Integral of $L^{p}(\mathbb{R}^{n})$ sequence outside a large ball

Question

I've seen the question

Integral of L^1 function outside a large ball

Could it be generalized for a sequence? Specifficaly, let $(u_{j})$ a bounded sequence in $L^{p}(\mathbb{R}^{n})$, I would like to prove (or give a counter-example) that

\begin{equation*} \lim_{R\rightarrow+\infty}\lim_{j\rightarrow +\infty}\int_{\mathbb{R}^{n}\setminus B_{R}(0)}|u_{j}|^{p}=0. \end{equation*}

My attempt: Since $(u_{j})$ is bounded in $L^{p}(\mathbb{R}^{N})$, there exists a constant $C>0$ such that

$$\int_{\mathbb{R}^{n}\setminus B_{R}(0)}|u_{j}|^{p}\leq C,$$

for all $j \in \mathbb{N}$ and some $R>0$. Now, choose $g \in C_{0}^{\infty}(\mathbb{R}^{N})$ with support $K$ outside of $B_{R}(0)$, and

$$ \int_{\mathbb{R}^{n}\setminus B_{R}(0)}g \geq C.$$

Hence,

$$ \lim_{R\rightarrow+\infty}\lim_{j\rightarrow +\infty}\int_{\mathbb{R}^{n}\setminus B_{R}(0)}|u_{j}|^{p} \leq \lim_{R\rightarrow+\infty} \int_{\mathbb{R}^{n}\setminus B_{R}(0)}g = 0.$$

I'm not sure if my argument is correct. I appreciate any help

Answer 1

Counter example: Let $u$ be a unit $L^p$ norm function with support in say $B(0,1)$ and consider $u_j(x):=u(x-j)$ which has support in $B(j,1)$. For any fixed $R$ the integral is $1$ for sufficiently large $j$ (any $j>R+2$ will do) and so this is the inner limit, which is independent of $R$. Hence the double limit is $1$, which not $0$.