Glivenko-Cantelli classes and NIP formulas

Abstract: 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: (i) $\phi$ 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 $\phi$-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.) (iv) For any model $M$ and any Keisler $\phi$-measure $\mu(x)$ over $M$, \begin{align*} \sup_{b\in M}|\frac{1}{k}\sum_1^k\phi(p_i,b)-\mu(\phi(x,b))|\to 0 \end{align*} for almost every $(p_i)\in S_{\phi}(M)^{\Bbb N}$ with the product measure $\mu^{\Bbb N}$. (Cf. Theorem 4.3.) (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.)