Let $u_n\rightharpoonup u$ in $L^2(\Omega)$, where $\Omega$ is any domain in $\mathbb R^n$. Is there any relationship between the supports of $u_n$ and $u$? The reason I am asking is that I remember seeing somewhere an assertion that, for a weakly (star) convergent sequence of measures, the support of the limit is a subset of the Kuratowski limit inferior (or superior, I cannot remember exactly) of the supports of the sequence members.

Edit: there is an "obvious" relation, namely $\mathrm{supp}\, u\subseteq \cup_n \mathrm{supp}\, u_n$. I would be interested in something more "optimal".


1 Answer 1


Ok, so we want something like $\operatorname{supp} u \subseteq \cap_{N=1}^\infty \cup_{n=N}^\infty \operatorname{supp} u_n$. Here we go:

$(u_n)_{n \in \mathbb{N}}$ converges weakly to $u$, and so does every restricted sequence $(u_n)_{n \geq N}$. $A_N = \{ f \in L^2; \operatorname{supp} f \subseteq \cup_{n=N}^\infty \operatorname{supp}u_n\}$ is a closed subspace of $L^2$, and closed implies weakly closed. Therefore $u$ needs to be in $A_N$ for each $N$. In particular,

$$ \operatorname{supp} u \subseteq \cap_{N=1}^\infty \cup_{n=N}^\infty \operatorname{supp} u_n.$$

Edit: A sketch of why I think the statement will be wrong if we replace the lim sup by lim inf:

Consider $L^2[0,1]$ and let the functions $f_{k,n}$ be defined by $$ f_{k,m} = 1_{\big [ \frac{k}{m},\frac{k+1}{m}\big ]}$$ for $m \in \mathbb{N}$ and $k\in \mathbb{N}, 0 \leq k < m$. Enumerate them somehow as $(f_n)_{n \in \mathbb{N}}$. Then $f_n \to 0$ in norm, hence $f_n \to 0$ weakly. Consider $$ u_n = 1 - f_n.$$ Then $u_n \to 1$ weakly. But on the other hand, for each $x \in [0,1]$ there are infinitely many $n \in \mathbb{N}$ such that there is an open interval $I(n)$ containing $x$ and $\operatorname{supp} u_n \cap I(n) = \emptyset$. Therefore $\lim \inf \operatorname{supp} u_n = \emptyset$ and $\operatorname{supp} u = [0,1]$.

  • $\begingroup$ True, but there is a relation: $\emptyset \subset [0,1]$ :-). I am more of thinking of the case where the supports of the sequence members wander around inside $\Omega$... $\endgroup$
    – tks
    May 12, 2017 at 6:58
  • 1
    $\begingroup$ Alright, in that sense you always have the relation $\operatorname{supp} u \subseteq \cup_{n} \operatorname{supp} u_n$ ;-) $\endgroup$
    – agb
    May 12, 2017 at 7:09
  • $\begingroup$ Yes, the question is if it is possible to put a \cap somewhere there in between.... $\endgroup$
    – tks
    May 12, 2017 at 7:11
  • $\begingroup$ I've modified the question to be more precise... $\endgroup$
    – tks
    May 12, 2017 at 7:14
  • $\begingroup$ Ok, I've modified my answer. Maybe this is more what you are looking for. $\endgroup$
    – agb
    May 12, 2017 at 7:19

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.