For (S, Σ) a measurable space, let ${\cal C}_1$ and ${\cal C}_2$ be convex, weak* closed sets of probability measures on Σ. We show that if ${\cal C}_1$ ∪ ${\cal C}_2$ satisfies the Lyapunov property , then there exists a set A ∈ Σ such that minμ1∈ ${\cal C}_1$ μ1(A) > maxμ2 ∈ ${\cal C}_2$ (A). We give applications to Maxmin Expected Utility (MEU) and to the core of a lower probability.

We establish a simple condition, based on the willingness to bet on events, under which two biseparable preferences have cardinally equivalent utilities.