Excessive kernels and Revuz measures

We consider a proper submarkovian resolvent of kernels on a Lusin measurabl e space and a given excessive measure xi. With every quasi bounded excessiv e function we associate an excessive kernel and the corresponding Revuz mea sure. Every finite measure charging no xi-polar set is such a Revuz measure , provided the hypothesis (B) of Hunt holds. Under a weak duality hypothesi s, we prove the Revuz formula and characterize the quasi boundedness and th e regularity in terms of Revuz measures. We improve results of Azema [2] an d Getoor and Sharpe [20] for the natural additive functionals of a Borel ri ght process.