Intuitively, generic sampling is a method for sampling types over monster models of a first-order theory with respect to a Keisler measure. In practice, this involves studying countably iterated Morley products of relatively tame Keisler measures. Given such a measure, one can straightforwardly derive a corresponding Borel measure on the space of L-structures over the natural numbers. In particular, if the original monster model is a graph (respectively, a hypergraph), then generic sampling yields measures on the space of all graphs (respectively, hypergraphs) on the natural numbers.
A graphon is a symmetric measurable map from $[0,1]^{2}$ to $[0,1]$. These graph-like objects also give rise to a notion of sampling. It turns out that all Sym(N)-invariant measures on the space of graphs (over the natural numbers) are mixtures of sampling procedures arising from graphons. We remark that graphons admit a higher-arity generalization to hypergraphs, called hypergraphons, which exhibit a similar property.
We prove a representation theorem: (hyper)graphon sampling can be encoded within the framework of generic sampling. We remark that generic sampling in practice can give rise to non-Sym(N)-invariant measures and so generic sampling is strictly more general than what arises from (hyper)graphons. This is joint work with Nathanael Ackerman, Cameron Freer, James Hanson, and Rehana Patel.

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