I encounter a problem when reading Struwe's book Variational Methods (4th ed). On page 38, it is assumed that 



> 
> $\|u\_m\|$ is a minimizing sequence for a functional $E$, i.e. $E(u\_m)\rightharpoonup I$ in $L^p(\mathbb{R}^n)$,
> 
> 
> 


and then it assume in addition that



> 
> $u\_m\rightharpoonup u$ weakly in $H^{1,2}(\mathbb{R}^n)$ and pointwise almost everywhere.
> 
> 
> 


My question is


why the pointwise convergence assumption is reasonable? Since $\mathbb R^n$ is not compact, the embedding theorem is not obviously valid.


Thanks in advance.



weak convergence in Sobolev spaces and pointwise convergence?
Just this: for all $r > 0$ the restriction of $u\_m$ to the ball $B\_r$ is compact in $L^1({B\_r})$, hence it has a subsequence converging a.e. there. By a standard diagonal argument, there is a single subsequence of $u\_m$ converging a.e. in $\mathbb{R}^n$.


