What is the measure of the following set of infinite binary words?

$S=\{w\in\{0,1\}^\omega\ \text{such that},\ \text{for every}\ N\in\mathbb{N},\, w\ \text{has a prefix of the form}\ pp\ \text{with}\ |p|\ge N \}$.

  • $\begingroup$ What context does this problem arise in? $\endgroup$ – Carl-Fredrik Nyberg Brodda Dec 14 '20 at 12:55
  • $\begingroup$ I am trying to understand what can be reasonably taken as the domain of the map $\mathcal{B}$ described in this other question: mathoverflow.net/q/377105/167834 (although the link may be not completely obvious, it is not too difficult to see that this question here is basic for the other one). $\endgroup$ – Alessandro Della Corte Dec 14 '20 at 13:58

For any given $p\in\{0,1\}^*$ (finite word), the measure of the set of $w\in\{0,1\}^\omega$ starting with $pp$ is $2^{-2|p|}$. So, summing over all the $p$ with $|p|=L$ (there are $2^L$ of them), given $L\in\mathbb{N}$, the measure of the set $S_L$ of $w\in\{0,1\}^\omega$ starting with $pp$ for some $p$ with $|p|=L$ is $2^{-L}$. So the measure of $\bigcup_{L=N}^{+\infty} S_L$ is $\leq \sum_{L=N}^{+\infty} 2^{-L} = 2^{1-N}$, and the measure of $S = \bigcap_{N=0}^{+\infty} \bigcup_{L=N}^{+\infty} S_L$ is $0$.


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