Before revision:  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 .
After revision:  We provide a variety of finiteness and continuity results for general risk measures, as well as dual representations for the convex case. We apply our results to risk measures based on Value-at-Risk and Tail Value-at-Risk on L^p spaces, as well as to shortfall risk measures based on utility functions on Orlicz spaces. We pay special attention to the case where the underlying space of positions is a Fr\'{e .