Letting $W\subset[0,2\pi]$ be measurable, and letting $(u_n)$ be a sequence of real numbers, how can it be shown that $$\int_W \cos^2(nx+u_n)dx\rightarrow \frac{\mu(W)}{2}$$ as $n\rightarrow \infty$, where $\mu$ is the Lebesgue measure? I know that you can begin by linearizing $\cos^2$.
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$\cos^{2}t=\frac 1 2 (1-\cos (2t))$ and $\int_W \cos (2nx+u_n) dx =\int_W [\cos (2nx) \cos (u_n)-\sin (2nx) \sin (u_n)]dx \to 0$ by Riemann Lebesgue Lemma.
Kavi Rama Murthy
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The trigonometric identity should be $\cos^2 (t)=\frac{1+\cos(2t)}{2}$. – Nugi Apr 14 '23 at 19:16