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Riesz Lemma

  1. Mar 14, 2006 #1
    Good Morning.

    I am reading the first pages of the "Lessons of Functional Analysis" of Riesz and Nagy, because I learned that this book was the main source for Apostol chapter about the Lebesgue Integral, and I am trying to find a proof of the Fundamental Theorem of Calculus for Lebesgue Integrals worked out within this approach (not Measure Theory).

    One of the first theorems of Riesz-Nagy's book states: "Every monotonic function f(x) posseses a finite derivative at every point x with the possible exception of the points x of a set of measure zero".

    To proof this T they use this lemma:

    "Let g(x) be a continuous function defined in the closed interval (a, b), and let E be the set of points x interior to this interval and such that there exists and e lying to the right of x with g(e) > g(x). Then the set E is either empty or an open set, i.e., it decomposes into a finite number or a denumerable infinity of open and disjoint intervals (a_k, b_k) (the underline is mine), and g(a_k) < g(b_k) for all these intervals".

    Well I suppose I can follow the lemma's proof on the book but in this
    statement there is something I did not know. Why to be E an opet set implies that it decomposes into a finite number or a denumerable infinity of open and disjoint intervals ???

    Thanks for your answers.
     
  2. jcsd
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