Does is hold that $f \ge 0$ almost everywhere if $\int_a^b f \varphi dx = 0$ for all $\varphi \in C^\infty_0((a,b); [0,1])$?

Question

Let $(a,b) \subseteq \mathbb{R}$. Suppose $f \in L^1_{\text{loc}}(a,b)$ is real-valued. Suppose that for all $\varphi \in C^\infty_0((a,b); [0,1])$ it holds that $\int_a^b f \varphi dx \ge 0$.

Does it hold that $f \ge 0$ almost everywhere on $(a,b)$?

For each compact subinterval $[c,d] \subseteq (a,b)$, $\int_c^d f dx \ge 0$. This follows by the dominated convergence theorem, from choosing a sequence of bump functions $\varphi_n \in C^\infty_0((a,b);[0,1])$ which are identically one on $[c,d]$ and supported in $(c- 1/n, d + 1/n)$.

Now I guess we need to approximate the set $\{f < 0\}$ using open intervals to conclude $\int_{\{f < 0\}} f dx = 0$. Is there some standard theory from measure theory that says $\{f < 0\}$ can be suitably approximated?

Answer 1

I'm going to assume that $f$ is integrable on $(a,b)$.

Consider the function $\mathbb{1}_{\{f<0\}}$. Multiplying by a cut-off function and mollifying, we see that there exists a sequence $(\varphi_n)\subseteq C^\infty_0((a,b);[0,1])$ that converge to $\mathbb{1}_{\{f<0\}}$ in $L^1((a,b))$. Thus, up to a subsequence, $\varphi_n \to \mathbb{1}_{f<0}$ almost everywhere.

By the dominated convergence theorem $$ \int_{(a,b)} f\varphi_n \to \int_{\{f < 0\}} f = 0. $$

So the theory you need is on mollifiers!