Signed measure that is positive over convex sets - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T14:47:25Z http://mathoverflow.net/feeds/question/77937 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/77937/signed-measure-that-is-positive-over-convex-sets Signed measure that is positive over convex sets Henrique de Oliveira 2011-10-12T17:24:00Z 2011-10-12T17:42:54Z <p>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.</p> <p>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.</p> <p>Do you know if this suffices to get that the measure is positive, or maybe have a counterexample?</p> http://mathoverflow.net/questions/77937/signed-measure-that-is-positive-over-convex-sets/77939#77939 Answer by Pietro Majer for Signed measure that is positive over convex sets Pietro Majer 2011-10-12T17:34:52Z 2011-10-12T17:42:54Z <p>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). </p>