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Automorphism–invariant measures on ℵ0-categorical structures without the independence property

  • Douglas E. Ensley (a1)

Abstract

We address the classification of the possible finitely-additive probability measures on the Boolean algebra of definable subsets of M which are invariant under the natural action of Aut(M). This pursuit requires a generalization of Shelah's forking formulas [8] to “essentially measure zero” sets and an application of Myer's “rank diagram” [5] of the Boolean algebra under consideration. The classification is completed for a large class of ℵ0-categorical structures without the independence property including those which are stable.

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[1]Albert, M., Measures on the random graph, Journal of the London Mathematical Society, vol. 50 (1994), pp. 417429.
[2]Baldwin, J., Fundamentals of stability theory, Springer-Verlag, 1988.
[3]Chang, C. C. and Keisler, H. J., Model theory, North Holland, Amsterdam, 1990.
[4]Ensley, D., Measures on ℵ0-categorical structures, Ph.D. thesis, Carnegie-Mellon University, 1993, Ph.D. thesis.
[5]Myers, D., Lindenbaum-tarski algebras, Handbook of Boolean algebras, North Holland, Amsterdam, 1989.
[6]Pillay, A., An introduction to stability theory, Oxford University Press, Oxford, 1983.
[7]Shelah, S., Simple unstable theories, Annals of Mathematical Logic, vol. 19 (1980), pp. 177203.
[8]Shelah, S., Classification theory and the number of non-isomorphic models, North-Holland, Amsterdam, 1990, revised edition.
[9]Vaughn, J., A note on forking and normalization, Arch. math. Logik, vol. 26 (1987), pp. 165168.

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