Generically stable and smooth measures in NIP theories
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- by Ehud Hrushovski, Anand Pillay and Pierre Simon PDF
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Abstract:
We formulate the measure analogue of generically stable types in first order theories with $NIP$ (without the independence property), giving several characterizations, answering some questions from an earlier paper by Hrushovski and Pillay, and giving another treatment of uniqueness results from the same paper. We introduce a notion of “generic compact domination”, relating it to stationarity of the Keisler measures, and also giving definable group versions. We also prove the “approximate definability” of arbitrary Borel probability measures on definable sets in the real and $p$-adic fields.References
- H. Adler, Introduction to theories without the independence property, to appear in Archive Math. Logic.
- Itaï Ben Yaacov, Continuous and random Vapnik-Chervonenkis classes, Israel J. Math. 173 (2009), 309–333 (English, with French summary). MR 2570671, DOI 10.1007/s11856-009-0094-x
- I. Ben-Yaacov, Transfer of properties between measures and random types, preprint.
- Itaï Ben Yaacov and H. Jerome Keisler, Randomizations of models as metric structures, Confluentes Math. 1 (2009), no. 2, 197–223. MR 2561997, DOI 10.1142/S1793744209000080
- Françoise Delon, Définissabilité avec paramètres extérieurs dans $\textbf {Q}_p$ et $\textbf {R}$, Proc. Amer. Math. Soc. 106 (1989), no. 1, 193–198 (French, with English summary). MR 953003, DOI 10.1090/S0002-9939-1989-0953003-8
- Deirdre Haskell, Ehud Hrushovski, and Dugald Macpherson, Stable domination and independence in algebraically closed valued fields, Lecture Notes in Logic, vol. 30, Association for Symbolic Logic, Chicago, IL; Cambridge University Press, Cambridge, 2008. MR 2369946
- Bradd Hart, Byunghan Kim, and Anand Pillay, Coordinatisation and canonical bases in simple theories, J. Symbolic Logic 65 (2000), no. 1, 293–309. MR 1782121, DOI 10.2307/2586538
- Ehud Hrushovski, Ya’acov Peterzil, and Anand Pillay, Groups, measures, and the NIP, J. Amer. Math. Soc. 21 (2008), no. 2, 563–596. MR 2373360, DOI 10.1090/S0894-0347-07-00558-9
- E. Hrushovski and A. Pillay, On $NIP$ and invariant measures, J. European Math. Soc, 13 (2011), 1005-1061.
- François Loeser, Geometry and non-Archimedean integrals, European Congress of Mathematics, Eur. Math. Soc., Zürich, 2010, pp. 277–292. MR 2648330, DOI 10.4171/077-1/13
- M. Karpinski and A. J. Macintyre, Approximating volumes and integrals in $o$-minimal and $p$-minimal theories, in Connections between Model Theory and Algebraic and Analytic Geometry, Quaderni di matematica, vol 6, Seconda Universita di Napoli, 2000.
- H. Jerome Keisler, Measures and forking, Ann. Pure Appl. Logic 34 (1987), no. 2, 119–169. MR 890599, DOI 10.1016/0168-0072(87)90069-8
- H. Jerome Keisler, Choosing elements in a saturated model, Classification theory (Chicago, IL, 1985) Lecture Notes in Math., vol. 1292, Springer, Berlin, 1987, pp. 165–181. MR 1033028, DOI 10.1007/BFb0082237
- D. Lascar and A. Pillay, Hyperimaginaries and automorphism groups, J. Symbolic Logic 66 (2001), no. 1, 127–143. MR 1825177, DOI 10.2307/2694914
- David Marker and Charles I. Steinhorn, Definable types in $\scr O$-minimal theories, J. Symbolic Logic 59 (1994), no. 1, 185–198. MR 1264974, DOI 10.2307/2275260
- A. Onshuus and A. Pillay, Definable groups and compact $p$-adic Lie groups, J. Lond. Math. Soc. (2) 78 (2008), no. 1, 233–247. MR 2427062, DOI 10.1112/jlms/jdn018
- Anand Pillay, Geometric stability theory, Oxford Logic Guides, vol. 32, The Clarendon Press, Oxford University Press, New York, 1996. Oxford Science Publications. MR 1429864
- Bruno Poizat, A course in model theory, Universitext, Springer-Verlag, New York, 2000. An introduction to contemporary mathematical logic; Translated from the French by Moses Klein and revised by the author. MR 1757487, DOI 10.1007/978-1-4419-8622-1
- Saharon Shelah, Dependent first order theories, continued, Israel J. Math. 173 (2009), 1–60. MR 2570659, DOI 10.1007/s11856-009-0082-1
- S. Shelah, Classification theory and the number of nonisomorphic models, 2nd ed., Studies in Logic and the Foundations of Mathematics, vol. 92, North-Holland Publishing Co., Amsterdam, 1990. MR 1083551
- P. Simon, Théories NIP, M.Sc. thesis, Paris VII.
- Alexander Usvyatsov, On generically stable types in dependent theories, J. Symbolic Logic 74 (2009), no. 1, 216–250. MR 2499428, DOI 10.2178/jsl/1231082310
- V.N. Vapnik and A.Y. Chervonenkis, On the uniform convergence of relative frequencies of events to their probabilities, Theory Probab. Appl., 16 (1971), 264-280.
- Frank O. Wagner, Simple theories, Mathematics and its Applications, vol. 503, Kluwer Academic Publishers, Dordrecht, 2000. MR 1747713, DOI 10.1007/978-94-017-3002-0
Additional Information
- Ehud Hrushovski
- Affiliation: Institute of Mathematics, Hebrew University of Jerusalem, 91904 Jerusalem, Israel
- Anand Pillay
- Affiliation: School of Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom
- MR Author ID: 139610
- Pierre Simon
- Affiliation: Institute of Mathematics, Hebrew University of Jerusalem, 91904 Jerusalem, Israel
- MR Author ID: 942320
- Received by editor(s): June 7, 2010
- Received by editor(s) in revised form: May 10, 2011
- Published electronically: December 13, 2012
- Additional Notes: The first author was supported by ISF grant 1048/07
The second author was supported by a Marie Curie Chair EXC 024052 and EPSRC grant EP/F009712/1 - © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 365 (2013), 2341-2366
- MSC (2010): Primary 03C68, 03C45, 22C05, 28E05
- DOI: https://doi.org/10.1090/S0002-9947-2012-05626-1
- MathSciNet review: 3020101