Given a topological space $X$ with a Borel $\sigma$-algebra $\mathcal{B}_X$, under what conditions can we say that a Radon meausre is determined by integrals of compactly supported continuous functions?
I'm pretty sure that under the Riesz-Markov-Kakutani theorem, locally compact Hausdorff spaces and regular measures are okay, but is this the weakest setting without which we have a counter-example?
My original question that I am intrested in is whether given a metric space $(X,d)$ and Radon meausre $\mu$ and $x_0\in X$, can we determine the measure $\mu$ if we know for all $r>0$, $\int_{X} f(x)d\mu(x)$ for all continuous $f\in C_c(X;\mathbb{R})$ such that $\text{supp}(f)\subseteq B_r(x_0)$?