(adsbygoogle = window.adsbygoogle || []).push({}); 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.

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# Homework Help: 2 Rudin Problems

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