Abstract: 

In search of a notion of graph limit for sequences of sparse graphs, Nešetřil and Ossona de Mendez introduced first-order convergence for sequences of finite structures (requiring that for every formula, the probability that it is satisfied by a tuple chosen uniformly at random converges) and corresponding analytic limit objects called modeling limits. A modeling limit is a Borel structure, i.e. its domain is a standard Borel space and every parameter-definable relation is Borel, equipped with a probability measure assigning the appropriate limiting value to definable relations (or rather, a sequence of measures for increasing powers of the structure). We show that a modeling limit can be constructed for any convergent sequence of finite structures from a monadically stable class. Behind this is a theorem about realizing measures from sufficiently saturated models of a monadically stable theory in Borel models. Joint work with Jarik Nešetřil and Patrice Ossona de Mendez.

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