# Homework Help: 2 Rudin Problems

1. Jul 15, 2010

### losiu99

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

2. Relevant equations
1. Hint: Rudin refers to previous results:
(a) For monotonic function $$\alpha$$, measure defined for intervals
$$\mu ((a,b))=\alpha(b-)-\alpha(a+)$$
$$\mu ((a,b])=\alpha(b+)-\alpha(a+)$$
$$\mu ([a,b])=\alpha(b+)-\alpha(a-)$$
$$\mu ([a,b))=\alpha(b-)-\alpha(a-)$$
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
$$\int_A \sin nx\,dx\rightarrow 0$$
for mesurable $$A$$, and
$$2\int_A \sin^2 nx\,dx\rightarrow m(A)$$.

3. The attempt at a solution
1. My guess is that set of discontinuities of $$f$$ must be measure zero with respect to measure defined as above for $$\alpha$$. 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