A basic measure theory question on lebesgue integral Let $\mu$ and $\nu$ are probability measures on a complete separable space $S$. Suppose, for every real-valued continuous function on $S$ we have that
$$\int fd\mu = \int fd\nu$$
does it imply $\mu = \nu$. looks like approximating characteristic functions by continuous functions. 

Let $U$ be open and consider
$$
f_n(x)=(n\cdot d(x,U^c))\wedge1,\quad n\geqslant 1.
$$
Then $f_n$ is non-negative and continuous for all $n\geqslant 1$ and $f_n\uparrow \mathbf{1}_U$. Thus by the monotone convergence theorem we have $\mu(U)=\nu(U)$. Since the Borel sigma-algebra is generated by the open sets (which are stable under intersection) we have that $\mu=\nu$.
