  In this paper, we show that the Euler characteristic of an even dimensional
closed projectively flat manifold is equal to the total measure which is
induced from a probability Borel measure on RP^n invariant under the holonomy
action, and then discuss its consequences and applications. As an application,
we show that the Chern's conjecture is true for a closed affinely flat manifold
whose holonomy group action permits an invariant probability Borel measure on
RP^n; that is, such a closed affinly flat manifold has a vanishing Euler
characteristic.
