lebesgue measurability of $\sum_{n=1}^\infty \frac{\cos nx}{n^2}$

Question

Can u show that Lebesgue measurability of $$f(x)=\sum_{n=1}^\infty \frac{\cos nx}{n^2}$$ and $$\int\limits_0^\pi f(x)\,dx=\text{?}$$

Answer 1

The series converges to the pointwise limit of a sequence of continuous functions, so is measurable. The function $f$ is bounded by $\sum_{n=1}^\infty\frac1{n^2}=\frac{\pi^2}6$, so the dominated convergence theorem implies you can swap the sum and integral: $$\int_0^\pi f(x)\ \mathrm{d}x=\sum_{n=1}^\infty\int_0^\pi\frac{\cos nx}{n^2}\ \mathrm{d}x=\sum_{n=1}^\infty\frac1n\sin nx\bigg|_{x=0}^\pi=0.$$