I'm supposed to show that the jordan content is $0$. The definition for a set $S$ having jordan content zero I have to work with is :$\forall\epsilon>0$ there is a finite collection of generalized rectangles in $\mathbb{R}^n$ that covers $S$ the sum of these rectangles volumes being less that $\epsilon$.
So I have the set $S = \{(x,y,z) \in \mathbb{R}^3\mid x^2 + y^2 + z^2 =1\}$. I can't just build $1 \cdot 1 \cdot \frac{1}{k}$ boxes (with $k > \frac{1}{\epsilon}$) and call it a day because the sum obviously isn't going to be less than epsilon for all epsilon.
I think another approach would be to try and build a sequence of generalized rectangles such that the first sequence has two boxes each covering half of the sphere and then sub divide and keep only the boxes that intersect the sphere. But I'm having trouble formalizing this idea.
Can somebody point me in the right direction / give any advice?
Thanks in advance.
Best Answer
Your rectangles need to get small in all three dimensions. Roughly speaking, you will have cubes of size $\frac 1k$, volume $\frac 1{k^3}$ you will need of order $4 \pi k^2$ of them, so the total volume is of order $\frac {4 \pi}k$. You should be able to find a collection of cubes that cover the sphere. Even if it is larger than that, as long as it doesn't grow too fast with $\frac 1k$ you are set.