Assume $(X,\tau)$ is a topological space and $\mathcal B (X)$ the corresponding Borel $\sigma$-Algebra. Let $\mu$ and $\nu$ be two Borel measures on $\mathcal B (X)$. Now let $Q$ be some set of measurable functions $f : X \to \mathbb C$.
Under which (minimal) assumptions on the topology $\tau$ or the set $Q$ can we conclude that from
$$ \int_X f \, d\mu = \int_X f \, d\nu \ \ \ \ \ \forall f \in Q $$
follows $\mu = \nu$?