Glivenko-Cantelli classes and NIP formulas

We give several new equivalences of NIP for formulas and new proofs of known results 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 phi(x,y), the following are equivalent:<br /> (i) phi has NIP (for theory T).<br /> (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.)<br /> (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, p is Baire-1/2 definable over M. (Cf. Definition 3.5.)<br /> (iv) For any model M and any Keisler phi-measure mu(x) over M, sup (b is an element of M)|(1)/(k)& sum;(k)(i=1)phi(p(i),b)-mu(phi(x,b))|-> 0 for almost every (pi)is an element of S-phi(M)N with the product measure mu(N). (Cf. Theorem 4.3.)<br /> (v) Suppose moreover that T is countable, then for any countable model M, the space of global M-finitely satisfied types/measures is a Rosenthal compactum. (Cf. Theorem A.1.)