Does $u_n(x) \subset L^p(-1,1)$ converge strongly-weak-weak* to something?

Question

Let $\lbrace u_n(x) := [ \sin(nx)]^+ \rbrace.$ Does this sequence converge in some sense to something in $L^{p}(-1,1)$?

My attempt: for $1<p< +\infty$

$u_n$ is limited so by Banach-Alaoglu + separability of space $\exists u \in L^p(-1,1)$ such that $u_n \rightharpoonup^* u$. Can't handle the case of $p=\infty$ because of is non-separability. For the strong and weak convergence I don't have any idea maybe caused by my lack of solid bases of measure theory and L^p spaces.

I'd appreciate if someone can provide me what are the results\knowledge needed to handle this kind of excercise because I fell like I'm lacking some results

Thanks in advance!

NOTATION: $f(x)^+$ means the positive part of the function.

Answer 1

Note that (the linear hull) of characteristic functions $\chi_{(a,b)}$ for all intervals $(a,b) \subset (-1,1)$ are (is) dense in $L^p(-1,1)$ for $1 \le p < \infty$. Now, try to identify $$ \lim_{n\to\infty} \int_a^b u_n \, dx. $$ Finally, find a function $u$ such that $$ \lim_{n\to\infty} \int_a^b u_n \, dx = \int_a^b u \, dx. $$ Finish by the appropriate density argument.