1. ## Measurable Set

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

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 $\displaystyle $0,2\pi$$. What do you know about $\displaystyle sin x$ in that interval? Where is it zero? Where is it one? Where is it somewhere in between? Let's try some points...
$\displaystyle x=0\text{ implies }sin(n_k x)=sin(0)=0$. Therefore, $\displaystyle 0 \in E$. What other points are in there? For what points will that limit not exist? For instance, what is $\displaystyle sin(\infty)$? I guess that depends on if infinity is $\displaystyle 2\pi$ periodic (hint: it usually is NOT. Unless it is $\displaystyle 2\pi$ periodic for all finite examples, then it will NOT be $\displaystyle 2\pi$ 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.