Suppose that $\mu$ and $\nu$ are sigma-finite measures on the Borel sigma-algebra over $\mathbb R$ such that $\int\_{\mathbb R}f\,d\mu=\int\_{\mathbb R}f\,d\nu$ for all nonnegative continuous functions $f$ on $\mathbb R$. Does it always follow that $\mu=\nu$? Some related information can be found at [math.stackexchange.com](https://math.stackexchange.com/questions/155776/to-define-a-measure-is-it-sufficient-to-define-how-to-integrate-continuous-func)



Is a sigma-finite Borel measure over $\mathbb R$ determined by its values on the continuous functions?
Let $\mu$ be counting measure on the rationals and $\nu$ the translate of $\mu$ by an irrational. Then for nonnegative continuous functions $f$ on $\mathbb R$,
$$ \int f \; d\mu = \int f \; d\nu = \cases{0 & if $f \equiv 0$\cr
 +\infty & otherwise\cr}$$


