Let X be a Borel right Markov process, let m be an excessive measure for X, and let
be the moderate Markov dual process associated with X and m. The potential theory of co-excessive measures (i.e., measures that are excessive for
) is developed with special emphasis on the Riesz decomposition. This is then applied to obtain the Riesz decomposition of excessive functions (of X) by exploiting the correspondence between such functions and co-excessive measures. The potential theory of co-excessive measures also enables us to discuss Walsh’s interior réduite under minimal conditions. Many of the tools of the theory of Markov processes are employed in this development. For example, Kuznetsov measures, Ray compactifications, h-transforms, and duality theory for Borel right processes.