I am currently doing some practice problems for the analysis qual. I have some thoughts on the following problem, and it would be great if someone could see if I am on the right track:
Problem:
Let $O\subset \mathbb{R}^n$ be an open set, and let $f\in L_{loc}^1(\mathbb{R}^n)$ (the set of locally integrable functions on $\mathbb{R}^n$). Assume that for all $\phi \in C_c(O)$ (the set of continuous functions with compact support in $O$), we have $$\int_Of\phi dm = 0,$$ where $dm$ stands for Lebesgue measure. Prove that $f(x) = 0$ for a.e. $x\in O$.
My thoughts are as follows: I want to show by contradiction that if there exists $f \in L_{loc}^1(\mathbb{R}^n)$ s.t. there is a Lebesgue measurable set $E$ with $m(E) > 0$ and $|f| > 0$ on $E$, then one can find some $\phi \in C_c(O)$ s.t. $$\int_Of\phi dm \ne 0.$$ Since $E$ is with positive measure, then by inner regularity of Lebesgue measure we can find a nonempty compact set $K$ s.t. $K \subset U$. Then by Urysohn lemma, there exists a nonnegative function $\phi \in C_c(O)$ s.t. $\phi|_K = 1$ and $\phi$ vanishes outside $O$. Now I claim that $\phi$ does the job: Note that $$\int_O|f|\phi = \int_E|f|\phi \ge \int_K|f|\phi = \int_K|f| > 0,$$ where the last inequality is by the fact that $K\subset E$. But now I don't find a good way to conclude that $$\int_Of\phi dm \ne 0.$$
Could anyone help me fill in this gap or point out which place goes wrong here? Much appreciated in advance!
Best Answer
Hint:
there are compacts sets $K_n$ such that $K_n\subset\operatorname{Int}(K_{n+1})$ and $O=\bigcup_nK_n$.
You may try to show that $f=0$ in $K_n$. Any measurable subset $E$ of $K_n$ can be approximated in $L_1$ by a sequence of continuous functions with support in $\operatorname{Int}(K_{n+1})$ and which are uniformly bounded. Then by dominated convergence $\int f\mathbb{1}_E=0$.
This implies that $f=0$ in $K_n$ for each $K_n$.