Determining measures by integrals What classes of functions are sufficient to determine whether two measures are equal? If 
$$\int_{R^d} f d\mu =\int_{R^d} f d\nu $$
for some functions $f$, when can we say that $\mu=\nu$?
Obviously, if $f$ can be any indicator function this is easy but is there a theorem for continuous functions say? Also, what about if we integrate over more general spaces, e.g. a Banach space?
Thanks.
EDIT: Extra question - According to Stefan in the comments, in the special case that $(S,d)$ is a metric space and μ and ν are Borel probability measures, then the space of non-negative bounded and continuous functions works. Please can somebody suggest a good reference for this or a reason why this is true?
 A: 
Let $(S,d)$ be a metric space and let $\mu$, $\nu$ be two Borel probability measures on $S$. Then
  $$
\int f\,\mathrm d\mu=\int f\,\mathrm d\nu,\;\forall\,f\in \mathrm{bC}(S)_+\iff\mu=\nu.
$$

Proof: Assume that the integrals of every function in $\mathrm{bC}(S)_+$ are identical. Then to show $\mu=\nu$, it is enough to show that $\mu(U)=\nu(U)$ for all $U\subseteq S$ open. For each $n\geq 1$ we define
$$
f_n(x):=n\cdot d(x,U^c)\wedge 1,\quad n\geq 1.
$$
Then the $f_n$'s are bounded, non-negative and continuous (they are in fact Lipschitz continuous) and $f_n\uparrow 1_U$ pointwise. Thus,
$$
\mu(U)=\int 1_U\,\mathrm d\mu=\lim_{n\to\infty}\int f_n\,\mathrm d\mu=\lim_{n\to\infty}\int f_n\,\mathrm d\nu=\nu(U).
$$

We can relax the assumption that $\mu$ and $\nu$ need to be probability measures. The above also holds in the more general setting where $\mu$ and $\nu$ are just measures such that there exists a sequence $(A_n)_{n\geq 1}$ of open Borel sets such that 
$$
S=\bigcup_{n\geq 1}A_n,\quad \text{and}\quad \mu(A_n)=\nu(A_n)<\infty,\quad n\geq 1.
$$
A: If is enough to check the equality of integrals with $f(x) = e^{i\langle x,\theta\rangle}$ for every $\theta \in \Bbb R^d$. This is the Fourier transform.
The proofs of both this and the general result given by Stefan can be found in Billingsley's Convergence of Probability Measures.
