This thesis explores the relationship between measures and amalgamation properties in simple ω-categorical structures. In particular, we prove that a stronger version of the independence theorem is satisfied by simple ω-categorical structures where a formula forks over ∅ if and only if it is assigned measure zero by every invariant Keisler measure. This will be used to show that certain classes of simple ω-categorical Hrushovski constructions have non-forking (over ∅) formulas which are assigned measure zero by every invariant Keisler measure. Until a recent article of Chernikov, Hrushovski, Kruckman, Krupinski, Moconja,
Pillay and Ramsey, it was unknown whether there were any simple structures where forking and being universally measure zero disagreed. We give the first simple ω-categorical example of this phenomenon.
We also study ω-categorical MS-measurable structures in the sense of Macpherson and Steinhorn. Our main results show that various classes of ω-categorical Hrushovski constructions and the generic tetrahedron-free 3-hypergraph fail to be MS-measurable in spite of being supersimple of finite rank. These examples answer negatively a question of Elwes and Macpherson on whether being supersimple ω-categorical of finite SU-rank implies being MS-measurable. Our results on ω-categorical Hrushovski constructions complement previous work of Evans who had given a counterexample to this question. Unlike Evans’ example, our Hrushovski constructions may satisfy independent n-amalgamation for all n over finite algebraically closed sets. We also develop tools to study the measures of higher amalgamations in MS-measurable structures. With these, we show that the generic tetrahedron-free 3-hypergraph is not MS-measurable giving the first one-based ω-categorical supersimple example of this phenomenon and showing that this structure cannot be elementarily equivalent to the ultraproduct of an asymptotic class of finite structures. Finally, we show that the universal homogeneous two-graph has a unique invariant Keisler measure not induced by any invariant type.