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).