Glivenko-Cantelli classes and NIP formulas
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2021
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| author | Khanaki, Karim |
| author_facet | Khanaki, Karim |
| contents | 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 $ϕ(x,y)$, the following are equivalent:
(i) $ϕ$ has $NIP$ (for theory $T$).
(ii) For any global $ϕ$-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 $M$, \begin{align*} \sup_{b\in M}|\frac{1}{k}\sum_1^kϕ(p_i,b)-μ(ϕ(x,b))|\to 0 \end{align*} for almost every $(p_i)\in S_ϕ(M)^{\Bbb N}$ with the product measure $μ^{\Bbb N}$. (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.) |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2103_10788 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Glivenko-Cantelli classes and NIP formulas Khanaki, Karim Logic 03C45 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 $ϕ(x,y)$, the following are equivalent: (i) $ϕ$ has $NIP$ (for theory $T$). (ii) For any global $ϕ$-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 $M$, \begin{align*} \sup_{b\in M}|\frac{1}{k}\sum_1^kϕ(p_i,b)-μ(ϕ(x,b))|\to 0 \end{align*} for almost every $(p_i)\in S_ϕ(M)^{\Bbb N}$ with the product measure $μ^{\Bbb N}$. (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.) |
| title | Glivenko-Cantelli classes and NIP formulas |
| topic | Logic 03C45 |
| url | https://arxiv.org/abs/2103.10788 |