Suppose that $F$ is a nonnegative function that is integrable on $\mathbb R$ and there is a constant $C$ such that
$\int_\mathbb R Ff \leq C\int_\mathbb R f$
whenever $f$ is a nonnegative continuous function on $\mathbb R$ having compact support. Prove that $F(x) \leq C$ for almost all $x$.
I feel like I should use the density of compactly supported continuous functions in $L_1$, but I'm having a tough time getting on the right track. Any help would be appreciated.