Submission date: 19 March 2021

Abstract:

We give some new equivalences of NIP for formulas and some new proofs of known result using [T87] and [HOR91]. We emphasize that Keisler measures are more complicated than types (even in NIP context), in an analytic sense. Among other things, we show that, for a first order theory T and formula φ(x,y), the following are equivalent: (i) φ has NIP (for theory T).
(ii) For any global \phi-type p(x) and any model M, if p is finitely satisfiable in M, then p is generalized DBSC definable over M. In particular, if M is countable, p is DBSC definable over M. (Cf. Definition 3.3, Fact 3.4.)
(iii) For any global Keisler φ-measure μ(x) and any model M, if μ is finitely satisfiable in M, then μ is generalized Baire-1/2 definable over M. In particular, if M is countable, p is Baire-1/2 definable over M. (Cf. Definition 3.5.)
(iv) For any model M and any Keisler φ-measure μ(x) over M,
sup_{b in M}|1/k ∑_1^k φ(p_i,b)-μ(φ(x,b))| → 0
for almost every (p_i) in S_{φ}(M)^ℕ with the product measure μ^ℕ. (Cf. Theorem 4.3.)

Mathematics Subject Classification: 03C45

Keywords and phrases:

Full text arXiv 2103.10788: pdf, ps.