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Riemman Stieltjes

  1. Nov 3, 2008 #1
    Let α>0, J:=[-a,a] and f:J→ℝ a bounded function.
    Let α an increases monotonically on J and P* the set of all partitions P of J containing 0 and such that are symmetric,i.e, x in P iff -x in P. Prove that
    fdα= sup L(P,f,α) with P in P*
  2. jcsd
  3. Nov 3, 2008 #2
    It is sufficient to show that for a given e>0, there exist a P* such that fdα-L(P*,f,α)<e.
    Find a P first and let P* be a refinement of P, which is also symmetric and containing 0. Then P* is what you want.
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