Construct an open subset of [0,1] of arbitrary measure dense therein

[benorin]

Q: Construct an open subset E of [0,1] having Lebesgue measure $m(E)=\epsilon$ such that $0<\epsilon<1$ which is dense [0,1].

A: The fat Cantor set. I need help proving it is dense in [0,1]. The usual Ternary expansion argument stuff won't work as the sets used are of length $$\frac{\epsilon}{3^k}$$ at the k^th iteration. Ideas?

The details are in the attached PDF:

 

[BerkMath]

Do you mean proving that [0,1]/C is dense?

 

[benorin]

Yes, sorry. The set at large, namely E, is the complement of the so-called fat Cantor set C in [0,1], i.e. [0,1]\C.

 

[NateTG]

Can you find a (not necessarily open) subset of [0,1] that is dense and has measure zero?