An example of a generalized Cantor set with positive Lebesgue measure

Question

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?

Answer 1

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.