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Let $1\leq p < \infty$ and $\Omega\subset\mathbb{R}^n$ be a measurable (Lebesgue) set. I know that $L_0^p(\Omega)\cap L^p(\Omega)$ is dense in $L^p(\Omega)$ when $m(\Omega)$ is finite. For the proof I used the absolute continuity of the integral and the fact that $\Omega$ can be approximated with a compact set.

Now I'm wondering if the claim is true when $m(\Omega)=\infty$. Obviously, I cannot use the same proof that I used in the case where $m(\Omega)<\infty$. If it is true, could you give a hint from where to start?

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Sure it is. Since $\mathbb{R}^n$ is $\sigma$-finite. Intersect your $\Omega$ with sets of the form $\{n<|x|<n+1\}$ and approximate on each annulus with error at most $\epsilon/2^n$, then you will see the total error will be less than $\epsilon$.

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  • $\begingroup$ Alright I give it a try. So I have $u\in L^p(\Omega)$ and I intersect my $\Omega$ with the sets you gave me. In each of those sets I can find a compact set $K_n$ such that $$ \int_{\{n < |x| <n+1\}} |u(x)-u(x)\chi_{K_n}(x)|^p \ dx < \varepsilon/2^n.$$ Since the sets create a partition of $\Omega$ I finally sum all the integrals and have $||u-u\chi_{\cup K_n}||^p_p<\varepsilon$. Since the countable union of compact sets is compact $u\chi_{\bigcup_{n=1}^{\infty}K_n} \in L_0^p(\Omega)$ . $\endgroup$ – xXxSniper666xXx Sep 21 '16 at 6:14
  • $\begingroup$ What are the $\chi$s? $\endgroup$ – Jacky Chong Sep 21 '16 at 6:15
  • $\begingroup$ $\chi_K$ is a characteristic function of the set $K$. $\endgroup$ – xXxSniper666xXx Sep 21 '16 at 6:31
  • $\begingroup$ How's $u\chi_{K_n} \in L^p_0(\Omega)$? $\endgroup$ – Jacky Chong Sep 21 '16 at 6:32
  • $\begingroup$ $$ \int_{\Omega} |u(x)\chi_{\bigcup_{n=0}^{\infty}K_n}(x)|^p \ dx \leq \int_{\Omega} |u(x)|^p \ dx = ||u||_{L^p(\Omega)}^p < \infty$$ which means that $$u \chi_{\bigcup_{n=0}^{\infty} K_n} \in L^p(\Omega)$$. Now $K_n\subset \Omega$ is compact for all $n\in \mathbb{N} \cup \{0\}$. Hence $$ K:=\bigcup_{n=0}^{\infty} K_n $$ is compact and $\subset \Omega$. Since $K$ is the support of $u\chi_{\bigcup_{n=0}^{\infty}K_n}$, $u\chi_{\bigcup_{n=0}^{\infty}K_n}\in L_0^p(\Omega)$. $\endgroup$ – xXxSniper666xXx Sep 21 '16 at 6:53

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