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Invariant measures concentrated on countable structures | | Nathanael Ackerman
; Cameron Freer
; Rehana Patel
; | | Date: |
18 Jun 2012 | | Abstract: | Let L be a countable language. Given a countable L-structure M with
underlying set the natural numbers N, we determine when there is a probability
measure on the space of all such L-structures that is invariant under the
action of Sym(N), and that assigns measure one to the isomorphism class of M.
In recent work, Petrov and Vershik [PV10] have proven the existence of such
invariant measures for Henson’s countable universal ultrahomogeneous K_n-free
graphs. Here we give a complete characterization of countable infinite
structures that admit invariant measures: There is an invariant measure
concentrated on the isomorphism class of M if and only if the "group-theoretic"
definable closure of every finite tuple of M is trivial, i.e., the pointwise
stabilizer in Aut(M) of an arbitrary finite tuple of M fixes no additional
points. When M is a Fraisse limit, this amounts to requiring that the age of M
have strong amalgamation. The proof makes use of the model theory of infinitary
logic to build upon Petrov and Vershik’s constructions. In the case when M is a
graph, these methods provide a new means of building dense graph limits, in the
sense of Lovasz and Szegedy [LS06]. Our result gives rise to new instances of
structures admitting invariant measures, such as the countable universal
ultrahomogeneous partial order, Henson’s family of continuum-many countable
ultrahomogeneous directed graphs, certain countable universal graphs forbidding
a finite homomorphism-closed set of finite connected graphs, and the rational
Urysohn space. | | Source: | arXiv, 1206.4011 | | Services: | Forum | Review | PDF | Favorites | |
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