Paradox Regarding Random (Normal) Numbers

I am investigating if some numbers like Pi or SQRT(2) are normal in base 2 or 10, that is, whether any sequence of digits appear with the same frequency in their decimal representation. Actually, I am even more interested in the nested square root representation (see here) where “digits” are either 0 (with probability 0.43), 1 (with probability 0.30) or 2 (with probability 0.23.) Pretty much all numbers have that particular distribution for the nested square root representation.

A famous result claims that pretty much all numbers are normal, in the sense that the set of non-normal numbers, though infinite, has a Lebesgue measure equal to 0. Few non-normal numbers are known, and these cases are artificially manufactured, such as the number 0.10110111011110… It is easy to prove, for instance, that numbers not containing the digit 5 in their base-10 representation (a subset of non-normal numbers) are very rare – so rare that the probability for any real number to have this property, is actually 0: Among the numbers with *n* digits, a proportion *p* = (9/10)^*n* do not contain the digit 5; and as *n* tends to infinity, *p* tends to zero. But this raises interesting questions (**paradoxes**):

- Since a uniform distribution for the digits, is just one example of distribution among many possible distributions, one would think that normal numbers are incredibly rare, as they would be far outnumbered by numbers having any kind of digit distribution, other than uniform. However the converse is true.
- Consider the set S’ of numbers in [0, 1] with all digits being duplicated, such as 0.22774499990033…, and let S be the interval [0,1]. There is a one-to-one mapping (bijection) between S and S’. Yet the Lebesgue measure of S is equal to 1. How come can the Lebesgue measure of S’ be zero? (S’ is a subset of non-normal numbers, for instance no number in S’ contain 123 anywhere in its decimal representation.)
- Thus it seems that pretty much all numbers are “random” in some sense, which is counter-intuitive.

How do you explain these paradoxes? Is it possible to have two sets S and S’ with a bijection between them, have one of them with a non-zero Lebesgue measure, and the other either with a zero measure or un-measurable? Is it possible that the probability for any real number to belong to S’ is zero, yet S’ is allowed to have a strictly positive measure?

These are questions for people interested in measure theory and random numbers.

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