I would like to show the following:
Suppose that $g \in L^1(\mathbb{R}^n)$ and $\int fg \,d \mu = 0$ for any $f \in C_0(\mathbb{R}^n)$. Then $g = 0$ $\mu$-a.e.
I'm stumped on trying to find an approach that works. I can't seem to figure out where the $C_0$ condition would be used. My idea was to show that $||g||_\infty < \infty$, and then by density of $C_c$ we can construct a sequence $s_k \to g$ in $L^1$. Then using Holder, $||s_kg - g^2||_1 \leq ||s_k - g||_1 ||g||_ \infty$, which would then go to $0$, so that $||g||_2 = 0$, which would imply $g = 0$ a.e.
However, I cannot show that $||g||_\infty < \infty$. How can I show this?
Define $g_M := \max(\min(g,M),-M) ∈ L^1 ∩ L^\infty$ and apply your argument modified so that $s_k → \operatorname{sgn} g_M$ in $L^1$ to find that $‖g_M‖_1 = 0$. On the other hand, $|g_M| \uparrow |g| ∈ L^1$, so MCT says $∫|g_M| → ∫ |g|$, which means $∫|g| = 0$.
(Actually since $g_M^2 \uparrow g^2$ as well you don't need to modify your $s_k$ but I'll just leave up the modification for diversity's sake.)