Find some extraordinary subset of [0,1]

[Maclaurin]

Homework Statement

I want to find a subset A of [0,1] such that A=cl(int A) and bd A does not have measure zero where cl means closure and int means interior.

Homework Equations

The Attempt at a Solution

I tried some sets, but failed.

 

[micromass]

Do you know the fat Cantor set? http://en.wikipedia.org/wiki/Smith–Volterra–Cantor_set

Try to do something with that. You may have to modify it.

 

[Maclaurin]

I tried that set but it was difficult for me to modify it. Can you give me more hint?

 

[micromass]

Well, you want the fat Cantor set to be the boundary of the set you want.

Every step you leave out some open intervals. Now let V be the union of the open intervals you leave out at the odd steps. Take $\overline{V}$ to be your set.

 

[Maclaurin]

I think bd of V is not the fat Cantor set... what should I do?

 

[Maclaurin]

ah instead of bd of V, boundary of closure of V.