(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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

2. Relevant equations

[tex]C[/tex] is Jordan-measurable iff [tex]\partial C[/tex] has measure zero.

The characteristic function of a set [tex]C[/tex] is the function [tex]\chi_{c}(x)=\left\{\begin{array}{cc}1 &x\in C\\ 0 &x\notin C\end{array}\right .[/tex]

3. The attempt at a solution

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

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

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

**Physics Forums | Science Articles, Homework Help, Discussion**