Higher amalgamation properties in measured structures

Using an infinitary version of the Hypergraph Removal Lemma due to Towsner, we prove a model-theoretic higher amalgamation result. In particular, we obtain an independent amalgamation property which holds in structures which are measurable in the sense of Macpherson and Steinhorn, but which is not generally true in structures which are supersimple of finite SU-rank. We use this to show that some of Hrushovski's non-locally-modular, supersimple $\omega$-categorical structures are not MS-measurable.