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

  1. Mar 3, 2008 #1
    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.
  2. jcsd
  3. Mar 4, 2008 #2
    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).
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