Please help in solving problem:

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?