1. ## Measurable Set

Let $n_1,n_2,n_3,...$ be an increasing sequence of positive integers. Let $E = \{x \in [0,2\pi] : lim_{k \to \infty} sin (n_k x)\}$ exists. Prove that $E$ is Lebesgue measurable and evaluate $m(E)$.

Hey this is something really new. How do I approach the question? Have been thinking for quite long and still couldn't get an idea on how to start.

Well, you are looking at a subset of the interval $$0,2\pi$$. What do you know about $sin x$ in that interval? Where is it zero? Where is it one? Where is it somewhere in between? Let's try some points...
$x=0\text{ implies }sin(n_k x)=sin(0)=0$. Therefore, $0 \in E$. What other points are in there? For what points will that limit not exist? For instance, what is $sin(\infty)$? I guess that depends on if infinity is $2\pi$ periodic (hint: it usually is NOT. Unless it is $2\pi$ periodic for all finite examples, then it will NOT be $2\pi$ periodic in the limit.) Figure out when sin is constant for all increasing integers (like we did for 0), and you will find when its limit exists.