What is a non-rectifiable bounded closed set in \mathbb{R}?

[GreyZephyr]

Homework Statement

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

Homework 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.

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.

 

[Dick]

Hint: fat Cantor set. Look it up.

 

[GreyZephyr]

Hint: fat Cantor set. Look it up.

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

 

[Dick]

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

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

 

[GreyZephyr]

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

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.