaining an infinite sequence of "heads" and "tails", which we may regard as an infinite sequence of 0s and 1s.
Borel's Theorem can then be regarded as saying that the coin tossing experiment will result in an asymptotically equal number of heads and tails, as the coin is tossed ad infinitum ---Borel's Theorem corresponds to the coin being unbiassed. Borel originally proved his Theorem in 1904, but it is widely considered that the proof he gave at that time was incorrect.
In his paper "Les ensembles de mesure nulle et leur role dans l'analyse", (Oeuvres Complètes}, I, Académie des Sciences de Hongrie , Budapest, 1960, 353-372), F. Riesz outlines a proof of Borel's Theorem using the Rademacher functions, an approach he attributes to Khintchine. This approach may also be found in Marc Kac's work "Statistical Independence in Probability, Analysis and Number Theory" ( Carus Mathematical Monographs, no. 12 , Wiley, New York 1959), and in G. Goodman's paper "Statistical independence and normal numbers: an aftermath to Marc Kac's Carus Monograph", (American Mathematical Monthly, 106 (1999), 112-126). Riesz's discussion is only an outline of the proof, while Kac and Goodman both use measure theory in their argument, the crucial step being where they interchange a limit with an integral sign, using the Lebesgue theory of measure and integration.
The purpose in my own paper " Normal numbers without measure theory" (American Mathematical Monthly , 107 (2000), 639-644), was to prove Borel's Theorem on normal numbers without recourse to measure theory. The proof follows Khintchine's idea of using the Rademacher functions, but it only requires a first year university knowledge of sequences and series, and knowledge of the integration of step functions over an interval.
The removal of the requirement for a previous knowledge of measure theory means that Borel's Theorem can be discussed at a second year university course on real analysis---in fact, as first year calculus covers both sequences and series and the integration of step functions, in theory, Borel's theorem can be discussed at first year university level.
In his paper, Goodman uses the Walsh functions as well as the Rademacher functions, to obtain results about the occurrence of a given finite sequence of 0s and 1s in the expansion of numbers to the base 2. Using the approach mentioned above, it is possible to obtain the same results without the necessity of using measure theory.
October 2005