I have a signed measure $\mu$ on a convex subset $C\subset \mathbb{R}^n$, and I want to prove that $\mu$ is a probability measure, most importantly that it is positive everywhere.


I do know that $\int f(x)d\mu(x)\geq 0$ for any positive CONVEX function $f$. So if I could get this inequality for indicator functions I'd be done.


Do you know if this suffices to get that the measure is positive, or maybe have a counterexample?



Signed measure that is positive over convex sets
A counterexample is a signed measure on the interval $I:=[-1,1]$ concentrated in the points $\{-1\}$, $\{0\}$, $\{1\}$ with
weights respectively $1/2$, $-1$, $1/2$. (Thus $\int\_If d\mu= f(1)/2+f(-1)/2 - f(0)\ge0$ is just the convexity inequality). 


