# Two different measures with equal support

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 !

• Multiply one measure by some density which is bounded from away and bounded away from zero. – Dirk Dec 10 '18 at 21:19
• What is your definition of support? – DisintegratingByParts Dec 10 '18 at 21:44

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.