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Lebesgue measure of Cantor set

An alternative way to define the Cantor set is the following:

    \[\mathfrak{C}=[0,1] \setminus \bigcup_{n=1}^\infty \;\bigcup_{k=0}^{3^{n-1}-1} \left(\frac{3k+1}{3^n},\frac{3k+2}{3^n}\right)\]

What is the Lebesgue measure of the Cantor set if we consider it as a subset of \mathbb{R}? Is \mathfrak{C} countable?


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