From a previous link in MSE: Prove the set of which sin(nx) converges has Lebesgue measure zero (from Baby Rudin Chapter 11), the question states
Suppose that $\{n_k\}$ is an increasing sequence of positive integers and $E$ is the set of all $x$ in $(-\pi,\pi)$ at which $\sin(n_kx)$ converges. Prove that $E$ has Lebesgue measure zero. Hint: For every subset $A$ of $E$, $\int_A \sin (n_kx) dx$ tends to zero, and $2\int_{A} \sin^2(n_k x)dx$ tends to the measure of $A$.
My question is that does this question assume that $E$ is measurable? How to check the measurability of $E$? And what will be the value of limit of such cosine functions?
$EDIT$: The answer is given in the comment...