Weak Closedness of Monotone Sets of Lotteries and Robust Representation of Risk Preferences

We consider a risk preference given by a total preorder < on the setM1,c of probability distributions on R with compact support, that is, a transitive binary relation such that for all μ, ν ∈ M1,c one has μ < ν or μ 4 ν or both. Elements μ ofM1,c are understood as lotteries, and μ < ν means that μ is at least as risky as ν. The goal of the paper is to provide conditions under which < has a numerical representation of the form ρ (μ) = sup l∈L R(l, 〈l, μ〉), (0.1)