Convergence in $L^p(B_r(0)) \implies$ convergence in $L^p(\mathbb{R}^N)$?

Question

I'm trying to understand a certain proof and the author prove that $f_n \to 0$ in $L^2(B_r(0)), \forall r>0$ and he conclude that $f_n \to 0$ in $L^2(\mathbb{R}^N)$. Initially I just think: "Oh, this is trivial". But now I'm not sure if that's really true, because to prove it I should be able to trade in the limits order, which in general needs some pretty strong hypotheses that aren't present in this problem. Can anyone help me?

Answer 1

The assumption $f_n \to 0$ in $L^2(B_r(0)), \forall r>0$ does not even guarantee boundedness in $\mathbb L^2$ of $\left(f_n\right)$: mass can escape at infinity.

To get a concrete counter-example, take $f_n(t)=1$ if $n\leqslant t\leqslant 2n$ and $0$ otherwise. In this way, $\left\lVert f_n\right\rVert_{L^2(B_r(0))}=0$ for $n>r$ but $\left\lVert f_n\right\rVert_{L^2(\mathbb R)}=\sqrt{n}$.