The author studies the structure and convergence properties of amarts consisting of random variables or countably additive set functions taking their values in a Fréchet space \({\mathbb{E}}\). In particular, amarts of countably additive set functions are characterized in terms of the difference property, the Riesz decomposition, and the existence of a finitely additive limit set function. This extends results obtained by the reviewer in the case where \({\mathbb{E}}\) is a Banach space [\textit{A. Gut} and the reviewer, Amarts and set function processes. Lect. Notes Math. 1042 (1983; Zbl 0525.60055)]. Furthermore, several characterizations of the nuclearity of \({\mathbb{E}}\) are given in terms of vector measures and potentials of random variables. These results are related to earlier characterizations of nuclearity obtained by \textit{L. Egghe} [J. Funct. Anal. 35, 207-214 (1980; Zbl 0422.46001) and J. Multivariate Anal. 12, 291-305 (1982; Zbl 0485.60003)].