Question on rectifiable sets

1. Aug 11, 2010

GreyZephyr

1. The problem statement, all variables and given/known data
I am trying to work my way through Analysis on manifolds by Munkres. Question 14.5 has me stumped. Any hints on how to tackle it would be appreciated. The question is:

Find a bounded closed set in $$\mathbb{R}$$ that is not rectifiable

2. Relevant equations

A subset S of $$\mathbb{R}$$ is rectifiable iff S is bounded and the boundary of S has measure zero.

The boundary of an interval in $$\mathbb{R}$$ has measure zero.

3. The attempt at a solution

I think I need a closed set who's boundary does not have measure zero. I presume it has to be an uncountable union of intervals of some description, but I have no idea how to approach the construction of such a thing.

2. Aug 11, 2010

Dick

Hint: fat Cantor set. Look it up.

3. Aug 11, 2010

GreyZephyr

Thank you. I should have thought of that, but only considered the standard Cantor set. Thanks again.

4. Aug 11, 2010

Dick

Wow! That was fast. I see that "a word to the wise is sufficient" is sometimes true.

5. Aug 11, 2010

GreyZephyr

I have come across fat cantor sets before and my problem was that I could not think of a closed set whose boundary had positive measure. As soon as you gave the hint the rest followed and I felt like a fool. Oh well such is the learning process. Thanks again for the help, I had been stuck on that for a couple of days.