{
  "id": "2103.10788",
  "version": "v1",
  "published": "2021-03-19T13:15:43.000Z",
  "updated": "2021-03-19T13:15:43.000Z",
  "title": "Glivenko-Cantelli classes and NIP formulas",
  "authors": [
    "Karim Khanaki"
  ],
  "categories": [
    "math.LO"
  ],
  "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 $\\phi(x,y)$, 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.)",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-19T13:15:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03C45"
    ],
    "keywords": [
      "nip formulas",
      "glivenko-cantelli classes",
      "first order theory",
      "product measure",
      "keisler measures"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}