Mokobodzki has shown that any kernel which satisfies the complete maximum principle and the absolute continuity hypothesis has the same excessive functions as a compact kernel. We start with a nice semigroup which satisfies the absolute continuity hypothesis, build for it a coresolvent with the same property, and apply Mokobodzki's result to the coresolvent in order to get a Martin compactification for the original process. Since the coresolvent may not separate points on the state space, we are obliged to throw away a polar Borel set before the Martin procedure applies. We prove a representation theorem for excessive functions, which implies in particular that any measure on the original state space, which does not charge semipolar sets, is equivalent to (i.e. has the same null sets as) a continuous additive functional. This implies the "fine support theorem": any finely perfect set is the fine support of a continuous additive functional, a theorem which has been proved by Mokobodzki and Azema using other methods.
"Une Theorie de la Dualite a Ensemble Polaire Pres I." Ann. Probab. 1 (2) 207 - 222, April, 1973. https://doi.org/10.1214/aop/1176996974