Risk Measure Duality Without Structure

We study risk measures on vector spaces of random variables which a priori have little structure, such as spaces lacking law invariance or a lattice structure. Ensuring the existence of a tractable dual representation (one which does not contain non-sigma-additive measures) is one of the main problems in risk measure theory, and we address it under minimal conditions. The existence of a tractable dual representation is shown to be equivalent to a Fatou-like property when the domain of the risk measure satisfies a topological regularity condition. Without the topological regularity condition, the Fatou property implies the existence of a tractable dual representation whenever the risk measure is viewed with constraints. We also present counterexamples demonstrating the sharpness of the assumptions made.