Take $\alpha >0$ and for all $x\in\mathbb{R}^{d}$ define the sequence:
$$u_{n}(x)=\min\{1,|x|^{-\alpha}\}\chi_{B(0,n)}(x)$$ where $\chi$ denotes the characteristic function. I have to study strong and if necessary also weak (weak* for $p=+\infty$).
It looks pretty clear to me that: $$u_{n}(x)\longrightarrow u(x)=\min\{1,|x|^{-\alpha}\}$$ that is to say there is pointwise convergence on all $\mathbb{R}^{d}$. We understand also that we may suppose $\alpha$ big enough so that indeed $u_{n}\in L^{p}(\mathbb{R}^{d})$. Set $B_{n}=B(0,n)\setminus B(0,1)$, now my problem is: are we entitled to say
$$\int_{B_{n}}|x|^{-\alpha}dx\longrightarrow \int_{\mathbb{R}^{d}\setminus B(0,1)}|x|^{-\alpha}dx$$ If yes then there should be no problems regarding strong convergence in $L^{p}(\mathbb{R}^{d})$ for $p\neq+\infty$, while I suppose in this last case we may have strong convergence as well but through a different idea