Before revision:  Moreover, it allows to unveil the key properties of acceptance sets and of eligible assetsdriving the effectiveness and the robustness of the associated capital requirements. In particular, if the underlying space is a Fr\'echet lattice, we provide new finiteness results for required capital, as well as a simplified proof of the Extended Namioka-Klee theorem for risk measures .
After revision:  Moreover, it allows to unveil the key properties of acceptance sets and eligible assets. We provide a variety of finiteness and continuity results when the eligible asset has a payoff with "interior-like" qualities, paying particular attention to the case where the underlying space of positions is a Fr\'{e .