Spivak's Calculus on Manifolds problem (I). Integration.

[ELESSAR TELKONT]

Homework Statement

If $$A\subset\mathbb{R}^{n}$$ is a rectangle show tath $$C\subset A$$ is Jordan-measurable iff $$\forall\epsilon>0,\, \exists P$$ (with $$P$$ a partition of $$A$$) such that $$\sum_{S\in S_{1}}v(S)-\sum_{S\in S_{2}}v(S)<\epsilon$$ for $$S_{1}$$ the collection of all subrectangles $$S$$ induced by $$P$$ such that $$S\cap C\neq \emptyset$$ and $$S_{2}$$ the collection of all subrectangles $$S$$ induced by $$P$$ such that $$S\subset C$$.

Homework Equations

$$C$$ is Jordan-measurable iff $$\partial C$$ has measure zero.
The characteristic function of a set $$C$$ is the function $$\chi_{c}(x)=\left\{\begin{array}{cc}1 &x\in C\\ 0 &x\notin C\end{array}\right .$$

The Attempt at a Solution

I have tried to proof directly (that means I suppose Jordan-measurable). Let $$\epsilon>0$$. Then I use the fact that if $$C$$ is Jordan-measurable then $$\chi_{C}(x)$$ is integrable in $$A$$. That means that $$U(\chi_{C},P)-L(\chi_{C},P)<\epsilon$$.

Now I could write $$U(\chi_{C},P)=\sum_{S\in S_{1}}v(S)+\sum_{S\in S_{2}}v(S)$$ because the supremum of the function at $$S_{1}$$ and $$S_{2}$$ is 1. In the same manner $$L(\chi_{C},P)=\sum_{S\in S_{2}}v(S)$$ because the infimum of the function at $$S_{1}$$ and $$S_{2}$$ is 0,1 respectively... Here my proof brokes since U-L as I have written U and L is $$U-L=\sum_{S\in S_{1}}v(S)<\epsilon$$. I have no idea how to put into that the other sum. Any help that you can provide is precious.

 

[Mathdope]

Does the boundary of C have content zero? That may be the angle to start from (at least for the implication that assumes C is Jordan-measurable).