Glivenko–Cantelli classes and NIP formulas

We give several new equivalences of NIP for formulas and new proofs of known results using Talagrand (Ann Probab 15:837–870, 1987) and Haydon et al. (in: Functional Analysis Proceedings, The University of Texas at Austin 1987–1989, Lecture Notes in Mathematics, Springer, New York, 1991). We emphasize that Keisler measures are more complicated than types (even in the NIP context), in an analytic sense. Among other things, we show that for a first order theory T and a formula \(\phi (x,y)\), the following are equivalent:

1. (i)

   \(\phi \) has NIP with respect to T.

2. (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, then p is DBSC definable over M. (Cf. Definition 3.7, Fact 3.8.)

3. (iii)

   For any global Keisler \(\phi \)-measure \(\mu (x)\) and any model M, if \(\mu \) is finitely satisfiable in M, then \(\mu \) is generalized Baire-1/2 definable over M. In particular, if M is countable, \(\mu \) is Baire-1/2 definable over M. (Cf. Definition 3.9.)

4. (iv)

   For any model M and any Keisler \(\phi \)-measure \(\mu (x)\) over M,

   $$\begin{aligned} \sup _{b\in M}\Big |\frac{1}{k}\sum _{i=1}^k\phi (p_i,b)-\mu (\phi (x,b))\Big |\rightarrow 0, \end{aligned}$$

   for almost every \((p_i)\in S_{\phi }(M)^{\mathbb N}\) with the product measure \(\mu ^{\mathbb N}\). (Cf. Theorem 4.4.)

5. (v)

   Suppose moreover that T is countable and NIP, then for any countable model M, the space of global M-finitely satisfied types/measures is a Rosenthal compactum. (Cf. Theorem 5.1.)