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  1. Let $E \subset \mathbb R$ be a set of finite (Lebesgue) measure and let $\alpha_n \to +\infty$. Then prove $$ \lim_{n} \int_E 2\sin^2{(\alpha_nt)} dt = \mathscr{L}^{1}(E) $$
  2. Using the previous point, if necessary, prove that is $a_n$ is a sequence of real numbers such that $\lim_n \sin(a_n x)$ exists on a set of positive measure, then $a_n$ has a finite limit.

I have some ideas concerning point 1 but I have not gota formal proof. I think we should use the identity $2\sin^2x=1-\cos(2x)$ and - obviusly - the dominated convergence theorem. If this approach is correct then it should be enough to prove that the integral of $\cos{\alpha_nt}$ over $E$ tends to zero...

Concerning the second point I have no idea.

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  • $\begingroup$ Concerning 1., the Riemann-Lebesgue lemma comes to mind. Concerning point 2., let $E = \{ x : \lim_{n\to\infty} \sin (\alpha_n x)\text{ exists}\}$. $\endgroup$ Dec 24, 2013 at 19:52

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For the first question: there is a result that says $E$ can be written as the union $G\cup N$ where $G$ is $G_\delta$ and $N$ has measure zero. Therefore, for any $\varepsilon>0$ there is an open set $U$ such that $|U-E|<\varepsilon.$ Write $U$ as a union of open intervals and on each open interval you have that the integral goes to $0$.

For question 2, try Daniel Fischer's comment.

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