Generic sampling and invariant measures on the space of $k$-uniform hypergraphs

We prove a model-theoretic representation theorem for the distribution of an ergodic exchangeable $k$-uniform hypergraph: every such measure arises as the pushforward of the countably-iterated Morley product of a global Borel-definable Keisler measure over the countable universal homogeneous $k$-uniform hypergraph. We show this by starting with a Borel $k$-hypergraphon $W$ and constructing a Keisler measure ${\mu }_W$ such that generic sampling with respect to ${\mu }_W$ yields the same invariant measure as does the standard hypergraphon sampling procedure with respect to $W$. When $k=2$, our results give a new representation theorem for ergodic exchangeable graphs via Keisler measures over a monster model of the Rado graph.