Two different measures with equal support

Question

Can there be two different complex (hence finite) measures whose support is equal? I presume no, as it may defy some separation theorems otherwise. But I need a concrete proof. Any help is hugely appreciated !

Answer 1

The support of a measure $\mu$ is the complement of the largest open set $U$ with $\mu (U)=0$. If $f$ and $g$ are strictly positive integrable functions on $\mathbb R$ and $\mu (A)=\int_A f(x)dx,\nu (A)=\int_A g(x)dx$ then the support of either measure is $\mathbb R$ (because the only open set $U$ of measure $0$ is the empty set). Take $f(x)=e^{-|x|}, g(x)=e^{-2|x|}$, for example to get a counterexample.