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Homework Help: 2 Rudin Problems

  1. Jul 15, 2010 #1
    1. The problem statement, all variables and given/known data
    1. Find necessery and sufficient condition for Riemann-Stieltjes integral [tex]\int_a ^b f\,d\alpha[/tex] to exist.
    2. Suppose [tex]E\sub (-\pi, \pi), m(E)>0, \delta > 0[/tex]. Use Bessel inequality to prove that there are at most finitely many integers n such that [tex]\sin nx > \delta[/tex] for all [tex]x\in E[/tex].

    2. Relevant equations
    1. Hint: Rudin refers to previous results:
    (a) For monotonic function [tex]\alpha[/tex], measure defined for intervals
    [tex]\mu ((a,b))=\alpha(b-)-\alpha(a+)[/tex]
    [tex]\mu ((a,b])=\alpha(b+)-\alpha(a+)[/tex]
    [tex]\mu ([a,b])=\alpha(b+)-\alpha(a-)[/tex]
    [tex]\mu ([a,b))=\alpha(b-)-\alpha(a-)[/tex]
    can be extended to regular measure on sigma-ring of sets.
    (b) Function is Riemann-integrable iff it's continuous almost everywhere.

    2. Don't know if it's related, but previous exercise established
    [tex]\int_A \sin nx\,dx\rightarrow 0[/tex]
    for mesurable [tex]A[/tex], and
    [tex]2\int_A \sin^2 nx\,dx\rightarrow m(A)[/tex].

    3. The attempt at a solution
    1. My guess is that set of discontinuities of [tex]f[/tex] must be measure zero with respect to measure defined as above for [tex]\alpha[/tex]. Still, I'm pretty much stuck when it comes to prove it. Rudin's proof of Lebesgue's criterion was based on easy approximation of Darboux sums, which doesn't work that well for Stieltjes:(

    2. No clue at all, I can't figure out what to bound with Bessel.

    I would be grateful for any help with these.

    Edit: I think I vanquished the first one. Any hints on the second appreciated.
     
    Last edited: Jul 15, 2010
  2. jcsd
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