#  Glivenko–Cantelli classes and $NIP$ formulas

Karim Khanaki
Arak University of Technology Partially supported by IPM grant 1400030118

###### Abstract

We give some new equivalences of $NIP$ for formulas and some 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 3.4, Fact 3.5.)
(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 3.6.)
(iv) For any model $M$ and any Keisler $\phi$-measure $\mu(x)$ over $M$,

$\displaystyle\sup_{b\in
M}|\frac{1}{k}\sum_{1}^{k}\phi(p_{i},b)-\mu(\phi(x,b))|\to 0$

for almost every $(p_{i})\in S_{\phi}(M)^{\mathbb{N}}$ with the product
measure $\mu^{\mathbb{N}}$. (Cf. Theorem 4.3.)

##  1 Introduction

This paper is a kind of companion-piece to [KP18] and [Kha20] although here we
study equivalences of $NIP$ formulas and local Keisler measures in classical
first order logic. We also give alternatives (or refinements) of some results
of [HP11] and [HPS13]. The main results/observations are Theorems 3.9, 4.3,
4.11, Lemma 3.8, and Corollaries 4.7, 4.9

We believe that the novelty of the results of this article is that it is
shown, assuming $NIP$ for formula $\phi$, that $\phi$-types correspond to a
proper subclass of Baire-1 functions (i.e. $DBSC$) and Keisler $\phi$-measures
correspond to another proper subclass of Baire-1 functions (i.e. Baire-1/2),
and these classes are drastically different, in an analytic sense. This means
that measures are definitely more complicated than types, in this sense. The
complexity can lead to the definition of new classes of $NIP$ theories in the
classical logic; that is, we can distinguish between theories that their types
and measures have the same complexity and those that do not. In the framework
of continuous logic this complexity has even more important consequences, as
this leads to the fact, contrary to a claim in the literature, that there is
no a perfect analog of Shelah’s theorem, namely a theory is unstable iff it
has $IP$ or $SOP$ (cf. [Kha20a]). On the other hand, using the Talagrand’s
characterization of Glivenko-Cantelli classes, we give some new equivalences
of $NIP$ for formulas that can have future applications.

Let us give some historical points on the subject. The notion dependence for a
family of sets/functions was introduced independently in model theory [Sh71],
learning theory [VC71], and function theory [Ros74]. The combinatorial nature
of the concept soon became apparent. The connection between the Vapnik-
Chervonenkis results in learning theory and the $NIP$ in model theory was
discovered by Laskowski [Las92]. In fact, [VC71] characterizes the Glivenko-
Cantelli classes using the notion independence. In function theory part, this
flow led to the invention of the concept Fremlin-Talagrand stability by David
Fremlin (cf. Definition 4.1 below). The connection of this concept with the
Glivenko-Cantelli classes was later identified and studied by Talagrand [T87].
His Characterization of the Glivenko-Cantelli classes has some advantages over
[VC71], as it is more suitable for studying (real-valued) functions and also
has no additional measurability conditions. In fact, all the power of that
concept lies in empirical measures or convexity. The importance of this issue
was understood in [HPP08], in which the concept of Keisler measure was
studied. Keisler measures had previously been invented and studied in [Kei87].
Thus all the power of the notion was used in model theory in [HPP08] and
[HP11]. In the present paper, the relationship between $NIP$ and the Glivenko-
Cantelli classes is explicitly studied and a completely functional expression
of the subject is presented.

Let us present the motivation of this article. Naturally, it can be
interesting to examine the relationship between one area of mathematics and
other areas. When a particular concept is introduced and studied independently
in different areas, that concept is inherently of special mathematical
importance. For example, we can refer to the stability and $NIP$ in model
theory, which also appear in various other fields such as functional analysis
and learning theory. This allows the tools, results, and approaches of one
field to be used in another field, and vice versa. This approach also improves
the previous results and leads to new results, and sheds light on what was
previously known.

Surprisingly, all tools already existed in functional analysis and probability
theory. Some of our results/observations– not all them– are just routine
translations of the results of [T87] and [HOR91] into the language of model
theory. However, there are several reasons that justify and even necessitate
the use of this translation: (1) Here we provide a useful dictionary (from
model theory to analysis and vice versa) that helps to understand the
connections between them and for future applications. (2) Many results in the
language of analysis can be more easily generalized to extensions of classical
logic (such as continuous logic) and these results have interesting
applications both for logic itself and for its applications in mathematics
(see for example [Kha20a] and [Kha21]).

On the other hand, it is worthwhile to provide more direct arguments and
model-theoretic proofs of the results of this paper, although we believe that
any model-theoretic proof of these analytical facts will necessarily express a
similar argument in the language of model theory and will not differ much in
nature.

This paper is organized as follows. In section 2, we provide all function
theory background and emphasize the relationship between some of the classes
of functions that we will use later.

In section 3, we present the relationship between types/measures and
analytical concepts, and provide a useful dictionary between model theory and
analysis. We also emphasize the distinction between types and measures (over
countable models or sets) in an analytic sense.

In section 4, we study the relationship between Glivenko–Cantelli-classes and
$NIP$ formulas, and using alternative proofs, we provide generalizations of
some of the results of section 3. We also provide results about ‘finite
approximation’ and ‘frequency interpretation’ for measures in $NIP$ and
arbitrary contexts.

In appendix A, we give a short proof of a new result on angelic spaces of
measures.

##  2 Preliminary of function spaces

If $f$ is a real-valued function on a set $X$, the symbol $[f>r]$ stands for
$\\{x\in X:f(x)>r\\}$, and $f^{-1}[F]$ stands for $\\{x\in X:f(x)\in F\\}$.
Similarly, we abbreviate $[f\geq r]$, $[f<r]$ and $[f\leq r]$. We define the
positive and negative parts of $f$ to be

$f^{+}(x)=\max(f(x),0),\ \ \ \ \ \ \ \ f^{-}(x)=\max(-f(x),0).$

Note that $f=f^{+}-f^{-}$ and $|f|=f^{+}+f^{-}$.

The topology of pointwise convergence on $X$ is that inherited from the usual
product topology of ${\mathbb{R}}^{X}$. A typical neighborhood of a function
$f$ is

$U_{f}(x_{1},\ldots,x_{n};\epsilon)=\\{g\in{\mathbb{R}}^{X}:|f(x_{i})-g(x_{i})|<\epsilon\text{
for }i\leq n\\},$

where $\epsilon>0$ and $\\{x_{1},\ldots,x_{n}\\}$ is a finite subset of $X$.
In this paper, the topology on subsets of ${\mathbb{R}}^{X}$ is pointwise
convergence. Otherwise, we explicitly state what is our desired topology.

Let $(f_{n})_{n<\omega}$ be a sequence of uniformly bounded real-valued
functions on $X$. (We always assume that $f_{0}(x)=0$ for all $x\in X$.) Let
$\cal D$ be a non-principal ultrafilter on $\omega$. Recall that a function
$f$ is called the $\cal D$-limit of $(f_{n})_{n<\omega}$, denoted by
$f=\lim_{\cal D}f_{n}$, if for every neighborhood $U$ of $f$ the set
$\\{n:f_{n}\in U\\}$ belongs to ${\cal D}$.111A basic fact in general topology
asserts that a topological space is compact and Hausdorff if and only if for
any net $(x_{\alpha})$ and any ultrafilter $\cal D$, there exists a unique
$\cal D$-limit of $(x_{\alpha})$. In this case, sometimes we write $f_{n}\to
f$.

######  Fact 2.1.

Let $(f_{n})_{n<\omega}$ and $\cal D$ be as above. Then the following are
equivalent:

  * (i)

There is $C>0$ such that $\sum_{n<\omega}|f_{n+1}(x)-f_{n}(x)|\leq C$ for all
$x\in X$.

  * (ii)

There is a function $f$ such that $f=\lim_{\cal D}f_{n}$ and

$f=\sum_{n<\omega}(f_{n+1}-f_{n})^{+}-\sum_{n<\omega}(f_{n+1}-f_{n})^{-}.$

###### Proof.

(i) $\Longrightarrow$ (ii): Note that $f_{n}=\sum_{k=0}^{n-1}(f_{k+1}-f_{k})$
and by (i), for all $x$, the infinite series
$\sum_{n<\omega}(f_{n+1}(x)-f_{n}(x))$ is absolutely convergent. Therefore,
the sequence $f_{n}(x)$ converges to $\sum_{n<\omega}(f_{n+1}(x)-f_{n}(x))$,
for all $x$. Let $f=\lim_{\cal D}f_{n}$. Clearly, $f=\lim_{\cal
D}f_{n}=\lim_{\cal D}f_{n}^{+}-\lim_{\cal D}f_{n}^{-}$. (Note that $\lim_{\cal
D}f_{n}^{+}\leq\lim_{\cal D}|f_{n}|\leq C$. Similarly for $\lim_{\cal
D}f_{n}^{-}$.)
(ii) $\Longrightarrow$ (ii): As $(f_{n})_{n<\omega}$ is uniformly bounded, $f$
is bounded. Therefore, there is $N>0$ such that for all $x$,
$\sum_{n<\omega}(f_{n+1}-f_{n})^{+}(x)<N$ and
$\sum_{n<\omega}(f_{n+1}-f_{n})^{-}(x)<N$. This means that

$\sum_{n<\omega}|f_{n+1}(x)-f_{n}(x)|=\sum_{n<\omega}(f_{n+1}-f_{n})^{+}(x)+\sum_{n<\omega}(f_{n+1}-f_{n})^{-}(x)<2N.$

∎

In the following, we give a definition for the metric spaces, but we will
shortly provide a general definition (i.e. without metrizability).

######  Definition 2.2.

Let $X$ be a compact metric space. The following is a list of the spaces of
real-valued functions defined on $X$. The important ones to note immediately
are (iv) and (v).

(i) $C(X)$ is the space of continuous real-valued functions on $X$.

(ii) $SC(X)$, the lower semi-continuous function, is the space of functions
$f:X\to\mathbb{R}$ such that for every $r\in\mathbb{R}$, $[f\leq r]$ is
closed.

(iii) $B_{1}(X)$, the Baire 1 class, is the space of functions
$f:X\to\mathbb{R}$ such that for every closed $F\subseteq\mathbb{R}$,
$f^{-1}[F]$ is $G_{\delta}$ in $X$, i.e. it is of the form
$\bigcap_{n=1}^{\infty}G_{n}$ where the $G_{n}$’s are open.

(iv) $DBSC(X)$, is the space of real-valued functions $f$ on $X$ such that
there are continuous functions $(f_{n})_{n<\omega}$ and a constant $C>0$ such
that $f_{\alpha}\to f$ pointwise and $\sum_{n<\omega}|f_{n+1}(x)-f_{n}(x)|<C$
for all $x$.

(v) $B_{1/2}(X)$ or Baire-1/2, is the space of real-valued functions $f$ on
$X$ such that $f$ is the uniform limit of a sequence $(F_{n})\in DBSC(X)$.

######  Fact 2.3.

Let $X$ be a compact metric space.

(i) Every $f\in DBSC(X)$ is the difference of bounded semi-continuous
functions; that is, there are semi-continuous functions $F_{1},F_{2}$ with
$f=F_{1}-F_{2}$.

(ii) Let $f$ be a bounded real-valued function on $X$. $f\in B_{1}(X)$ if and
only if it is the poinwise limit of a sequence of continuous functions.

(iii) A real-valued function $f$ on $X$ is lower semi-continuous if and only
if there is a sequence $(f_{n})$ of continuous functions such that $f_{1}\leq
f_{2}\leq\cdots$ and $(f_{n})$ converges pointwise to $f$ (for short we write
$f_{n}\nearrow f$).

(iv) $f\in DBSC(X)$ if and only if there are semi-continuous functions
$F_{1},F_{2}$ with $f=F_{1}-F_{2}$.
(v) For any simple222A function is called simple if its range is a finite set.
function $f$ on $X$, $f\in B_{1/2}(X)$ if and only if it is $DBSC$.
(vi) For uncountable compact metric space $X$,

$C(X)\subsetneqq SD(X)\subsetneqq DBSC(X)\subsetneqq B_{1/2}(X)\subsetneqq
B_{1}(X).$

###### Proof.

(i): Set $F_{1}=\sum\big{(}f_{n+1}(y)-f_{n}(y)\big{)}^{+}$ and
$F_{2}=\sum\big{(}f_{n+1}(y)-f_{n}(y)\big{)}^{-}$ and use Fact 2.1.

(ii) This is a classical result due to Lebesgue and Hausdorff (see [Kec95,
Theorem 24.10].) Note that the limit of any sequence of continuous functions
is Baire 1 in any topological space. For a generalization of (ii), see [Kur66,
p. 393] and also [Jay74].

(iii) is a classical result in functional analysis, essentially due to Baire
(cf. [Hau62, p. 274]).

(iv) follows from (i), (iii). (See [HOR91].)

(v) follows from [CMR96, Proposition 2.2].

(vi) [HOR91, Proposition 5.1] and [CMR96]. ∎

Note that the difference between the given classes $B_{1/2}$ and $DBSC$ is
fundamental, and in fact there are many other classes between them (cf.
[HOR91]).

######  Fact 2.4.

Let $X$ be a compact metric space and $f$ be a simple function on $X$. $f\in
DBSC$ iff it is strongly Borel, that is, there exist disjoint differences of
closed sets $W_{1},\ldots,W_{n}$ and real numbers $r_{1},\ldots,r_{n}$ such
that $f=\sum_{i=1}^{n}r_{i}\chi_{W_{i}}$.

###### Proof.

This is equivalence (3) $\Longleftrightarrow$ (4) of [CMR96, Proposition 2.2].
Indeed, suppose that $f\in DBSC$. For simplicity, assume that $f$ is
$\\{0,1\\}$-valued. By Fact 2.3(iv), there are semi-continuous functions
$F_{1},F_{2}$ with $f=F_{1}-F_{2}$. It is easy to see that
$f^{-1}(1)=\bigcup_{0\leq n\leq C}([F_{1}>n+1/2]\cap[F_{2}\leq n])$. Note that
$f$ is $\\{0,1\\}$-valued and $F_{1},F_{2}$ are valued in
$\\{0,1,\ldots,C\\}$. As $F_{1},F_{2}$ are lower semi-continuous, the
$[F_{1}>n+1/2]$’s are open and the $[F_{2}\leq n]$’s are closed. The converse
is evident. ∎

We define a norm on $DBSC$.

######  Definition 2.5.

Let $X$ be a compact metric space.
Suppose that $f\in DBSC(X)$. The $D$-norm of $f$, denoted by $\|f\|_{D}$, is
defined by

$\|f\|_{D}=\inf\\{C:(f_{n})\in C(X),\ f_{n}\to f\text{ pointwise},\
\sum(f_{n+1}-f_{n})\leq C\\}.$

######  Fact 2.6.

Let $X$ be a compact metric space. [HOR91, page 3]: The classes $DBSC$ and
$B_{1/2}$ are Banach algebras with respect to the $D$-norm and the uniform
norm, respectively.

We now generalize Definition 2.2 for arbitrary compact spaces.

######  Definition 2.7.

Let $X$ be a compact space (not necessarily metric).

(i) A simple function $f$ on $X$ is called a generalized $DBSC$ function (or
short generalized $DBSC$) if there exist disjoint differences of closed sets
$W_{1},\ldots,W_{n}$ and real numbers $r_{1},\ldots,r_{n}$ such that
$f=\sum_{i=1}^{n}r_{i}\chi_{W_{i}}$.

(ii) A real-valued functions $f$ on $X$ is called generalized Baire-1/2 if $f$
is the uniform limit of a sequence $(F_{n})$ of simple $DBSC$ functions.

Note that in the general case, i.e. when $X$ is not metric, a generalized
$DBSC$ (or generalized Baire-1/2) function is not Baire-1, even assuming that
it is $\\{0,1\\}$-valued. (Of course, such functions are Borel measurable.) In
this case, the question of whether or not it is Baire-1 is related to
descriptive set theory.

##  3 Types and measures

In this section we give the model-theoretic results using ideas from
functional analysis as presented in the previous section.

We work in classical ($\\{0,1\\}$-valued) model theory context. Our model
theory notation is standard, and a text such as [S15] will be sufficient
background for the model theory part of the paper.

We fix an $L$-formula $\phi(x,y)$, a complete $L$-theory $T$ and a subset $A$
of the monster model of $T$. We let $\tilde{\phi}(y,x)=\phi(x,y)$. Let
$X=S_{\tilde{\phi}}(A)$ be the space of complete $\tilde{\phi}$-types on $A$,
namely the Stone space of ultrafilters on Boolean algebra generated by
formulas $\phi(a,y)$ for $a\in A$. Each formula $\phi(a,y)$ for $a\in A$
defines a function $\phi(a,y):X\to\\{0,1\\}$, which takes $q\in X$ to 1 if
$\phi(a,y)\in q$ and to 0 if $\phi(a,y)\notin q$. Note that $X$ is compact and
these functions are continuous, and as $\phi$ is fixed we can identify this
set of functions with $A$. So, $A$ is a subset of all bounded continuous
functions on $X$, denoted by $A\subseteq C(X)$. Just as we did above, one can
define $B_{1}(X)$, Baire-1/2 and $DBSC(X)$.

Convention. In the this paper, a type over the monster model $\cal U$ is
called a “global type”, a type over a small set $A$ is called a “non-global
type”, a consistence collection of Boolean combinations of the instances
$\phi(x,b),b\in\cal U$, is called a “global $\phi$-type”, and a consistence
collection of Boolean combinations of the instances $\phi(x,b),b\in A$, is
called a “non-global $\phi$-type.”

######  Definition 3.1 (Keisler measures).

Let $T$ be a complete theory, $\cal U$ the moster model of $T$, and
$A\subseteq\cal U$.
(i) A (Keisler) measure $\mu$ over $A$ in the variable $x$ is a finitely
additive probabilty measure on all $L(A)$-formulas with the variable $x$. We
will sometimes write $\mu$ as $\mu(x)$ to emphasize that $\mu$ is a measure on
the variable $x$. We say $\mu$ is a global (Keisler) measure, if $A=\cal U$,
and is non-global (Keisler) measure, in otherwise.
(ii) Let $\phi(x,y)$ be a formula. A (Keisler) $\phi$-measure $\mu$ over $A$
in the variable $x$ is a finitely additive probabilty measure on the Boolean
combinations of the instances $\phi(x,b),b\in A$. We will sometimes write
$\mu$ as $\mu_{\phi}$ (or $\mu_{\phi}(x)$) to emphasize that $\mu$ is a
$\phi$-measure (on the variable $x$). A (Keisler) $\phi$-measure $\mu$ is a
global (Keisler) $\phi$-measure, if $A=\cal U$, and is non-global (Keisler)
$\phi$-measure, in otherwise.

We let $S(\cal U)$ be the space of global types, and ${\mathfrak{M}}({\cal
U})$ the collection of global Keisler measures. We let $S_{\phi}(\cal U)$ be
the space of global $\phi$-types, and ${\mathfrak{M}}_{\phi}({\cal U})$ the
collection of global Keisler $\phi$-measures. Similarly, for $A\subseteq\cal
U$, the symbols $S(A)$, ${\mathfrak{M}}({A})$, $S_{\phi}(\cal U)$ and
${\mathfrak{M}}_{\phi}(A)$ are defined.

For a formula $\phi(x,y)$, we let $\Delta_{\phi}$ be the set of all Boolean
combinations of the instances of $\phi(x,y)$ (with the variable $x$). If
$\mu\in{\mathfrak{M}}({A})$, we also define
$\mu|_{\phi}(\theta(x,b)):=\mu(\theta(x,b))$ for all $\theta\in\Delta_{\phi}$
and $b\in A$. Note that $\mu|_{\phi}\in{\mathfrak{M}}_{\phi}({A})$ and so we
will sometimes write $\mu|_{\phi}$ as $\mu_{\phi}$ (or $\mu$ again).

In the following, we give a useful dictionary that can also be used in future
work.

######  Fact 3.2.

Let $T$ be a complete theory, $M$ a model of $T$ and $\phi(x,y)$ a formula.
(i)(Pillay) There is a correspondence between global $M$-finitely satisfiable
$\phi$-types $p(x)$ and the functions in the pointwise closure of all
functions $\phi(a,y):S_{\tilde{\phi}}(M)\to\\{0,1\\}$ for $a\in M$, where
$\phi(a,q)=1$ if and only if $\phi(a,y)\in q$.
(ii) The map $p\mapsto\delta_{p}$ is a correspondence between global
$\phi$-types $p(x)$ and Dirac measures $\delta_{p}(x)$ on $S_{\phi}({\cal
U})$, where $\delta_{p}(A)=1$ if $p\in A$, and $=0$ in otherwise. Moreover,
$p(x)$ is finitely satisfiable in $M$ iff $\delta_{p}(x)$ is finitely
satisfiable in $M$.
(iii) There is a correspondence between global $\phi$-measures $\mu(x)$ and
regular Borel probability measures on $S_{\phi}({\cal U})$. Moreover, a global
$\phi$-measures is finitely satisfiable in $M$ iff its corresponding regular
Borel probability measure is finitely satisfiale in $M$.
(iv) The closed convex hull of Dirac measures $\delta(x)$ on $S_{\phi}({\cal
U})$ is exactly all regular Borel probability measures $\mu(x)$ on
$S_{\phi}({\cal U})$. Moreover, the closed convex hull of Dirac measures on
$S_{\phi}({\cal U})$ which are finitely satisfiable in $M$ is exactly all
regular Borel probability measures $\mu(x)$ on $S_{\phi}({\cal U})$ which are
finitely satisfiable in $M$.
(v) There is a correspondence between global $M$-finitely satisfiable
$\phi$-measures $\mu(x)$ and the functions in the pointwise closure of all
functions of the form $\frac{1}{n}\sum_{1}^{n}r_{i}\cdot\theta(a_{i},y)$ on
$S_{\tilde{\theta}}(M)$, where $\theta\in\Delta_{\phi}$, $a_{i}\in M$,
$r_{i}\in[0,1]$, and $\theta(a_{i},q)=1$ if and only if $\theta(a_{i},y)\in
q$.

###### Proof.

(i) is due to Pillay (cf. [P18, Remark 2.1]).
(ii): Let $X=S_{\phi}({\cal U})$. For $p(x)\in X$, set the Dirac’s measure
$\delta_{p}(x)$ on $X$, defined by $\delta_{p}(A)=1$ if $p\in A$, and $=0$ in
otherwise. It is easy to verify that the map $p\mapsto\delta_{p}$ is a
bijection between $X$ and all Dirac measures on $X$. Moreover, $p(x)$ is
finitely satisfiable in $M$ iff $M\models\psi(x)$ for all $\psi\in p(x)$ iff
$M\models\psi(x)$ for all $\psi$ such that $\delta_{p}(\psi(x))=1$.
(iii) is well-known (cf. [S15, Section 7.1]). For every global $M$-finitely
satisfiable $\phi$-measures $\mu$, suppose that $\mu_{r}$ is its corresponding
regular Borel probability measure. Clearly, for any $\psi(x)$ if
$\mu(\psi(x))>0$ iff $\mu_{r}(\psi(x))>0$. Therefore, $\mu$ is $M$-finitely
satisfiable iff $\mu_{r}$ is so.
(iv) is a well-known result in Functional Analysis. Indeed, let
$X=S_{\phi}({\cal U})$ and $D(X)$ be the set of all Dirac measures on $X$;
that is, $D(X)=\\{\delta_{x}:x\in X\\}$, where $\delta_{x}(A)=1$ if $x\in A$,
and $=0$ in otherwise. Let $M(X)$ be the set of all regular Borel probability
measures on $X$. By Proposition 437P of [Fre06], every $\delta_{x}\in D(X)$ is
an extreme point of $M(X)$. By Riesz Representation Theorem ([Fre06, Theorem
436J]), there is a correspondence between regular Borel probability measures
$\mu\in M(X)$ and positive linear functionals $I_{\mu}$ on $C(X)$ such that
$\|I_{\mu}\|=1$. (Therefore, $M(X)$ is a compact convex set of $C(X)^{*}$ with
the weak* topology.) By Krein–Milman theorem ([Con90, Theorem 7.4]), the
closed convex hull $\overline{conv}(D(X))$ of $D(X)$ is $M(X)$. Moreover, it
is easy to verify that the set of all $M$-finitely satisfiable measures in
$M(X)$ is a closed convex subset of $M(X)$, and $M$-finitely satisfiable Dirac
measures in $D(X)$ are its extreme points. Again, by Krein–Milman theorem, the
proof is completed.
(v) follows from (i)–(iv). (Indeed, note that the functions of the form
$\frac{1}{n}\sum_{1}^{n}r_{i}\cdot\phi(a_{i},y)$ alone are not enough to
define a measure, because we need information about the Boolean combinations
of the instances $\phi(x,b)$’s. Therefore, for every $\theta\in\Delta_{\phi}$
we need to have a function $f_{\theta}$ on $S_{\tilde{\theta}}(M)$. Also, for
any $\theta_{1},\theta_{2}\in\Delta_{\phi}$, we should have
$f_{\theta_{1}\vee\theta_{2}}=f_{\theta_{1}}+f_{\theta_{2}}-f_{\theta_{1}\wedge\theta_{2}}$.)
∎

######  Remark 3.3.

Note that all statements of Fact 3.2 can be proved for global types/measures
instead of $\phi$-types/$\phi$-measures. Because the purpose of this article
is to study local properties, we refrain from stating them precisely.

Let $A$ be a (small) set of the monster model $\cal U$. A set $F$ is called a
closed set over $A$ if it is the set of realizations in $\cal U$ of a partial
type over $A$. The complements of closed sets over $A$ are called the opens
sets over $A$. The Borel set over $A$ is the $\sigma$-algebra generated by the
open sets over $A$. Alternatively, these are corespondes to the Borel sets of
the relevant Stone space of complete types over $A$.

######  Definition 3.4.

(i)[HP11] A global type $p(x)$ is called Borel definable over $A$ if for any
$L$-formula $\phi(x,y)$, the set $\\{b\in{\cal U}:\phi(x,b)\in p(x)\\}$ is a
Borel set over $A$.
(ii)[HP11] A global type $p(x)$ is called strongly Borel definable over $A$
(or $DBSC$ definable over $A$) if for any $L$-formula $\phi(x,y)$, the set
$\\{b\in{\cal U}:\phi(x,b)\in p(x)\\}$ is a finite Boolean combination of
closed sets over $A$.
(iii) A global (Kiesler) measure $\mu(x)$ is called Borel definable over $A$
if for any $L$-formula $\phi(x,y)$ and any closed subset $F\subseteq[0,1]$,
the set $\\{b\in{\cal U}:\mu(\phi(x,b))\in F\\}$ is a Borel set over $A$.
(iv) Let $\phi(x,y)$ be a formula and $p(x)$ a global $\phi$-type. Then Borel
definability and strong Borel definability are defined similarly. The Borel
definability of a global $\phi$-measure is defined similarly.
(v) For non-global types and non-global $\phi$-types, the Borel definability
and strong Borel definability are defined similarly. For non-global measures
and non-global $\phi$-measures, the Borel definability is defined similarly.
(We will shorlty give the suitable notion to measures, cf. 3.6.)

######  Fact 3.5.

(a) Let $A$ be a (small) set and $p(x)$ a global type which is finitely
satisfiable in $A$. For any $L$-formula $\phi(x,y)$, define the function
$p_{\phi}:S_{\tilde{\phi}}({\cal U})\to\\{0,1\\}$, denoted by $p_{\phi}(q)=1$
iff $\phi(x,b)\in p$ for some (any) $b\models q$. Then the following are
equivalent:
(i) $p(x)$ is strongly Borel definable.
(ii) For any $L$-formula $\phi(x,y)$, the function $p_{\phi}$ is $DBSC$. (Cf.
Definitions 2.2, 2.7.)

(b) The same holds for global $\phi$-types, non-global types and non-global
$\phi$-types.

###### Proof.

This follows from Facts 2.4 and 3.2(i). ∎

We now give the suitable notion to measures.

######  Definition 3.6 (Baire-1/2 definable measure).

(i) Let $A$ be a (small) set and $\mu(x)$ a global (Kiesler) measure which is
$A$-finitely satisfiable. For any $L$-formula $\phi(x,y)$, define the function
$\mu_{\phi}:S_{\tilde{\phi}}({\cal U})\to[0,1]$, denoted by
$\mu_{\phi}(q):=\mu(\phi(x,b))$ for some (any) $b\models q$. $\mu(x)$ is
called (generalized) Baire-1/2 over $A$ if for any $L$-formula $\phi(x,y)$,
the function $\mu_{\phi}$ is (generalized) Baire-1/2.
(ii) For global $\phi$-measures, non-global measures and non-global
$\phi$-measures, the (generalized) Baire-1/2 definability is defined
similarly.

In the next lemma we give a new characterization of $NIP$ for formulas. Later
we will even provide a generalization of it. (Cf. Corollary 4.9 below.) First,
we need some definitions. (Note the distinct in form between the two
definitions below. In fact, the part (i) of Definition 3.7 coresponds to $NIP$
in classical logic and the part (ii) corresponds to $NIP$ in continuous
logic.)

######  Definition 3.7.

Let $X$ be a set and $(f_{i})$ a sequence of $[0,1]$-valued functions on $X$.

(i) We say the independence property is uniformly blocked on $(f_{i})$ if
there is a natural number $N$ such that for each $r<s$ there is a set
$E\subset\\{1,\ldots,N\\}$ such that for every sequence $(a_{i})$ and each
$i_{1}<\cdots<i_{N}<\omega$, the following does not hold

$\exists y\big{(}\bigwedge_{j\in E}f_{i_{j}}(y)\leq
r~{}\wedge~{}\bigwedge_{j\in N\setminus E}f_{i_{j}}(y)\geq s\big{)}.$

(ii) We say the independence property is semi-uniformly blocked on $(f_{i})$
if for each $r<s$ there is a natural number $N_{r,s}$ and a set
$E\subset\\{1,\ldots,N_{r,s}\\}$ such that for every sequence $(a_{i})$ and
each $i_{1}<\cdots<i_{N_{r,s}}<\omega$, the following does not hold

$\exists y\big{(}\bigwedge_{j\in E}f_{i_{j}}(y)\leq
r~{}\wedge~{}\bigwedge_{j\in N\setminus E}f_{i_{j}}(y)\geq s\big{)}.$

######  Lemma 3.8.

Let $T$ be complete theory and $\phi(x,y)$ an $NIP$ formula. Suppose that
$(a_{n})$ is a sequence (in the monster model of $T$) and $(g_{n})$ is a
sequence of functions of the form
$g_{n}=\sum_{i=1}^{k_{n}}r_{i}\cdot\phi(a_{i},y)$ where $\sum r_{i}=1$ and
$n\in\mathbb{N}$. Then the following properties hold (are even equivalent):

(i) There is a subsequence $(a_{n}^{\prime})\subseteq(a_{n})$ such that:

(a) There is a real number $C$ such that for all $b$ in the monster model,

$\sum_{n<\omega}\big{|}\phi(a_{n+1}^{\prime},b)-\phi(a_{n}^{\prime},b)\big{|}\leq
C.$

(b) The independence property is uniformly blocked on
$(\phi(a_{n}^{\prime},y))$.

(ii) There is a subsequence $(g_{n}^{\prime})\subseteq(g_{n})$ such that the
independence property is semi-uniformly blocked on $(g_{n}^{\prime})$.

(iii) $(\phi(a_{n}^{\prime},y))$ converges pointwise to a function $f$ which
is strongly $DBSC$.

(iv) $(g_{n}^{\prime})$ converges pointwise to a function $g$ which is
Baire-1/2.

###### Proof.

(i)(a) and (i)(b) follow from Proposition 2.14(iii) and Lemma 2.8 of [Kha19b].
(In fact, they are equivalent.)

(ii): We use some result of [Ben09] and [BK09]. Indeed, Let
$A=\\{a_{n}:n<\omega\\}$, $M^{*}$ a model such that every type over $A$ is
realized in it, and $T_{*}=Th(M^{*})$. Note that every $g_{n}$ can be extended
to a global Keisler measure $\nu_{n}$. (Indeed, if $g_{n}=\sum
r_{i}\phi(a_{i},y)$, define $\nu_{n}(x)=\sum r_{i}\text{tp}(a_{i}/M^{*})$.
That is, $\nu(\psi(x))=\sum r_{i}\psi(a_{i})$ for all $L(M^{*})$-formula
$\psi(x)$.) Note that $\nu_{n}$ is a measure on the Stone space $S(T_{*})$
(cf. [S15]). Then, by Corollary 2.10 of [BK09], every measure $\nu_{n}$ in
$T_{*}$ corresponds to a type ${\bf p}_{n}$ in the randomization $T_{*}^{R}$
of $T_{*}$. In this case, for every $L(M^{*})$-formula $\psi(x)$,
$\nu_{n}(\\{q:\psi(x)\in q\\})=(\mu[[\psi(x)]])^{{\bf P}_{n}}$ where
$\mu[[\psi(x)]]$ is the corresponding formula in $T_{*}^{R}$. By Theorem 5.3
(or Theorem 4.1) of [Ben09], as $\phi(x,y)$ is $NIP$ in $T$, its corresponding
formula $\mu[[\phi(x,y)]]$ is dependent in $T_{*}^{R}$. Note that the Ben
Yaacov’s argument is essentially local (formula-by-formula), although the
statement of Theorem 5.3 is global, it is easy to check that the local result
holds too. Let ${\bf a}_{n}$ be a realization of ${\bf p}_{n}$. Now, as
$T_{*}^{R}$ is a continuous theory and the formula $\mu[[\phi(x,y)]]$ is
dependent in it, there is a subsequence $({\bf a}_{n}^{\prime})\subseteq({\bf
a}_{n})$ such that the independence property is semi-uniformly blocked on
$(\mu[[\phi({\bf a}_{n}^{\prime},y)]])$. If not, by compactness (of continuous
logic) and Ramsey’s theorem, one can show that the formula $\mu[[\phi(x,y)]]$
has $IP$, a contradiction. (See also Theorem 4.3(iv) of [Kha20a].) This means
that the independence property is semi-uniformly blocked on the corresponding
sequence $(\nu_{n}^{\prime})$ of measures in $T$. (Note that our terminology
is different from [Ben09] that this notion is called the uniformly dependence.
Because the natural number $N_{r,s}$ depends to $r,s$, we believe that semi-
uniformly dependence is more suitable.)

(iii) is Proposition 2.14(iv) in [Kha19b].

(iv): First, notice that $(g_{n}^{\prime})$ converges to a Baire-1 function,
denoted by $g$. This is a well known result in functional analysis, due to
Rosenthal [Ros74]. Suppose, for a contradiction, that $g$ is not Baire-1/2.
Equivalently, suppose that the limit $f$ of $(\mu[[\phi({\bf
a}_{n}^{\prime},y)]])$ is not Baire-1/2. So, by the equivalence (1)
$\Leftrightarrow$ (5) of Proposition 2.3 of [HOR91], there are $r<s$ such that
the ordinal index $\alpha(f;r,s)$ is infinite; equivalently, for all natural
numbers $m$, $K_{m}(f;r,s)\neq\emptyset$. (See [HOR91, page 7] for the
definitions of $K_{m}$ and $\alpha$.) By Lemma 3.1 in [HOR91], for all $m$
there are $r<r^{\prime}<s^{\prime}<s$ and a subsequence $(\mu[[\phi({\bf
a}_{j_{i}},y)]]:i<\omega)$ so that

$\exists y\big{(}\bigwedge_{i\in E}\phi({\bf a}_{i},y)\leqslant
r^{\prime}~{}\wedge~{}\bigwedge_{i\in F}\phi({\bf a}_{i},y)\geqslant
s^{\prime}\big{)}$

holds, for all disjoint subsets $E,F$ of $\\{1,\ldots,m\\}$. So, by the
compactness theorem, $\mu[[\phi(x,y)]]$ has $IP$, a contradiction. This means
that the limit of $(\nu_{n}^{\prime})$ (or $(g_{n}^{\prime})$) is Baire-1/2. ∎

In the following we give a result on definability of measures on countable
models (sets), assuming $NIP$. In the next section, we give an alternative
proof which is also a generalization to the general case (i.e. without
assuming countability).

Recall that $\Delta_{\phi}$ is the set of all Boolean combinations of the
instances of $\phi(x,y)$ (with the variable $x$).

######  Theorem 3.9.

The following are equivalent:

  * (i)

$\phi$ has $NIP$ (for theory $T$).

  * (ii)

For any $\mu(x)\in{\mathfrak{M}}_{\phi}({\cal U})$ and any countable model
$M$, if $\mu$ is finitely satisfiable in $M$, then $\mu$ is Baire-1/2
definable over $M$.

  * (iii)

For any $p(x)\in S_{\phi}({\cal U})$ and any countable model $M$, if $p$ is
finitely satisfiable in $M$, then $p$ is strongly Borel definable over $M$.

###### Proof.

(i) $\Longrightarrow$ (ii): By Fact 3.2(v) above, for any
$\theta(x)\in\Delta_{\phi}$, there is function $f$ in the pointwise closure of
the convex hull $\overline{\textrm{conv}}(A)$ of
$A=\\{\theta(a,y):S_{\tilde{\theta}}({\cal U})\to\\{0,1\\}|~{}a\in M\\}$ such
that $f(q)=\mu(\theta(x,b))$ for some (any) $b\models q$. (Note that we can
assume the functions in $A$ (and so the function $f$) are on
$S_{\tilde{\theta}}(M)$. Therefore, as $M$ is countable,
$S_{\tilde{\theta}}(M)$ is Polish and these functions are defined on a Polish
space.) As $\theta$ is $NIP$, by Proposition 5J of [BFT78], there is a
sequence $(f_{n})$ in the convex hull $\textrm{conv}(A)$ of $A$ such that
$f_{n}\to f$; that is, there is a sequence $(f_{n})$ of the form
$f_{n}(y)=\sum_{i=1}^{k_{n}}r_{i}\cdot\theta(a_{i},y)$ (where $a_{i}\in M_{0}$
and $\sum r_{i}=1$) such that $f_{n}\to f$ pointwise. (Note that since
Proposition 5J of [BFT78] needs countability, one can consider rational convex
hull of $A$; that is, the set of functions
$\sum_{i=1}^{k}r_{i}\cdot\theta(a_{i},y)$ where
$r_{i}\in{\mathbb{Q}}\cap[0,1]$ and $\sum_{i=1}^{k}r_{i}=1$.) As $\theta$ is
$NIP$, by Lemma 3.8(ii) above, there is a subsequence
$(f_{n}^{\prime})\subseteq(f_{n})$ such that the independence property is
semi-uniformly blocked on $(f_{n}^{\prime})$, and consequently the sequence
$(f_{n}^{\prime})$ converges to a Baire-1/2 function, by Lemma 3.8(iv). This
means that $f$ is Baire-1/2 and so $\mu$ is Baire-1/2 definable.

(i) $\Longrightarrow$ (iii) is similar to (i) $\Longrightarrow$ (ii) and even
easier. By Fact 3.2(v) above, there is function $f$ in the pointwise closure
of $A=\\{\phi(a,y):S_{\tilde{\phi}}({\cal U})\to\\{0,1\\}|~{}a\in M\\}$ such
that $f(q)=\phi(p,b)$ for some (any) $b\models q$. As $\phi$ is $NIP$, by
Proposition 5J (or even Theorem 4D) of [BFT78], there is a sequence
$(\phi(a_{n},y))$ in $A$ such that $\phi(a_{n},y)\to f(y)$ pointwise. By Lemma
3.8(iii), $f$ is strongly $DBSC$ or equivalently strongly Borel.

(iii) $\Longrightarrow$ (i): By the equivalence (iv) $\Longleftrightarrow$
(vi) of [BFT78, Theorem 4D], we need to show that for any countable model $M$,
the family $\phi(a,y),a\in M$, is relatively compact in $M_{r}(X)$, where
$X=S_{\tilde{\phi}}(M)$ and $M_{r}(X)$ is the space of all functions which are
measurable with respect to all Radon measures on $X$ (cf. [BFT78, Definition
1A(l)]). Suppose that $M$ is a countable model, $(a_{i})_{i\in I}\in M$ and
$\phi(a_{i},y)\to f(y)$ pointwise on $X=S_{\tilde{\phi}}(M)$. Note that $f(y)$
defines the type $p(x)=\lim_{i}tp(a_{i}/{\mathcal{U}})$. As $p$ is finitely
satisfiable in $M$, by (iii), $f$ is Borel measurable. Therefore, by the
equivalence (iv) $\Longleftrightarrow$ (vi) of [BFT78, Theorem 4D], the proof
is completed.

(ii) $\Longrightarrow$ (i): As every type is a measure, the argument (iii)
$\Longrightarrow$ (i) above works well. ∎

######  Remark 3.10.

Note that Theorem 3.9 can be proved for global types/measures instead of
$\phi$-types/$\phi$-measures. That is, the following are equivalent:
(i) $T$ has $NIP$.
(ii) For any global measure $\mu(x)\in{\mathfrak{M}}({\cal U})$ and any
countable model $M$, if $\mu$ is finitely satisfiable in $M$, then $\mu$ is
Baire-1/2 definable over $M$.
(iii) For any global type $p(x)\in S({\cal U})$ and any countable model $M$,
if $p$ is finitely satisfiable in $M$, then $p$ is strongly Borel definable
over $M$.
(Cf. Remark 3.3 above.)

##  4 Glivenko–Cantelli classes

The classical theorem of Glivenko and Cantelli is a generalization of the law
of large numbers in probability theory. The families of sets/functions for
which the consequence of the Glivenko–Cantelli theorem holds are called
Glivenko–Cantelli classes. Vapnik and Chervonenkis gave a characterization of
such classes [VC71]. As mentioned before, Talagrand’s characterization [T87]
of Glivenko–Cantelli classes has some adventures over [VC71]. In this case,
the fundamental notion has been invented by David H. Fremlin, namely the
Fremlin–Talagrand stability. From a logical point of view, this notion was
first studied in [Kha16] in the framework of integral logic.

######  Definition 4.1 (Fremlin–Talagrand stability, [Fre06], 465B).

Let $A\subseteq C(X)$ be a pointwise bounded family of real-valued continuous
functions on $X$. Suppose that $\mu$ is a measure on $X$. We say that $A$ is
$\mu$-stable, if $A$ is a stable set of functions in the sense of Definition
465B in [Fre06], that is, whenever $E\subseteq M$ is measurable, $\mu(E)>0$
and $s<r$ in $\mathbb{R}$, there is some $k\geqslant 1$ such that
$(\mu^{2k})^{*}D_{k}(A,E,s,r)<(\mu E)^{2k}$ where

$\displaystyle D_{k}(A,E,s,r)=\bigcup_{f\in A}\big{\\{}w\in$ $\displaystyle
E^{2k}:f(w_{2i})\leqslant s,~{}f(w_{2i+1})\geqslant r\textrm{ for
}i<k\big{\\}}.$

######  Remark 4.2.

Recall from [Kha20, Corollary 3.15] that a formula $\phi(x,y)$ has $NIP$ if
and only if for any model $M$ and any Keisler $\phi$-measure $\mu(x)$ over $M$
the family $\phi(x,b),b\in M$ is $\mu$-stable.

The following theorem is a translation of [Fre06, Theorem 465M] and asserts
that a formula $\phi(x,y)$ has $NIP$ if and only if for any model $M$ and any
Keisler $\phi$-measure $\mu(x)$ over $M$ the family $\phi(x,b),b\in M$ is a
Glivenko–Cantelli classe with respect to $\mu$.

######  Theorem 4.3.

The following are equivalent:

  * (i)

$\phi$ has $NIP$ (for theory $T$).

  * (ii)

For any model $M$ and any measure $\mu\in{\mathfrak{M}}_{\phi}(M)$,
$\sup_{b\in M}|\frac{1}{k}\sum_{1}^{k}\phi(p_{i},b)-\mu(\phi(x,b))|\to 0$ as
$k\to\infty$ for almost every $(p_{i})\in S_{\phi}(M)^{\mathbb{N}}$ with the
product measure $\mu^{\mathbb{N}}$.

###### Proof.

(i) $\Longrightarrow$ (ii): Let $M$ be a model of $T$, and $\mu(x)$ a Keisler
measure over $M$. By Corollary 3.15 of [Kha20], as $\phi$ is $NIP$ (for theory
$T$), the set $A=\\{\phi(x,b):b\in M\\}$ is $\mu$-stable for every Radon
measure on $S_{\phi}(M)$, in the sense of [Kha20, Definition 3.5]. (Note that
as $\phi$ is $NIP$, we do not need to assume $M$ is $\aleph_{0}$-saturated and
the argument of [Kha20, Corollary 3.15] works well.) By Theorem 465M of
[Fre06], $\sup_{b\in
M}|\frac{1}{k}\sum_{1}^{k}\phi(p_{i},b)-\mu(\phi(x,b))|\to 0$ for almost every
$(p_{i})\in S_{\phi}(M)^{\mathbb{N}}$ with the product measure.

(ii) $\Longrightarrow$ (i): Again, by Theorem 465M of [Fre06], for any Keisler
measure $\mu(x)$ over $M$, the set $A$ above is $\mu$-stable, and so $\phi$ is
$NIP$, by Corollary 3.15 of [Kha20]. ∎

One may demand a better result, that is, a results for Keisler measures (i.e.
non only the measures on the Boolean combinations of the instances of
$\phi(x,y)$). We respond positively to his/her demand:

######  Proposition 4.4.

Let $T$ be a countable theory. The following are equivalent:

  * (i)

$T$ is $NIP$.

  * (ii)

For any model $M$ and any measure $\mu\in{\mathfrak{M}}(M)$, there is a set
$X\subseteq S(M)^{\mathbb{N}}$ of the full measure 1 such that for any formula
$\theta(x,\bar{y})$,

$\sup_{{\bar{b}}\in
M}|\frac{1}{k}\sum_{1}^{k}\theta(p_{i},{\bar{b}})-\mu(\theta(x,{\bar{b}}))|\to
0$

as $k\to\infty$ for all $(p_{i})\in X$.

###### Proof.

Let $\theta(x,\bar{y})$ be a formula. By Theorem 4.3, it is easy to verify
that the set $X_{\theta}$ of all $(p_{i})\in S(M)^{\mathbb{N}}$ such that
$\sup_{{\bar{b}}\in
M}|\frac{1}{k}\sum_{1}^{k}\theta(p_{i},{\bar{b}})-\mu(\theta(x,{\bar{b}}))|\to
0$ for all $(p_{i})\in X_{\theta}$ has the measure 1 with respect to
$\mu^{\mathbb{N}}$. On the other hand, as measures are continuous of above
(cf. [Fol99, Theorem 1.8]), there is a set $X\subseteq S(M)^{\mathbb{N}}$ of
the full measure 1 such that for any formula $\varphi(x,\bar{y})$,
$\sup_{{\bar{b}}\in
M}|\frac{1}{k}\sum_{1}^{k}\varphi(p_{i},{\bar{b}})-\mu(\varphi(x,{\bar{b}}))|\to
0$ as $k\to\infty$ for all $(p_{i})\in X$. Indeed, note that the set of all
formulas (with the variable $x$) is countable. Let
$\\{\theta_{1}(x),\theta_{2}(x),\ldots\\}$ be an enumeration of them and
$X=\bigcap_{n}X_{\theta_{n}}$. Therefore,
$\mu^{\mathbb{N}}(X)=\lim_{n}\mu^{\mathbb{N}}(\bigcap_{1}^{n}X_{\theta_{i}})=1$.
∎

######  Remark 4.5.

In Theorem 4.3, suppose that $N\prec M$ is a small model and $\mu$ is
$N$-finitely satisfiable. Let $X$ be the set of all $N$-finitely satisfiable
$\phi$-types over $M$. Then it is easy to verify that for each $b\in M$,
$\mu(\phi(x,b))=\mu(\phi(x,b)\cap X)$. In fact, $\mu(X)=1$. Assuming $NIP$ for
theory, the same holds if $\mu$ is $N$-invariant and $X$ is the set of all
$N$-invariant $\phi$-types over $M$ (cf. [HP11, Proposition 4.6]), but not in
general.

######  Remark 4.6.

(i) In Theorem 4.3 above, if we consider $X_{1},\ldots,X_{m}$ a finite
collection of Borel over $M$ sets, then for each $i\leq m$, the set
$A_{i}=\\{\phi(x,b)\cap X_{i}:b\in M\\}$ is $\mu$-stable (for all Radon
measures $\mu$). So, the above argument implies the consequences of Corollary
2.6 of [HPS13].
(ii) Note that for a measure $\mu$ the direction (i) $\Longrightarrow$ (ii)
holds by a weaker assumption. That is, a measure is $\mu$-stable if and only
if it is frequency interpreted.

######  Corollary 4.7.

The following are equivalent:

  * (i)

$\phi$ has $NIP$ (for theory $T$).

  * (ii)

For any $\mu(x)\in{\mathfrak{M}}_{\phi}({\cal U})$ and any (not necessarily
countable) model $M$, if $\mu$ is finitely satisfiable in $M$, then $\mu$ is
(generalized) Baire-1/2 definable over $M$. (Cf. Definition 3.6.)

  * (iii)

For any $p(x)\in S_{\phi}({\cal U})$ and any (not necessarily countable) model
$M$, if $p$ is finitely satisfiable in $M$, then $p$ is (generalized) $DBSC$
over $M$. (Cf. Definition 3.4 and Fact 3.5.)

###### Proof.

(i) $\Longrightarrow$ (ii): By Theorem 4.3, for each $\epsilon>0$, there is
$k$ such that $\sup_{b\in\cal
U}|\frac{1}{k}\sum_{1}^{k}\phi(p_{i},b)-\mu(\phi(x,b))|\leq\epsilon$ for
almost every $(p_{i})\in S_{\phi}({\cal U})^{k}$ with the product measure
$\mu^{k}$. By Remark 4.5, note that $\mu(S_{\phi}({\cal U}))=\mu(X)$ where $X$
is the set of all all $M$-finitely satisfiable global $\phi$-types. Recall
from [HP11, Proposition 2.6] that every invariant global type is strongly
Borel definable. It is easy to check that every finitely satisfiable
$\phi$-type (in not necessarily $NIP$ theory) can be extended to a global
finitely satisfiable type. Therefore, using an argument similar to Proposition
2.6 of [HP11], one can show that every $p\in S_{\phi}({\cal U})$ is strongly
Borel definable over $M$. For each $k$, let $(p_{1}^{k},\ldots,p_{k}^{k})\in
X^{k}$ such that $\sup_{b\in\cal
U}|\frac{1}{k}\sum_{1}^{k}\phi(p_{i}^{k},b)-\mu(\phi(x,b))|\leq\epsilon$. Let
$f_{k}(y)=\frac{1}{k}\sum_{1}^{k}g_{i,k}(y)$ where $g_{i,k}$ defines
$p_{i}^{k}$. Clearly, as $g_{i,k}$’s are $DBSC$, $f_{k}$ is $DBSC$. Therefore,
the functions $f_{k}$’s are strongly Borel and the sequence $(f_{k})$
uniformly converse to a fuction $f$ which defines $\mu$. This means that $\mu$
is (generalized) Baire-1/2 definable.

(i) $\Longrightarrow$ (iii) follows from Proposition 2.6 of [HP11].

(ii) $\Longrightarrow$ (i) and (iii) $\Longrightarrow$ (i) follow from the
directions (ii) $\Longrightarrow$ (i) and (iii) $\Longrightarrow$ (i) of
Theorem 3.9 above. ∎

######  Remark 4.8.

(i) Assuming $NIP$ for the theory, the above result holds for invariant
measures rather than coheirs.

(ii) When $M$ is countable, the above corollary gives an alternative proof of
Theorem 3.9.

The following is a generalization of Lemma 3.8 above.

######  Corollary 4.9.

Let $T$ be complete theory, $M$ a model, and $\phi(x,y)$ an $NIP$ formula.
Suppose that $(\mu_{n})$ is a sequence of Keisler $\phi$-measures over $M$.
Then there is a subsequence $(\mu_{n}^{\prime})\subseteq(\mu_{n})$ such that
$(\mu_{n}^{\prime})$ converges to a measure $\mu$ over $M$. Moreover, if the
$\mu_{n}$’s are finitely satisfiable in $N\preceq M$, then $\mu$ is
generalized Baire-1/2 definable over $N$. In particular, if $N$ is also
countable, $\mu$ is Baire-1/2 definable over $N$.

###### Proof.

Use Lemma 3.8, Theorem 4.3 and Remark 4.5. Indeed, for each $n$, by Theorem
4.3 (or Proposition 4.4) there is a measure $\nu_{n}(\theta(x,b))=\sum
r_{i}\theta(p_{i},b)$ for all $\theta(x)\in\Delta_{\phi}$ (where $p_{i}\in
S(M)$ and $\sum r_{i}=1$) such that $\sup_{b\in
M}|\nu_{n}(\theta(x,b))-\mu_{n}(\theta(x,b))|<1/n$. By Lemma 3.8, there is a
subsequence $(\nu_{n}^{\prime})\subseteq(\nu_{n})$ which converges to a
measure $\mu$. (Indeed, note that the set of all Boolean combinations of the
instances of $\phi(x,y)$ is countable, and so using an easy diagonal argument
(and Lemma 3.8(ii)), we can find the desired subsequence $(\nu_{n}^{\prime})$,
i.e. for any $\theta(x,y)\in\Delta_{\phi}$, the sequence
$\big{(}\nu_{n}^{\prime}(\theta(x,y))\big{)}$ converges pointwise.) Clearly,
its corresponding subsequence $(\mu_{n}^{\prime})$ converges to $\mu$.
Moreover, if the $\mu_{n}$’s are finitely satisfiable in $N\preceq M$, by
Remark 4.5, $\mu$ is generalized Baire-1/2 definable over $N$. When $N$ is
countable, by Lemma 3.8(iv), $\mu$ is Baire-1/2 definable over $N$. ∎

######  Remark 4.10.

We reiterate that the global case is also established. That is, assuming $T$
is countable and $NIP$, the argument of Corollary 4.9 works well. On the other
hand, it seems that an adaptation of the argument of Lemma 3.8 must prove a
similar result.

The following is a generalization of an observation due to Kyle Gannon [G];
that is, we do not assume countability.

######  Theorem 4.11.

Assume that $\phi(x,y)$ is $NIP$ and $\mu\in{\mathfrak{M}}_{\phi}({\cal U})$
is finitely satisfiable in a (not necessarily countable) model $M$. Let $\nu$
be any local Keisler measure in $S_{\tilde{\phi}}(M)$ and $\epsilon>0$.

(i) There exists a measurable set $C\subseteq S_{\tilde{\phi}}(M)$ such that:

1) $\nu(C)>1-\epsilon$, and

2) there exist $\bar{a}_{1},\ldots,\bar{a}_{k}\in M$ such that for all $b$ (in
the monster model) such that $tp_{\tilde{\phi}}(b/M)\in C$, we have:

$\Big{|}\frac{\sum_{i=1}^{k}\theta_{i}(\bar{a}_{i},b)}{m}-\mu(\phi(x,b))\Big{|}<\epsilon,$

where $\theta_{i}$ is of the form $\bigwedge_{j}\phi(a_{j},y)$ and $a_{j}\in
M$ and $k\leq m\in{\mathbb{N}}$.

(ii) Suppose moreover that $M$ is countable, then

(a) there exist a measurable set $C$ with the condition (1) and
$a_{1},\ldots,a_{n}\in M$ such that for all $b$ (in the monster model) such
that $tp_{\tilde{\phi}}(b/M)\in C$, we have:
$|\sum_{1}^{n}\phi(a_{i},b)-\mu(\phi(x,b))|<\epsilon$.

(b) Consequently, there is a measurable subset $E\subseteq
S_{\tilde{\phi}}(M)$ with $\nu(E)>1-\epsilon$ and Baire-1/2 function $f$ on
$S_{\tilde{\phi}}(M)$ such that $\mu(\phi(x,b))=f(b)$ for all $b\in\cal U$,
$tp_{\tilde{\phi}}(b/M)\in E$.

###### Proof.

(i): By Theorem 4.3 and Remark 4.5, there are types $p_{1},\ldots,p_{n}\in
S_{\phi}({\cal U})$, which are finitely satisfiable in $M$ and

$|\sum_{1}^{n}\phi(p_{i},b)-\mu(\phi(x,b))|<\epsilon.$

Recall from Fact 3.2 that every $p_{i}$ corresponds to a function on
$S_{\tilde{\phi}}(M)$, denoted by $p_{i}$ again, which defines it. By Lusin’s
theorem ([Fol99, Theorem 7.10]), there are continuous functions $f_{i}$’s on
$S_{\tilde{\phi}}(M)$ such that (for $i\leq n$) $f_{i}=p_{i}$ expect on a set
$S_{\tilde{\phi}}(M)\setminus C_{i}$ of measure $<\epsilon$. By the
Stone–Weierstrass theorem, for any $i\leq n$ there is a function $\psi_{i}(y)$
of the form $\sum_{1}^{k}r_{i}.\theta_{i}(\bar{a}_{i},y)$ (where $\theta_{i}$
is of the form $\bigwedge_{j}\phi(a_{j},y)$ and $a_{j}\in M$) such that
$\sup_{b\in\cal U}|f_{i}(b)-\psi_{i}(b)|<\epsilon$. So putting everything
together the desirable result follows.

(ii): Suppose moreover that $M$ is countable.

(a): Note that every $p_{i}$ (above) is in the closure of the convex hull of
the $\phi(a_{i},y)$’s ($a_{i}\in M$). As $M$ is countable, by Proposition 5J
of [BFT78], any $p_{i}$ is the limit of a sequence $(g_{n})$ where $g_{n}$ is
of the form $\sum r_{i}.\phi(a_{i},y)$ ($a_{i}\in M$ and $\sum r_{i}=1$). By
Egorov’s theorem, there exists a measurable set $C$ with the condition (1)
such that $(g_{n})$ uniformly converges to $p_{i}$ on $C$. Again, putting
everything together the desirable result follows.

(b): As $M$ is countable, by the previous paragraph, set
$\bar{a}_{n}=(a_{1}^{n},\ldots,a_{k}^{n})\in M$ and measurable $C_{n}\subseteq
S_{\tilde{\phi}}(M)$ such that $\nu(C_{n})>1-\epsilon/2^{n}$ and
$|\sum_{1}^{k}\phi(a_{i}^{n},b)-\mu(\phi(x,b))|<\epsilon$ for $b\in\cal U$,
$tp_{\tilde{\phi}}(b/M)\in C_{n}$. Let $E=\bigcap_{n}C_{n}$. By Lemma 3.8, the
sequence $\big{(}Av(\bar{a}_{n})(\phi(x,y))\big{)}$ converges to a Baire-1/2
function $f(y)$ on $S_{\tilde{\phi}}(M)$. Clearly, $\mu(\phi(x,b))=f(b)$ for
all $b\in\cal U$, $tp_{\tilde{\phi}}(b/M)\in E$. ∎

######  Remark 4.12.

(i) As mentioned above, the part (ii)(a) of Theorem 4.11 (i.e. without
Baire-1/2 definability) was suggested by Gannon [G].

(ii) The regularity of $\nu$ has been used for the general case (i.e., not
necessarily countable models). However, in countable case, regularity is
established automatically; that is, every Borel probability measure on a
metric space is regular.

The following is another application of the Stone–Weierstrass theorem:

######  Proposition 4.13.

Assume that $\phi(x,y)$ is a (not necessarily $NIP$) formula,
$\mu\in{\mathfrak{M}}_{\phi}({\cal U})$ and $M$ a model. If $\mu$ is finitely
satisfiable in and definable over $M$, then for any $\epsilon>0$ there exist
$\bar{a}_{1},\ldots,\bar{a}_{k}\in M$ such that for all $b$ (in the monster
model), we have:

$\Big{|}\frac{\sum_{i=1}^{k}\theta_{i}(\bar{a}_{i},b)}{m}-\mu(\phi(x,b))\Big{|}<\epsilon,\
\ \ \ \ \ \boxtimes$

where $\theta_{i}$ is of the form $\bigwedge_{j}\phi(a_{j},y)$ and $a_{j}\in
M$ and $k\leq m\in{\mathbb{N}}$.

###### Proof.

As $\mu$ is definable over $M$, its corresponding functions on
$S_{\tilde{\phi}}(M)$ is continuous. By the Stone–Weierstrass theorem, it is
in the closure of the algebra generated by formulas $\phi(a,y),a\in{\cal U}$.
So $\boxtimes$ holds. ∎

######  Remark 4.14.

(i) The notion “$NIP$ of $\phi(x,y)$ in a model” was introduced in [Kha20] and
more applications of it were presented in [KP18]. Some of the above results in
the present article– not all them– can be proved by the weaker assumption
‘$NIP$ in a model’. For this, one can use Fact 3.2 above and Propositions 5I
and 5J of [BFT78].
(ii) As the types in continuous logic correspond to the measures in classical
logic, all results of [Kha20, Section 3] can be translated for measures in
classical logic. These ideas, as well as the original ideas in functional
analysis, seem to have led to results in model theory, which are sometimes not
explicitly and deservedly mentioned in the literature!

######  Question 4.15.

We say that a theory $T$ is $DBSC$ if for any countable model $M$ and formula
$\phi(x,y)$, every $M$-finitely satisfiable Keisler $\phi$-measure over $M$ is
$DBSC$ definable over $M$. Some questions arise. Are there any unstable $NIP$
theories which are $DBSC$? If yes, Are there interesting such theories? Is
there any characterization of $DBSC$ theories? Does it separate interesting
theories?

##  Appendix A Angelic spaces of global types/measures

We apply the work of Bourgain-Fremlin-Talagrand on Rosenthal compacta to give
a short proof (and a refinement) of a very recent result of Gannon (which is
based on a result due to Simon).

We emphasize that our approach allows us to make easy generalizations as
follows:
(i) The following argument works well in the framework of continuous logic.
(ii) The argument can be adapted to invariant types/measures. That is, given a
countable $NIP$ theory $T$ and countable model $M$, the space of global
$M$-invariant types/measures is a Rosenthal compactum. (This generalizes
another result of Simon that is about $\phi$-types.)

However, to make it easier to read, we limit ourselves to the classic logic
and finitely satisfied types/measures.

The key here is how to capture a “global” type/measure with a suitable
function. By “global” we mean for all formulas, not just one. (Ben Yaacov and
Pillay had previously proposed a suitable function for the local types and
local coheirs, respectively.)

Recall that, given types $p_{1}(x),\ldots,p_{n}(x)$, the average measure of
them, denoted by $Av(p_{1},\ldots,p_{n})$ is defined as follows:

$Av(p_{1},\ldots,p_{n})(\theta(x))=\frac{|\\{i:~{}\theta(x)\in p_{i}\\}|}{n}\
\ \ \ \text{for all formula }\theta(x).$

######  Theorem A.1.

Let $T$ be a countable $NIP$ theory and $M$ a countable model of it. Let
$p(x)$ and $\mu(x)$ be global type and global measure which are finitely
satisfiable in $M$, respectively. Then

(i) (Simon) There is a sequence $(a_{n})\in M$ such that
$\lim_{n}tp(a_{n}/{\cal U})=p$.

(ii) (Gannon) There is a sequence $(\bar{a}_{n})\in M^{<\omega}$ such that
$\lim_{n}Av(\bar{a}_{n})=\mu$.

Moreover,

(iii) the limits in (i),(ii) above are $DBSC$ and Baire-1/2, respectively.
That is, for every formula $\theta(x,y)$, the function
$\lim_{n}\theta(a_{n},y)$ (resp. $\lim_{n}Av(\bar{a}_{n})(\theta(x,y))$) is
$DBSC$ (resp. Baire-1/2) on $S_{\tilde{\theta}}(M)$.

(iv) The space of global $M$-finitely satisfied types/measures is a Rosenthal
compactum.

###### Proof.

We want to store the information of each $M$-finitely satisfied type $p$
(resp. measure $\mu$) by a function of the form $f_{p}:X^{*}\to[0,1]$ (resp.
$f_{\mu}:X^{*}\to[0,1]$) where $X^{*}$ is a Polish space.

Let $\theta_{1}(x),\theta_{2}(x),\ldots$ be an enumeration of all formulas
with the variable $x$. For each $n$, let $X_{n}$ be the Stone space
$S_{\tilde{\theta}_{n}}(M)$. (Recall that the $X_{n}$’s are Polish.) For each
$n$, we adjoin an isolated point $q_{n}^{*}$ to $X_{n}$ and define
$X_{n}^{*}:=X_{n}\cup\\{q_{n}^{*}\\}$ with the topological sum.333See
https://math.stackexchange.com/questions/1275577/lusin-space-isolated-point
Set $X^{*}=\prod_{n}X_{n}^{*}$ with the product topology. (Recall that the
product of a countable number of Polish spaces is Polish.)

Let $a\in M$ and $q\in S_{\tilde{\theta}}(M)$. We let $\theta(a,q)=1$ if for
some (any) $b\models q$ we have $\models\theta(a,b)$, and $\theta(a,q)=0$ in
otherwise. For the isolated points $q_{n}^{*}$, we always assume that
$\theta_{n}(a,q_{n}^{*})=1$.

The following function $f_{a}$ stores the information of the type $tp(a/\cal
U)$ (for $a\in M$). For $a\in M$, we define a function $f_{a}:X^{*}\to[0,1]$
by $(q_{n})\mapsto\sum_{n=1}^{\infty}\frac{1}{2^{n}}\theta_{n}(a,q_{n})$.

We point out that for every $a\in M$ the information of the type $tp(a/\cal
U)$ is stored in the function $f_{a}$. (Why?) Therefore, every $M$-finitely
satisfiable global type corresponds to the limit of a net of functions in
$\\{f_{a}:a\in M\\}$. (Cf. Fact 3.2).

Since the functions $f_{a}$, $a\in M$, may not be continuous, we define the
following functions: for any $a\in M$ and $k\in\mathbb{N}$, we define a
function $f_{a}^{k}:X^{*}\to[0,1]$ by
$(q_{n})\mapsto\sum_{n=1}^{k}\frac{1}{2^{n}}\theta_{n}(a,q_{n})$.

Let $A=\\{f_{a}^{k}:a\in M,k\in{\mathbb{N}}\\}$. It is easy to verify that:
(1) $X^{*}$ is Polish.
(2) $f_{a}^{k}$ is continuous (for any $a\in M$, $k\in{\mathbb{N}}$).
(3) $f_{a}^{k}\to f_{a}$ as $k\to\infty$ (for all $a\in M$). Therefore,
$f_{a}$ is in the pointwise closure $\bar{A}$ of $A$.
(4) The information of every global $M$-finitely satisfied type $p(x)$ is
stored in a function $f_{p}\in\bar{A}$.

Claim: $A$ is relatively sequentially compact in $[0,1]^{X^{*}}$. That is,
every sequence in $A$ has a pointwise convergent subsequence.

###### Proof of Claim.

Let $(f_{a_{n}})$ be a sequence in $A$. As $\theta_{1},\theta_{2},\ldots$ are
$NIP$, by a diagonal argument, it is easy to verify that there is a convergent
subsequence. Indeed, let $(a_{n}^{1})$ be a subsequence of $(a_{n})$ such that
the sequence $(\theta_{1}(a_{n}^{1},y)$) converges as $n\to\infty$. Similarly,
by induction, let $(a_{n}^{i})$ be a subsequence of $(a_{n}^{i-1})$ such that
the sequence $(\theta_{i}(a_{n}^{i},y)$) converges as $n\to\infty$. Let
$(c_{n})$ be the diagonal of these sequences, i.e. $c_{n}=a_{n}^{n}$ for all
$n$.

It is easy to check that the subsequence $(f_{c_{n}})$ converges.

Alternatively, as Simon pointed out to us, the sequence $tp(a_{n}/\cal U)$ has
a convergent subsequence $tp(c_{n}/\cal U)$. Again, it is easy to check that
the subsequence $(f_{c_{n}})$ converges. ∎

By [BFT78, Theorem 4D], every $f$ in the closure of $A$ is the limit of a
sequence in $A$. By our translation (cf. Fact 3.2 and Remark 3.3), this means
that there is a sequence $(a_{n})\in M$ such that $\lim_{n}tp(a_{n}/{\cal
U})=p$. This proves (i).

(ii): Let $conv(A)$ be the convex hull of $A$. That is,

$conv(A)=\Big{\\{}\frac{f_{1}+\cdots+f_{n}}{n}:f_{i}\in A,~{}i\leq
n\in{\mathbb{N}}\Big{\\}}.$

It easy to verify that the information of $Av(a_{1},\ldots,a_{n})$ is stored
in $\lim_{k}\frac{1}{n}(f_{a_{1}}^{k}+\cdots+f_{a_{n}}^{k})$.

Again, by [BFT78, Proposition 5J], as $A$ is relatively sequentially compact,
$conv(A)$ is relatively sequentially compact, and so every $f$ in the closure
of $conv(A)$ is the limit of a sequence in $conv(A)$. By our translation (cf.
Fact 3.2 and Remark 3.3), this proves (ii).

(iii) follows from Lemma 3.2.

(iv) follows from the argument of (i), (ii). ∎

Acknowledgements. I want to thanks David Fremlin, Gilles Godefroy and Pierre
Simon for their helpful comments.

I would like to thank the Institute for Basic Sciences (IPM), Tehran, Iran.
Research partially supported by IPM grant no 1400030118.

## References

  * [Ben09] I. Ben Yaacov. Continuous and Random Vapnik-Chervonenkis Classes. Israel Journal of Mathematics 173 (2009), 309-333
  * [BK09] I. Ben Yaacov, and H.J. Keisler, Randomizations of models as metric structures. Confluentes Mathematici, 1:197–223 (2009).
  * [BFT78] J. Bourgain, D. H. Fremlin, and M. Talagrand. Pointwise compact sets of baire-measurable functions. American Journal of Mathematics, 100(4):pp. 845-886, 1978.
  * [CMR96] F. Chaatit, V. Mascioni and H. Rosenthal, On functions of finite Baire index, J. Funct. Anal. 142 (1996), no. 2, 277–295
  * [Con90] J. B. Conway. A course in functional analysis. Vol. 96. Springer Science & Business Media, 1990.
  * [Fol99] G. B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd edition, John Wiley & Sons Inc. (1999).
  * [Fre06] D. H. Fremlin, Measure Theory, vol.4, (Topological Measure Spaces, Torres Fremlin, Colchester, 2006).
  * [G] K. Gannon. Private communication.
  * [Hau62] F. Hausdorff, Set Theory, Chelsea, New York, 1962.
  * [HOR91] R. Haydon, E. Odell, H. Rosenthal, On certain classes of Baire-1 functions with applications to Banach space theory, Functional Analysis Proceedings, The University of Texas at Austin 1987-89, Lecture Notes in Math., vol. 1470, Springer-Verlag, New York, 1991, pp. 1-35.
  * [HPP08] E. Hrushovski, Y. Peterzil, and A. Pillay, Groups, measures, and the NIP, Journal of the American Mathematical Society, vol. 21 (2008), no. 2, pp. 563–596.
  * [HP11] E. Hrushovski, A. Pillay, On $NIP$ and invariant measures, Journal of the European Mathematical Society, 13 (2011), 1005-1061.
  * [HPS13] Hrushovski, Ehud, Anand Pillay, and Pierre Simon. Generically stable and smooth measures in NIP theories. Transactions of the American Mathematical Society 365.5 (2013): 2341-2366.
  * [Jay74] J. E. Jayne, Space of Baire functions. I, Annales de l’institut Fourier, tome 24, no 4 (1974), p. 47-76
  * [Kec95] A.S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, Springer, (1995).
  * [Kei87] H. J. Keisler, Measures and forking, Annals of Pure and Applied Logic, vol. 34 (1987), no. 2, pp. 119-169.
  * [Kha21] K. Khanaki, $\aleph_{0}$-categorical Banach spaces contain $\ell_{p}$ or $c_{0}$, (on arXiv 2016) accepted, 2021.
  * [Kha16] K. Khanaki, Amenability, extreme amenability, model-theoretic stability, and NIP in integral logic, Fundamenta Mathematicae 234 (2016), 253-286
  * [Kha19b] K. Khanaki, _Dividing lines in unstable theories and subclasses of Baire 1 functions_ , arXiv:1904.09486v6, 2019.
  * [Kha17] K. Khanaki, _NIP formulas and Baire 1 definability_ , unpublished note, arXiv:1703.08731, 2017.
  * [Kha20a] K. Khanaki, _On classification of continuous first order theories_ , submitted, 2020.
  * [Kha20] K. Khanaki, _Stability, the NIP, and the NSOP; Model Theoretic Properties of Formulas via Topological Properties of Function Spaces_ , Math. Log. Quart. 66, No. 2, 136-149 (2020) / DOI 10.1002/malq.201500059
  * [KP18] K. Khanaki, A. Pillay, Remarks on $NIP$ in a model, Math. Log. Quart. 64, No. 6, 429-434 (2018) / DOI 10.1002/malq.201700070
  * [Kur66] K. Kuratowski, Topology, vol. I. Academic, 1966.
  * [Las92] Michael C. Laskowski, Vapnik-Chervonenkis classes of definable sets, Journal of the London Mathematical Society, vol. 45 (1992), pp. 377-384.
  * [P18] A. Pillay, Generic stability and Grothendieck, South American Journal of Logic Vol. 2, n. 2,(2016), p. 1-6.
  * [Ros74] H. P. Rosenthal, A characterization of Banach spaces containing $l^{1}$, Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 2411-2413.
  * [Sh71] S. Shelah, Stability, the f.c.p., and superstability; model theoretic properties of formulas in first order theory, Annals of Mathematical Logic, vol. 3 (1971), no. 3, pp. 271-362.
  * [S15] P. Simon. A guide to NIP theories. Cambridge University Press, 2015.
  * [S15a] Pierre Simon, Invariant types in NIP theories, Journal of Mathematical Logic, (2015):
  * [T87] M. Talagrand, The Glivenko–Cantelli problem, Ann. Probab. 15 (1987), 837-870.
  * [VC71] V. N. Vapnik and A. Ya. Chervonenkis, On the uniform convergence of relative frequencies of events to their probabilities Theory of Probability and its Applications, vol. 16 (1971), no. 2, pp. 264-280.