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



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


