A question on locally integrable function on $mathbb{R}^n$

Mathematics Asked by Kevin_H on December 8, 2020

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 $$Osubset mathbb{R}^n$$ be an open set, and let $$fin L_{loc}^1(mathbb{R}^n)$$ (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 $$int_Ofphi dm = 0,$$ where $$dm$$ stands for Lebesgue measure. Prove that $$f(x) = 0$$ for a.e. $$xin O$$.

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. $$int_Ofphi dm ne 0.$$ 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 $$Ksubset E$$. But now I don’t find a good way to conclude that $$int_Ofphi dm ne 0.$$
Could anyone help me fill in this gap or point out which place goes wrong here? Much appreciated in advance!

Hint:

there are compacts sets $$K_n$$ such that $$K_nsubsetoperatorname{Int}(K_{n+1})$$ 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 $$int fmathbb{1}_E=0$$.

This implies that $$f=0$$ in $$K_n$$ for each $$K_n$$.

Correct answer by Oliver Diaz on December 8, 2020