
Integral inequality
Please help in solving problem:
We consider the npoint space of elementary outcomes $\displaystyle \Omega$, it is given a convex set of nonnegative functions $\displaystyle \xi$. The probability of each point is $\displaystyle 1/n$. It is known that $\displaystyle P\{\xi>1\}<=1/n$, where $\displaystyle \{\xi>1\}=\{x\in \Omega  \xi(x)>1\}$. Prove that there exist a point $\displaystyle z$ such that $\displaystyle \int_{\Omega\setminus z}\xi dP<=1$.
$\displaystyle P\{\xi>1\}<=1/n$ 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?