Suppose I have a class of functions $\mathcal{F}$ with the property that
$\int f(x) g(x) = \int f(x) h(x)$ for all $f \in \mathcal{F}$ implies $g = h$.
What's the correct name for this property? If $g$ and $h$ are in $L^p$, do I say that $\mathcal{F}$ is a seperating set, or that it seperates points in $L^p$, or something else?
If $\mathcal{F}$ is the class of smooth functions with compact support and $g$ and $h$ live in $L^p$, is the implication correct? If so, what's that result called?