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Register a new userInvariant measures in simple and in small theories
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We give examples of (i) a simple theory with a formula (with parameters) which does not fork over the empty set but has mu measure 0 for every automorphism invariant Keisler measure mu, and (ii) a definable group G in a simple theory such that G is not definably amenable, i.e. there is no translation invariant Keisler measure on G We also discuss paradoxical decompositions both in the setting of discrete groups and of definable groups, and prove some positive results about small theories, including the definable amenability of definable groups, and nontriviality of the graded Grothendieck ring.
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We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of $NIP$ (not the independence property), continuing aspects of math.LO/0607442. Among key results are: (i) if $p = tp(b/A)$ does not fork over $A$ then the Lascar strong type of $b$ over $A$ coincides with the compact strong type of $b$ over $A$ and any global nonforking extension of $p$ is Borel definable over $bdd(A)$ (ii) analogous statements for Keisler measures and definable groups, including the fact that $G^{000} = G^{00}$ for $G$ definably amenable, (iii) definitions, characterizations and properties of generically stable types and groups (iv) uniqueness of translation invariant Keisler measures on groups with finitely satisfiable generics (vi) A proof of the compact domination conjecture for definably compact commutative groups in $o$-minimal expansions of real closed fields.
We characterize nonforking (Morley) sequences in dependent theories in terms of a generalization of Poizats special sequences and show that average types of Morley sequences are stationary over their domains. We characterize generically stable types in terms of the structure of the eventual type. We then study basic properties of strict Morley sequences, based on Shelahs notion of strict nonforking. In particular we prove Kims lemma for such sequences, and a weak version of local character.
We study the randomness properties of reals with respect to arbitrary probability measures on Cantor space. We show that every non-computable real is non-trivially random with respect to some measure. The probability measures constructed in the proof may have atoms. If one rules out the existence of atoms, i.e. considers only continuous measures, it turns out that every non-hyperarithmetical real is random for a continuous measure. On the other hand, examples of reals not random for any continuous measure can be found throughout the hyperarithmetical Turing degrees.
We study the question, ``For which reals $x$ does there exist a measure $mu$ such that $x$ is random relative to $mu$? We show that for every nonrecursive $x$, there is a measure which makes $x$ random without concentrating on $x$. We give several conditions on $x$ equivalent to there being continuous measure which makes $x$ random. We show that for all but countably many reals $x$ these conditions apply, so there is a continuous measure which makes $x$ random. There is a meta-mathematical aspect of this investigation. As one requires higher arithmetic levels in the degree of randomness, one must make use of more iterates of the power set of the continuum to show that for all but countably many $x$s there is a continuous $mu$ which makes $x$ random to that degree.
We answer two open questions about the model theory of valued differential fields introduced by Scanlon. We show that they eliminate imaginaries in the geometric language introduced by Haskell, Hrushovski and Macpherson and that they have the invariant extension property. These two result follow from an abstract criterion for the density of definable types in enrichments of algebraically closed valued fields. Finally, we show that this theory is metastable.
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