A probabilistic representation of exact games on σ-algebras

The purpose of this paper is to establish the intrinsic relations between the cores of exact games on σ-algebras and the extensions of exact games to function spaces. Given a probability space, exact functionals are defined on L∞ as an extension of exact games. To derive a probabilistic representation for exact functionals, we endow them with two probabilistic conditions: law invariance and the Fatou property. The representation theorem for exact functionals lays a probabilistic foundation for nonatomic scalar measure games. Based on the notion of P-convexity, we also investigate the equivalent conditions for the representation of anonymous convex games.