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Changing the lengths of the intervals excluded during the construction of the ternary Cantor set, show that is possible to build a compact, totally disconnected and perfect set (a Cantor set) with positive Lebesgue measure.

I have no idea on how to solve this question. It seems also counter-intuitive to me there is a Cantor set whose Lebesgue measure is positive.

Question:

How should I solve the exercise?

Thanks in advance!

Pedro Gomes
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  • Such a set is called a "fat Cantor set," and can be constructed by removing the middle $1/2^{n+1}$ of each interval, where $n$ is the stage in the construction; i.e., first the middle fourth, then the middle eighth, then the middle sixteenth, and so on. – Plutoro Nov 30 '18 at 19:02
  • @AlexS But the measure is going to be zero which is not positive right? – Pedro Gomes Nov 30 '18 at 19:09
  • Do a simple Google search for "fat Cantor set." you will find many sources showing how such a thing is constructed, and how the measure is positive, not zero. – Plutoro Nov 30 '18 at 19:14

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