The concave integral over large spaces

This paper investigates the concave integral for capacities defined over large spaces. We characterize when the integral with respect to capacity v can be represented as the infimum over all integrals with respect to additive measures that are greater than or equal to v. We introduce the notion of loose extendability and study its relation to the concave integral. A non-additive version for the Levi theorem and the Fatou lemma are proven. Finally, we provide several convergence theorems for capacities with large cores.