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I want to know if there exist a set $ X\subset \mathbb R$ such that $X$ is

$i)$ Perfect

$ii)$ Compact

$iii)$ Has empty interior

$iv)$ Totally disconnected

$v)$ Is not countable

But $X$ has positive Lebesgue measure.

The sets that are defined with the above properties are called generalized Cantor sets.

Please could you tell me how to construct an explicit example?


marked as duplicate by Cameron Buie, Martin, Brian M. Scott, srijan, Mark Bennet May 30 '13 at 7:06

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  • $\begingroup$ The Wikipedia page on the Smith-Volterra-Cantor set contains an explicit construction with pictures. $\endgroup$ – Martin May 30 '13 at 7:35

Yes there is. Simply modify the construction of the standard ternary set in such a way that you remove an increasing proportion of the remaining intervals with each step.

Note that it's not enough to simply remove a greater proportion than 1/3. You've got to remove an increasing proportion so that the total length of the removed intervals is less than one.

There's a nice discussion of this in Stromberg's Classical Real Analysis.


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