Some generalizations of Kajii's theorem to games with infinitely many players

In this paper, we first generalize Kajii's (1992) result in Hausdorff topological vector spaces. Second, we prove the existence of the finite-coalition α−core for games with infinitely many players. Third, by strengthening some assumptions, we prove the nonemptiness of the weak α−core for games with infinitely many players, Finally, we also characterize the weak α−core by providing a coincidence of the weak α−core and the closed-coalition α−core.