If $(f_n)$ is bounded in $L^p$ then $\int_{B_r^C}|f_n|^p dx\to 0$ uniformly in $n$ if $r\to \infty$?

Question

Let $(f_n)$ be bounded in $L^p(\mathbb{R}^n)$. Is it true that $\int_{B_r^C}|f_n|^p dx\to 0$ if $r\to \infty$ uniformly in $n$? If we take $f\in L^p(\mathbb{R}^n)$, then this should be true. Define $c_k=\int_{B_k(0)^C} |f|^p dx$. Then \begin{align} \int_{\mathbb{R}^n} |f|^p dx \geq \sum_{k=1}^{\infty} \int_{B_k^C(0)\setminus B_{k+1}^C(0)} |f|^p dx=\sum_{k=1}^{\infty} c_k-c_{k+1}. \end{align} Now since $c_k-c_{k+1}\geq 0$ for all $k$ we know that $c_k-c_{k+1}\to 0$ as $k\to \infty$. Hence $c_k\geq c_{k+1}$ and $c_k<\infty$ for all $k$ so that $c_k\to c$ for some $c$. It follows \begin{align} c_k=\int_{B_k^C(0)}|f|^p dx = \sum_{l=k}^{\infty} \int_{B_{l}^C(0)\setminus B_{l+1}^C(0)} |f|^p dx=\sum_{l=k}^{\infty} c_l-c_{l+1}\to 0. \end{align} Now, if we try to give this proof in the case of seqences, define $c_k^n=\int_{B_k(0)^C} |f_n|^p dx$. Then \begin{align} \int_{\mathbb{R}^n} |f_n|^p dx \geq \sum_{k=1}^{\infty} \int_{B_k^C(0)\setminus B_{k+1}^C(0)} |f_n|^p dx=\sum_{k=1}^{\infty} c_k^n-c^n_{k+1}. \end{align} Now since $c_k^n-c_{k+1}^n\geq 0$ for all $k$ and all $n$ we know that $c_k^n-c_{k+1}^n\to 0$ as $k\to \infty$ for any fixed $$. Hence $c_k^n\geq c_{k+1}^n$ and $c_k^n<\infty$ for all $k,n$ so that $c_k^n\to c^n$ for some $c^n$. It follows \begin{align} c_k^n=\int_{B_k^C(0)}|f|^p dx = \sum_{l=k}^{\infty} \int_{B_{l}^C(0)\setminus B_{l+1}^C(0)} |f_n|^p dx=\sum_{l=k}^{\infty} c_l^n-c_{l+1}^n\to 0 \end{align} only for fixed $n$. Do we need uniform convergence of $c_k^n$? I don't think that the boundedness is enough, but I would be grateful for any help!

Answer 1

In $\mathbb R$ take $f_k=k\chi_{(k,k+\frac1 k)}$. Take $p=1$. Then, $\|f_k||_p=1$ for all $n$ but $\sup_k \int_{(-r,r)^{c}} |f_k|^{p}=1$ for each $r$.