Joint work with Colin Jahel and Samuel Braunfeld.

We study ways in which we can expand a homogeneous relational structure in a "random" way. The most basic example of this, studied by Ackerman, Freer and Patel (2016) and many others, are measures on the space of \(\mathcal{L}\)-structures on a countable set which are invariant under the action of \(S_\infty\). For a fixed homogeneous structure we study \(\mathrm{Aut}(\mathcal{M})\)-invariant measures on the space of expansions of \(\mathcal{M}\) to a language \(\mathcal{L}'\). In particular, we are interested in the case of \(\mathcal{M}\) being a homogeneous \(k\)-hypergraph, for which we study \(\mathrm{Aut}(\mathcal{M})\)-invariant expansions by a lower arity hypergraph. Heavily modifying techniques of Angels, Kechris and Lyons (2014), we are able to prove that in various cases these \(\mathrm{Aut}(\mathcal{M})\)-invariant measures are actually \(S_\infty\)-invariant. Invariant Keisler measures on a given homogeneous structure are a special case of the context we study. Hence, we are able to describe the spaces of invariant Keisler measures for various homogeneous structures. In particular, we prove that many simple homogeneous structures, such as the generic tetrahedron-free 3-hypergraph, have non-forking formulas which are universally measure zero.