# Integral inequality

We consider the n-point space of elementary outcomes $\Omega$, it is given a convex set of nonnegative functions $\xi$. The probability of each point is $1/n$. It is known that $P\{\xi>1\}<=1/n$, where $\{\xi>1\}=\{x\in \Omega | \xi(x)>1\}$. Prove that there exist a point $z$ such that $\int_{\Omega\setminus z}\xi dP<=1$.
$P\{\xi>1\}<=1/n$ this implies that no more than one point function may be greater than 1, so the conclusion seemed obvious. But it comes to throwing a universal reference point for all functions at once, so this evidence does not pass. What then?