Integral with parameters, limit permutation and weak convergence in $L^2$

Question

Let $u_n$ a sequence in $L^{2}(\mathbb{R})$ with

$\bullet ~ u_n \overset{*}{\underset{n \to +\infty} \rightharpoonup} u \in L^{2}$ (weakly)

$\bullet \Vert u_n \Vert_{2} < C< +\infty$

Is there a way to prove that

$$ \underset{A \to +\infty}\lim\underset{n \to \infty}\lim \int_{\mathbb{R}\setminus [-A,A]}u_n^{2}=0.$$ (up to a subsequence of $u_n$?)

I think it might be true but I cannot prove it.

Answer 1

This isn't true. Take $u_n = \displaystyle \frac1n \chi_{[-n^2,n^2]}$. This has $\|u_n\|_2^2 =2$ for all $n$ but of course converges weakly to $0$ (check against test functions). But of course, for each $R>0$ we have for $n$ large enough $$\int_{\mathbb{R} \setminus [-R,R]} |u_n|^2 = \int_{[-n^2, -R]}\frac{1}{n^2} + \int_{[R, n^2]} \frac{1}{n^2}$$ $$= \frac{2(n^2-R)}{n^2} \to 2.$$