For (S, Σ) a measurable space, let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$${\cal C}_1$$</EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$${\cal C}_2$$</EquationSource> </InlineEquation> be convex, weak<Superscript>*</Superscript> closed sets of probability measures on Σ. We show that if <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$${\cal C}_1$$</EquationSource> </InlineEquation> ∪ <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$${\cal C}_2$$</EquationSource> </InlineEquation> satisfies the Lyapunov property , then there exists a set A ∈ Σ such that min<Subscript>μ1</Subscript>∈<InlineEquation ID="IEq5"> <EquationSource Format="TEX">$${\cal C}_1$$</EquationSource> </InlineEquation> μ<Subscript>1</Subscript>(A) > max<Subscript>μ2</Subscript> ∈ <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$${\cal C}_2$$</EquationSource> </InlineEquation>(A). We give applications to Maxmin Expected Utility (MEU) and to the core of a lower probability. Copyright Springer 2006
