# Sequential Approximations for Types and Keisler Measures

Kyle Gannon  Department of Mathematics
University of California, Los Angeles
Los Angeles, CA, 90095, USA<EMAIL_ADDRESS>
###### Abstract.

This paper is a modified chapter of the author’s Ph.D. thesis. We introduce
the notions of sequentially approximated types and sequentially approximated
Keisler measures. As the names imply, these are types which can be
approximated by a sequence of realized types and measures which can be
approximated by a sequence of “averaging measures” on tuples of realized
types. We show that both generically stable types (in arbitrary theories) and
Keisler measures which are finitely satisfiable over a countable model (in NIP
theories) are sequentially approximated. We also introduce the notion of a
smooth sequence in a measure over a model and give an equivalent
characterization of generically stable measures (in NIP theories) via this
definition. In the last section, we take the opportunity to generalize the
main result of [8].

###### Key words and phrases:

Keisler measures, NIP, generic stability; MSC2020: 03C45, 03C68

##  1\. Introduction

One of the joys of working in a metric space is that the closure of a set
coincides with its sequential closure. In particular, if $X$ is a metric
space, $A$ is a subset of $X$, and $b$ is in the closure of $A$, then there
exists a sequence of elements in $A$ which converges to $b$. In [17], Simon
showed that global types which are finitely satisfiable over a countable model
of a countable NIP theory admit a similar property. Let $T$ be a complete,
first-order theory, $\mathcal{U}$ a monster model of $T$, and $M$ a small
submodel of $\mathcal{U}$. Simon proved the following ([17, Lemma 2.8]):

######  Theorem 1.1.

Let $T$ be a countable NIP theory. Suppose $p$ is a type in
$S_{x}(\mathcal{U})$ and $p$ is finitely satisfiable over $M$ where
$|M|=\aleph_{0}$. Then there exists a sequence of points
$(a_{i})_{i\in\omega}$ in $M^{x}$ such that
$\lim_{i\to\infty}\operatorname{tp}(a_{i}/\mathcal{U})=p$.

One of the goals of this paper is to morally generalize the proof of the above
theorem in two different directions. By mimicking Simon’s proof, we are able
to prove the following,

  1. (T$1$)

Let $T$ be any countable theory. Suppose $p$ is a type in $S_{x}(\mathcal{U})$
and $p$ is generically stable over $M$. Then there exists a sequence of points
$(a_{i})_{i\in\omega}$ in $M^{x}$ such that
$\lim_{i\to\infty}\operatorname{tp}(a_{i}/\mathcal{U})=p$.

  2. (T$2$)

Let $T$ be a countable NIP theory. Suppose $\mu$ is a Keisler measure in
$\mathfrak{M}_{x}(\mathcal{U})$ and $\mu$ is finitely satisfiable over $M$
where $|M|=\aleph_{0}$. Then there exists a sequence of points
$(\overline{a}_{i})_{i\in\omega}$ in $(M^{x})^{<\omega}$ such that
$\lim_{i\to\infty}\operatorname{Av}(\overline{a}_{i})=\mu$. More explicitly,
for any formula $\varphi(x)$ in $\mathcal{L}_{x}(\mathcal{U})$, we have that

$\lim_{i\to\infty}\operatorname{Av}(\overline{a}_{i})(\varphi(x))=\mu(\varphi(x)).$

The proofs of both of these theorems are slightly more enjoyable than one
would anticipate. For example, we already know many diverse and useful
approximation theorems for measures in NIP theories (and some for generically
stable types in arbitrary theories) and so one might expect that our proofs
rely on composing approximation techniques. However, stringing together
different approximation methods can result in an array with some kind of
modes-of-convergence problem.

As stated previously, the technique used to prove both these theorems mimics
the argument used in [17, Lemma 2.8]. In the generically stable case, the set
up is identical: Suppose $p$ is in $S_{x}(\mathcal{U})$ where $p$ is
generically stable over $M$ and $I$ is a Morley sequence in $p$ over $M$. As
in Simon’s proof, we use both $M$ and $I$ to find an eventually indiscernible
sequence of points in $M^{x}$ which converge to $p|_{MI}$. The eventual EM-
type of this sequence over $M$ is precisely $p^{(\omega)}|_{M}$. Using generic
stability and compactness, we conclude that this sequence must converge to
$p$.

Our proof of the Keisler measure case is slightly more exotic since there is
no standard notion of a “Morley sequence in a Keisler measure”. The proof we
provide is essentially done in first order model theory (with an important
exceptional lemma following from Ben Yaacov’s work on randomizations [2]). We
expect that there exists other proofs using other methods such as continuous
model theory111In fact, after this paper was posted to arXiv, another proof
was discovered by Khanaki using BFT on an infinite product space [14].. The
proof we give here embraces the ideology first developed in [10] and shows
that this can be resolved by replacing the Morley sequence (in Simon’s proof)
by a smooth sequence in $\mu$ over $M$. This provides more evidence for the
intuition that smooth measures can play the role of realized types, at least
in the NIP context. After constructing a countable model $N_{\omega}$
“containing this sequence”, we find a sequence of points in
$(M^{x})^{<\omega}$ such that the corresponding average measures on these
tuples converge to $\mu|_{N_{\omega}}$. After constructing an eventually
indiscernible subsequence in this context, we are able to readapt most of
Simon’s proof technique by making use of known approximation theorems,
symmetry properties, and some basic integration techniques.

It is interesting to note that one can give another equivalent
characterization of generically stable measures in NIP theories using smooth
sequences. This characterization highlights the connection between generically
stable types and generically stable measures. Recall that a type $p$ is
generically stable over a model $M$ if for every Morley sequence
$(a_{i})_{i\in\omega}$ in $p$ over $M$,
$\lim_{i\to\infty}\operatorname{tp}(a_{i}/\mathcal{U})=p$. We show that in an
NIP theory, a measure $\mu$ is generically stable over a model $M$ if and only
if for every smooth sequence in $\mu$ over $M$, the limit of this sequence is
precisely $\mu$.

In addition to proving these theorems, we also introduce the classes of
sequentially approximated measures and sequentially approximated types. These
definitions can be seen as the global analogue to Khanaki’s definition of
Baire 1 definability for local types (see [13]). Sequentially approximated
measures should be thought of as a “halfway point” between finitely
approximated measures and Keisler measures which are finitely satisfiable over
a small model. For instance, we show that a Keisler measure is finitely
approximated if and only if it is both definable and sequentially approximated
(Proposition 3.4) and sequentially approximated measures commute with
definable measures (Proposition 3.7). Sequentially approximated types remain a
little more mysterious. We show that there exists a type such that its
corresponding Keisler measure is sequentially approximated (even finitely
approximated), but the type itself is not sequentially approximated
(Proposition 4.14).

In the last section, we consider connections to the local measure case and
generalize the main result in [8] (Theorem 6.4). Explicitly, the main result
in [8] demonstrates that if a formula $\varphi$ is NIP and $\mu$ is a
$\varphi$-measure which is $\varphi$-definable and finitely satisfiable over a
countable model, then $\mu$ is $\varphi$-finitely approximated in said model.
Here, we demonstrate that countable can be replaced by small.

This paper is structured as follows: In section 2, we discuss preliminaries.
In section 3, we describe sequentially approximated measures and sequentially
approximated types. In section 4, we show that if $p$ is generically stable
over $M$, then $p$ is sequentially approximated over $M$. We also give some
examples of types which are which are not sequentially approximated at the end
of the section. In section 5, we show that if $T$ is a countable NIP theory,
and $\mu$ is finitely satisfiable over a countable model $M$, then $\mu$ is
sequentially approximated over $M$. We then give an equivalent
characterization of generically stable measures in NIP theories using smooth
sequences. In section 6, we generalize the main theorem in [8].

### Acknowledgements

We would like to thank Gabriel Conant, James Hanson, Karim Khanaki, Pierre
Simon and our Ph.D. defense committee Daniel Hoffmann, Anand Pillay, Sergei
Starchenko, and Minh Chieu Tran for helpful discussions and comments. Thanks
also to the referee for many helpful comments. This paper was also partially
supported by the NSF research grant DMS-1800806 as well as the NSF CAREER
grant DMS-1651321.

##  2\. Preliminaries

If $r$ and $s$ are real numbers and $\epsilon$ is a real number greater than
$0$, then we write $r\approx_{\epsilon}s$ to mean $|r-s|<\epsilon$.

Fix $\mathcal{L}$ a countable language. Throughout this paper, we always have
a countable, complete, first-order theory $T$ and a monster model
$\mathcal{U}$ of $T$ in the background. The letters $M$ and $N$ will be used
to denote small elementary submodels of $\mathcal{U}$. The letters $x,y,z$
will denote tuples of variables. If $A\subseteq\mathcal{U}$, we let
$\mathcal{L}(A)$ be the collection of formulas with parameters from $A$
(modulo logical equivalence). A formula in $\mathcal{L}(A)$ is called an
“$\mathcal{L}(A)$-formula”. If $x_{0},...,x_{k}$ is a finite sequence of
pairwise disjoint tuples of variables, we let
$\mathcal{L}_{x_{0},...,x_{k}}(A)$ be the collection of
$\mathcal{L}(A)$-formulas with free variables in these tuples. We write
$\mathcal{L}_{x_{0},...,x_{k}}(\emptyset)$ simply as
$\mathcal{L}_{x_{0},...,x_{k}}$. If $(x_{i})_{i\in\omega}$ is a countable
sequence of pairwise distinct tuples of variables, we let
$\mathcal{L}_{(x_{i})_{i\in\omega}}(A)=\bigcup_{k\in\omega}\mathcal{L}_{x_{0},...,x_{k}}(A)$.
For a tuple $x$, let $A^{x}=\\{(a_{0},...,a_{|x|-1}):a_{i}\in
A,i\leq|x|-1\\}$. We let $(A^{x})^{<\omega}$ be the collection of all finite
sequences of points in $A^{x}$. If we call $\varphi(x,y)$ a partitioned
$\mathcal{L}_{x,y}(\mathcal{U})$-formula, we treat $x$ as object variables and
$y$ as parameter variables. The formula $\varphi^{*}(y,x)$ denotes the exact
same formula as $\varphi(x,y)$, but with the roles exchanged for parameters
and object tuples. Generally speaking, in any instance where we have multiple
tuples of variables (e.g. $x$ and $y$, or $(x_{1},x_{2},x_{3},...)$), we will
always assume they are pairwise distinct without comment.

Unlike similar papers about Keisler measures, we do not identify a type and
its corresponding Keisler measure. We let $S_{x}(A)$ denote the usual type
space over $A$ and $\mathfrak{M}_{x}(A)$ the space of Keisler measures over
$A$. We let $\mathfrak{M}_{(x_{i})_{i\in\omega}}(\mathcal{U})$ be the
collection of finitely additive probability measures on
$\mathcal{L}_{(x_{i})_{i\in\omega}}(\mathcal{U})$. For any (tuple of)
variable(s) $x$, and any subset $A\subseteq\mathcal{U}$, we have a map
$\delta:S_{x}(A)\to\mathfrak{M}_{x}(A)$ via $\delta(p)=\delta_{p}$ where
$\delta_{p}$ is the Dirac measure at the type $p$. We sometimes refer to
$\delta_{p}$ as the corresponding Keisler measure of $p$. If
$\overline{a}=(a_{1},...,a_{n})$ is a sequence of points in $\mathcal{U}^{x}$,
then we let $\operatorname{Av}(\overline{a})$ be the associated average
measure in $\mathfrak{M}_{x}(\mathcal{U})$. Explicitly, for any
$\psi(x)\in\mathcal{L}_{x}(\mathcal{U})$, we define

$\operatorname{Av}(\overline{a})(\psi(x))=\frac{|\\{1\leq i\leq
n:\mathcal{U}\models\psi(a_{i})\\}|}{n}.$

###  2.1. Basics of convergence

Recall that if $A\subseteq\mathcal{U}$, then both $S_{x}(A)$ and
$\mathfrak{M}_{x}(A)$ carry a natural compact Hausdorff topology. For
$S_{x}(A)$, we have the usual Stone space topology. Similarly,
$\mathfrak{M}_{x}(A)$ admits a compact Hausdorff topology. There are two ways
to describe this topology. First, this topology is the topology induced from
the compact Hausdorff space $[0,1]^{\mathcal{L}_{x}(A)}$ where we identify
each measure with the obvious map from $\mathcal{L}_{x}(A)$ to $[0,1]$. This
topology on $\mathfrak{M}_{x}(A)$ can also be described as the coarsest
topology such that for any continuous function $f:S_{x}(A)\to\mathbb{R}$, the
map $\int f:\mathfrak{M}_{x}(A)\to\mathbb{R}$ is continuous.

We will routinely need to keep track of which sets of parameters our types and
measures are converging over. Hence, we establish the following conventions.

######  Definition 2.1.

Fix $A\subseteq\mathcal{U}$, $p\in S_{x}(A)$ and $\mu\in\mathfrak{M}_{x}(A)$.

  1. $(i)$

We say that a sequence of types $(p_{i})_{i\in\omega}$, where each $p_{i}$ is
in $S_{x}(A)$, converges to $p$ if it converges in the Stone space topology on
$S_{x}(A)$, which we write as “$\lim_{i\to\infty}p_{i}=p$ in $S_{x}(A)$” or
simply as “$\lim_{i\to\infty}p_{i}=p$” when the underlying space is obvious.
We recall that $\lim_{i\to\infty}p_{i}=p$ if for every $\psi(x)\in p$, there
exists some natural number $N_{\psi}$ such that for any $n>N_{\psi}$,
$\psi(x)\in p_{n}$.

  2. $(ii)$

We say that a sequence of measures $(\mu_{i})_{i\in\omega}$, where each
$\mu_{i}$ is in $\mathfrak{M}_{x}(A)$, converges to $\mu$ if this sequence
converges in the compact Hausdorff topology on $\mathfrak{M}_{x}(A)$, which we
write as “$\lim_{i\to\infty}\mu_{i}=\mu$ in $\mathfrak{M}_{x}(A)$” or simply
as “$\lim_{i\to\infty}\mu_{i}=\mu$” when there is no possibility of confusion.
Notice that $\lim_{i\to\infty}\mu_{i}=\mu$ if for every
$\psi(x)\in\mathcal{L}_{x}(A)$ and $\epsilon>0$, there exists some natural
number $N_{\psi,\epsilon}$ such that for any $n>N_{\psi,\epsilon}$,

$|\mu_{n}(\psi(x))-\mu(\psi(x))|<\epsilon.$

We now observe the relationship between finitely satisfiable types and
measures and topological closure in their respective spaces.

######  Fact 2.2.

Suppose $p\in S_{x}(\mathcal{U})$, $\mu\in\mathfrak{M}_{x}(\mathcal{U})$ and
$M\prec\mathcal{U}$. Assume that $p$ and $\mu$ are finitely satisfiable over
$M$. Then the following are true.

  1. ($i$)

The type $p$ is in the closure of $\\{tp(a/\mathcal{U}):a\in M^{x}\\}$ in
$S_{x}(\mathcal{U})$.

  2. ($ii$)

The associated Keisler measure $\delta_{p}$ is in the closure of
$\\{\delta_{a}:a\in M^{x}\\}$ in $\mathfrak{M}_{x}(\mathcal{U})$.

  3. ($iii$)

The measure $\mu$ is in the closure of

$\Big{\\{}\sum_{i=1}^{n}r_{i}\delta_{a_{i}}:n\in\mathbb{N},r_{i}>0,\sum_{i=1}^{n}r_{i}=1,a_{i}\in
M^{x}\Big{\\}}$

in $\mathfrak{M}_{x}(\mathcal{U})$.

  4. ($iv$)

The measure $\mu$ is in the closure of
$\\{\operatorname{Av}(\overline{a}):\overline{a}\in(M^{x})^{<\omega}\\}$ in
$\mathfrak{M}_{x}(\mathcal{U})$.

We remark that the proof of $(i)$ is a standard exercise and the proof of
$(ii)$ follows directly from $(i)$. A proof of $(iii)$ can be found at [4,
Proposition 2.11] and $(iv)$ follows directly from $(iii)$.

###  2.2. Types

We recall some basic definitions and facts about special kinds of types (e.g.
generically stable types). Our notion of an EM-type is not defined in complete
generality since we are only concerned with countable sequences in this paper.

######  Definition 2.3.

Let $(a_{i})_{i\in\omega}$ be a sequence of points in $\mathcal{U}^{x}$ and
let $B\subseteq\mathcal{U}$. Then the Ehrenfeucht-Mostowski type or EM-type of
the sequence $(a_{i})_{i\in\omega}$ over $B$, denoted
$\operatorname{EM}((a_{i})_{i\in\omega}/B)$, is the following partial type:

$\\{\varphi(x_{0},...,x_{k})\in\mathcal{L}_{(x_{i})_{i\in\omega}}(B):\mathcal{U}\models\varphi(a_{i_{0}},...,a_{i_{k}})\text{
for any }i_{0}<...<i_{k}\\}.$

We remark that this partial type corresponds to a subset of
$S_{(x_{i})_{i\in\omega}}(B)$.

######  Observation 2.4.

It is clear from the definition above that for any sequence of points
$(a_{i})_{i\in\omega}$ in $\mathcal{U}^{x}$ and any $B\subseteq\mathcal{U}$,
the type $\operatorname{EM}((a_{i})_{i\in\omega}/B)$ is complete if and only
if the sequence $(a_{i})_{i\in\omega}$ is indiscernible over $B$.

The general notion of a generically stable type was introduced by Pillay and
Tanović in [15]. The definition of a generically stable type provided below
was proved to be equivalent in [6] (see Proposition 3.2). We also provide the
definition of a $\operatorname{dfs}$ type which will be important throughout
this paper. In general, the class of $\operatorname{dfs}$ types strictly
contains the class of generically stable types.

######  Definition 2.5.

Suppose that $p\in S_{x}(\mathcal{U})$.

  1. $(i)$

We say that $p$ is dfs if there exists a small model $M\prec\mathcal{U}$ such
that $p$ is both definable and finitely satisfiable over $M$. In this case, we
say that $p$ is dfs over $M$.

  2. $(ii)$

We say that $p$ is generically stable if there exists a small model
$M\prec\mathcal{U}$ such that $p$ is invariant over $M$ and for any Morley
sequence $(a_{i})_{i\in\omega}$ in $p$ over $M$, we have that
$\lim_{i\to\infty}\operatorname{tp}(a_{i}/\mathcal{U})=p$. In this case, we
say that $p$ is generically stable over $M$.

Finally, we provide a collection of standard facts about these classes of
types.

######  Fact 2.6.

Let $p$ be in $S_{x}(\mathcal{U})$ and $M\prec\mathcal{U}$.

  1. $(i)$

If $p$ is generically stable over $M$, then $p$ is $\operatorname{dfs}$ over
$M$ $($[15, Proposition 1]$)$.

  2. $(ii)$

If $p$ is $\operatorname{dfs}$ over $M$, then any Morley sequence in $p$ over
$M$ is totally indiscernible over $M$ $($[9, Proposition 3.2], proof does not
use NIP$)$.

  3. $(iii)$

If $p$ is generically stable/$\operatorname{dfs}$ over $M$ and
$M_{0}$-invariant, then $p$ is respectively generically
stable/$\operatorname{dfs}$ over $M_{0}$ $($generically stable case follows
from $(i)$ of [15, Proposition 1]; $\operatorname{dfs}$ case can be found in
[16, Lemma 2.8]$)$.

  4. $(iv)$

$($T is countable$)$ If $p$ is generically stable/$\operatorname{dfs}$ over
$M$, there exists an elementary submodel $M_{0}$ such that
$|M_{0}|=\aleph_{0}$ and $p$ is generically stable/$\operatorname{dfs}$ over
$M_{0}$ $($Easy to check from $(iii)$$)$.

  5. $(v)$

$($T is NIP$)$ If $p$ is $\operatorname{dfs}$ over $M$ then $p$ is generically
stable over $M$ $($e.g. [16, Theorem 2.29]$)$.

###  2.3. Keisler measures

In this subsection, we will briefly recall some important definitions and
facts about these measures. As with any paper about Keisler measures, we
provide the following standard atlas.

######  Definition 2.7.

Let $\mu\in\mathfrak{M}_{x}(\mathcal{U})$.

  1. ($i$)

$\mu$ is invariant if there exists a model $M\prec\mathcal{U}$ such that for
every partitioned $\mathcal{L}$-formula $\varphi(x,y)$ and
$b,b^{\prime}\in\mathcal{U}^{y}$ such that $b\equiv_{M}b^{\prime}$,
$\mu(\varphi(x,b))=\mu(\varphi(x,b^{\prime}))$. In this case, we say that
$\mu$ is $M$-invariant or invariant over $M$.

  2. ($ii$)

If $\mu$ is invariant over $M$, then for every partitioned
$\mathcal{L}(M)$-formula $\varphi(x,y)$, we can define the map
$F_{\mu,M}^{\varphi}:S_{y}(M)\to[0,1]$ via
$F_{\mu,M}^{\varphi}(q)=\mu(\varphi(x,b))$ where $b\models q$. When $M$ is
obvious we will simply write $F_{\mu,M}^{\varphi}$ as $F_{\mu}^{\varphi}$.

  3. ($iii$)

$\mu$ is Borel-definable if there exists a model $M\prec\mathcal{U}$ such that
$\mu$ is $M$-invariant and for every partitioned $\mathcal{L}$-formula
$\varphi(x,y)$, the map $F_{\mu,M}^{\varphi}$ is Borel. In this case, we say
that $\mu$ is Borel-definable over $M$.

  4. ($iv$)

$\mu$ is definable if there exists a model $M\prec\mathcal{U}$ such that $\mu$
is $M$-invariant and for every partitioned $\mathcal{L}$-formula
$\varphi(x,y)$, the map $F_{\mu,M}^{\varphi}$ is continuous. In this case, we
say that $\mu$ is $M$-definable or definable over $M$.

  5. ($v$)

$\mu$ is finitely satisfiable over a small model if there exists
$M\prec\mathcal{U}$ such that for every formula
$\varphi(x)\in\mathcal{L}_{x}(\mathcal{U})$, if $\mu(\varphi(x))>0$ then there
exists $a\in M^{x}$ such that $\mathcal{U}\models\varphi(a)$. In this case, we
say that $\mu$ is finitely satisfiable over $M$.

  6. ($vi$)

$\mu$ is finitely approximated if there exists a model $M\prec\mathcal{U}$
such that for every partitioned $\mathcal{L}$-formula $\varphi(x,y)$ and every
$\epsilon>0$, there exists $\overline{a}\in(M^{x})^{<\omega}$ such that

$\sup_{b\in\mathcal{U}^{y}}|\mu(\varphi(x,b))-\operatorname{Av}(\overline{a})(\varphi(x,b))|<\epsilon.$

In this case, we say that $\mu$ is finitely approximated over $M$.

  7. ($vii$)

$\mu$ is smooth if there exists a model $M\prec\mathcal{U}$ such that for any
$\lambda\in\mathfrak{M}_{x}(\mathcal{U})$ if $\lambda|_{M}=\mu|_{M}$, then
$\lambda=\mu$. If this is the case, we say that $\mu$ is smooth over $M$.

We now provide a collection of basic facts. Statements $(i)$, $(iii)$, $(iv)$,
and $(v)$ in Fact 2.8 are relatively straightforward to prove and so we leave
them as exercises.

######  Fact 2.8.

Assume that $T$ is any theory and $\mu\in\mathfrak{M}_{x}(\mathcal{U})$ with
$M\prec\mathcal{U}$.

  1. $(i)$

If $\mu=\operatorname{Av}(\overline{a})$ for some
$\overline{a}\in(M^{x})^{<\omega}$, then $\mu$ is smooth over $M$.

  2. $(ii)$

If $\mu$ is smooth over $M$, then $\mu$ is finitely approximated over $M$,
$($e.g. [16, Proposition 7.10]$)$.

  3. $(iii)$

If $\mu$ is finitely approximated over $M$, then $\mu$ is both definable and
finitely satisfiable over $M$.

  4. $(iv)$

If $\mu$ is definable or finitely satisfiable over $M$, then $\mu$ is
$M$-invariant.

  5. $(v)$

The measure $\mu$ is definable over $M$ if and only if for every partitioned
$\mathcal{L}(M)$-formula $\varphi(x,y)$ and for every $\epsilon>0$, there
exists formulas $\psi_{1}(y),...,\psi_{n}(y)\in\mathcal{L}_{y}(M)$ and real
numbers $r_{1},...,r_{n}\in[0,1]$ such that

$\sup_{q\in
S_{y}(M)}|F_{\mu,M}^{\varphi}(q)-\sum_{i=1}^{n}r_{i}\mathbf{1}_{\psi_{i}(y)}(q)|<\epsilon.$

where $\mathbf{1}_{\psi_{i}(y)}$ is the characteristic function of the clopen
set $[\psi_{i}(y)]$.

Moreover, if $T$ is NIP then the following also hold.

  1. $(vi)$

If $\mu$ is invariant over $M$, then $\mu$ is Borel-definable $M$ $($e.g. [16,
Proposition 7.19]$)$.

  2. $(vii)$

Any measure $\mu$ is definable and finitely satisfiable over $M$ if and only
if $\mu$ is finitely approximated over $M$ $($[10, Proposition 3.2]$)$.

  3. $(viii)$

Every measure has a “smooth extension”. In particular, for any given
$M\prec\mathcal{U}$ and $\mu\in\mathfrak{M}_{x}(\mathcal{U})$, there exists
some $N$ such that $M\prec N\prec\mathcal{U}$ and a measure
$\lambda\in\mathfrak{M}_{x}(\mathcal{U})$ such that $\lambda$ is smooth over
$N$ and $\lambda|_{M}=\mu|_{M}$ $($[10, Lemma 2.2]$)$.

######  Proposition 2.9 (T is countable).

If $\mu$ is definable, finitely approximated, smooth or $\operatorname{dfs}$,
then there exists a countable model $M_{0}$ such that $\mu$ is definable,
finitely approximated, smooth or $\operatorname{dfs}$ over $M_{0}$
$($respectively$)$.

###### Proof.

We notice that the properties of definability and smoothness only require the
existence of $\aleph_{0}$-many $\mathcal{L}(M)$-formulas (by [10, Lemma 2.3]
and (v) of Fact 2.8 respectively). If we choose an elementary submodel $M_{0}$
of $M$ containing the parameters from these formulas, then $\mu$ will have the
desired property over $M_{0}$. Finitely approximated measures only require the
existence of $\aleph_{0}$-many elements of $M$. Choosing an elementary
submodel $M_{0}$ of $M$ with these elements demonstrates that $\mu$ is
finitely approximated over $M_{0}$.

Finally, if $\mu$ is $\operatorname{dfs}$ then $\mu$ is definable over a
countable model $M_{0}$. In particular, $\mu$ is invariant over $M_{0}$ and so
$\mu$ is also finitely satisfiable over $M_{0}$ by the same argument as in [8,
Proposition 4.13]. ∎

######  Remark 2.10.

Assuming $T$ is countable, there are measures (even types) which are finitely
satisfiable over a small submodel, but are not finitely satisfiable over a
countable submodel. See Proposition 4.11 and Remark 4.12 for an explicit
example.

######  Definition 2.11.

Let $\mu\in\mathfrak{M}_{x}(\mathcal{U})$,
$\nu\in\mathfrak{M}_{y}(\mathcal{U})$ and assume that $\mu$ is Borel-definable
over $M$. Then we define the Morley product of $\mu$ and $\nu$ (denoted
$\mu\otimes\nu)$ is the unique Keisler measure in
$\mathfrak{M}_{x,y}(\mathcal{U})$ with the following property: for any formula
$\varphi(x,y)\in\mathcal{L}_{x,y}(\mathcal{U})$,

$\mu\otimes\nu(\varphi(x,y))=\int_{S_{y}(N)}F_{\mu}^{\varphi}d(\nu|_{N}),$

where $N$ is any small elementary submodel of $\mathcal{U}$ containing $M$ and
any parameters from $\varphi$ and $\nu|_{N}$ is the associated regular Borel
probability measure of the restriction of $\nu$ to $N$ on the type space
$S_{y}(N)$.

We remark that this this product is well-defined and the computation does not
depend on our choice of $N$ (assuming $N$ contains $M$ and all parameters in
$\varphi(x,y)$) (see discussion after [16, Proposition 7.19]). This
observation allows us to grow or shrink the space in which we are integrating
over and we will make substantial use of this property in section 5. We end
this section with a list of facts about measures and products.

######  Fact 2.12.

Assume that $T$ is any theory and $\mu\in\mathfrak{M}_{x}(\mathcal{U})$,
$\nu\in\mathfrak{M}_{y}(\mathcal{U})$, and
$\lambda\in\mathfrak{M}_{z}(\mathcal{U})$. Assume that $\mu$ and $\nu$ are
both $M$-invariant.

  1. $(i)$

If $\mu$ is smooth and $\nu$ is Borel definable, then
$\mu\otimes\nu=\nu\otimes\mu$ $($see [10, Corollary 2.5]$)$.

  2. $(ii)$

If $\mu$ and $\nu$ are definable (over $M$), then $\mu\otimes\nu$ is definable
(over $M$) and $\mu\otimes(\nu\otimes\lambda)=(\mu\otimes\nu)\otimes\lambda)$
$($see [6, Proposition 2.6]$)$.

  3. $(iii)$

If $\mu$ and $\nu$ are smooth (over $M$), then $\mu\otimes\nu$ is smooth (over
$M$) $($e.g. [5, Corollary 3.1]$)$.

  4. $(iv)$

If $\mu$ is Borel definable (over $M$) and $\nu$ is invariant (over $M$), then
$\mu\otimes\nu$ is invariant (over $M$) $($discussion before [16, Exercise
7.20]$)$.

  5. $(v)$

If $\mu$ and $\nu$ are $\operatorname{dfs}$ (over $M$), then $\mu\otimes\nu$
is $\operatorname{dfs}$ (over $M$) $($e.g. [6, Proposition 2.10]$)$.

Moreover, if $T$ is NIP then the following also hold.

  1. $(a)$

If $\mu,\nu$ are invariant then
$\mu\otimes(\nu\otimes\lambda)=(\mu\otimes\nu)\otimes\lambda$ $($see [5]$)$.

  2. $(b)$

If $\mu$ is $\operatorname{dfs}$ and $\nu$ is invariant, then
$\mu\otimes\nu=\nu\otimes\mu$ $($see [10, Theorem 3.2]$)$.

######  Definition 2.13 (T is NIP).

Suppose that $\mu\in\mathfrak{M}_{x}(\mathcal{U})$ and $\mu$ is invariant.
Then, we define the following measures:

  1. (1)

$\mu^{(0)}(x_{0})=\mu(x_{0})$.

  2. (2)

$\mu^{(n)}=\mu(x_{n})\otimes\mu^{(n-1)}(x_{0},...,x_{n-1})$.

  3. (3)

$\mu^{(\omega)}=\bigcup_{i\in\omega}\mu^{(n)}$ (where
$\mu^{(\omega)}\in\mathfrak{M}_{(x_{i})_{i\in\omega}}(\mathcal{U})$).

We note that $\mu^{(n)}$ and $\mu^{(\omega)}$ are well-defined by Fact 2.12,
and moreover we do not need to worry about the ordering of the parentheses in
the product.

##  3\. Sequentially approximated types and measures

We begin this section by isolating the property of sequential approximability.
We again remark that these classes of objects are a global version of
Khanaki’s Baire 1 definability [13]. We assume that $T$ is countable, but make
no other global assumptions about $T$. As usual, $\mathcal{U}$ is a fixed
sufficiently saturated model of $T$. We now define sequentially approximated
types and measures.

######  Definition 3.1.

Let $p\in S_{x}(\mathcal{U})$ and $\mu\in\mathfrak{M}_{x}(\mathcal{U})$. We
say that,

  1. (1)

$p$ is sequentially approximated if there exists $M\prec\mathcal{U}$ and a
sequence of points $(a_{i})_{i\in\omega}$ in $M^{x}$ such that
$\lim_{i\to\infty}\operatorname{tp}(a_{i}/\mathcal{U})=p$ in
$S_{x}(\mathcal{U})$. In this case, we say $p$ is sequentially approximated
over $M$.

  2. (2)

$\mu$ is sequentially approximated if there exists $M\prec\mathcal{U}$ and a
sequence of points $(\overline{a}_{i})_{i\in\omega}$ in $(M^{x})^{<\omega}$
such that $\lim_{i\to\infty}\operatorname{Av}(\overline{a}_{i})=\mu$ in
$\mathfrak{M}_{x}(\mathcal{U})$. In this case, we say $\mu$ is sequentially
approximated over $M$.

We warn the reader that Definition 3.1 is only meaningful in the context of
types and measures over large models. Indeed, if $M$ is a countable model and
$T$ is a countable theory, then for every $p\in S_{x}(M)$, there exists a
sequence of points in $M^{x}$ such that
$\lim_{i\to\infty}\operatorname{tp}(a_{i}/M)=p$ in $S_{x}(M)$. The analogous
statement also holds for measures.

We also emphasize to the reader that there is a real distinction between a
type $p$ being sequentially approximated over a model $M$ and its associated
Keisler measure $\delta_{p}$ being sequentially approximated over $M$.
Proposition 4.14 gives an example of a type which is not sequentially
approximated while its associated Keisler measure is sequentially
approximated. However, the other implication holds almost trivially.

######  Observation 3.2.

If a type $p$ in $S_{x}(\mathcal{U})$ is sequentially approximated over a
model $M$, then the associated Keisler measure $\delta_{p}$ is sequentially
approximated over $M$.

###### Proof.

If $\lim_{i\to\infty}\operatorname{tp}(a_{i}/\mathcal{U})=p$ in
$S_{x}(\mathcal{U})$, then $\lim_{i\to\infty}\delta_{a_{i}}=\delta_{p}$ in
$\mathfrak{M}_{x}(\mathcal{U})$ since
$\delta:S_{x}(\mathcal{U})\to\mathfrak{M}_{x}(\mathcal{U})$ is a topological
embedding. ∎

###  3.1. Basic properties

We now connect sequentially approximated types and measures to standard model-
theoretic properties. For the reader’s intuition, sequential approximability
(at least in the case of measures) should be thought of as a strong version of
finite satisfiability over a small model or a weak version of finite
approximability. Sequentially approximated types remain a little more
mysterious.

######  Proposition 3.3.

Assume that $p\in S_{x}(\mathcal{U})$ and
$\mu\in\mathfrak{M}_{x}(\mathcal{U})$.

  1. ($i$)

If $p$ and $\mu$ are sequentially approximated over $M$, then $p$ and $\mu$
are finitely satisfiable over $M$. Even more, $p$ and $\mu$ are finitely
satisfiable over a countable elementary submodel of $M$.

  2. ($ii$)

If $p$ and $\mu$ are sequentially approximated over $M$, then $p$ and $\mu$
are Borel-definable over $M$.

  3. ($iii$)

If $\mu$ is finitely approximated over $M$, then $\mu$ is sequentially
approximated over $M$. $($Warning: In general, this fails for types.$)$

  4. ($iv$)

If $T$ is NIP, then $p$ is sequentially approximated over $M$ if and only if
$\delta_{p}$ is sequentially approximated over $M$.

  5. ($v$)

Assume that $k\subseteq\\{1,2,...,n\\}$ and let $\pi_{k}:S_{n}(\mathcal{U})\to
S_{k}(\mathcal{U})$ and
$\rho_{k}:\mathfrak{M}_{n}(\mathcal{U})\to\mathfrak{M}_{k}(\mathcal{U})$ be
the obvious projection maps. If $p\in S_{n}(\mathcal{U})$ and $p$ is
sequentially approximated over $M$, then $\pi_{k}(p)$ is sequentially
approximated over $M$. Similarly, if $\mu\in\mathfrak{M}_{n}(\mathcal{U})$ is
sequentially approximated over $M$ then so is $\rho_{k}(\mu)$.

###### Proof.

We prove the claims.

  1. ($i$)

The first part of $(i)$ is obvious. For the second part, we only need to
choose a submodel containing a sequence which sequentially approximates our
type or measure. Since $T$ is countable, we can choose a countable model.

  2. ($ii$)

The proofs for both the type and measure cases are similar, so we prove the
measure case. Assume that $(\overline{a}_{i})_{i\in\omega}$ is a sequence of
points in $(M^{x})^{<\omega}$ such that
$\lim_{i\to\infty}\operatorname{Av}(\overline{a}_{i})=\mu$ in
$\mathfrak{M}_{x}(\mathcal{U})$. By part $(i)$, $\mu$ is finitely satisfiable
over $M$ and hence $M$-invariant. So, for any partitioned formula
$\varphi(x,y)$ in $\mathcal{L}$, the map $F_{\mu}^{\varphi}:S_{y}(M)\to[0,1]$
is well-defined. By sequential approximability, the sequence of continuous
functions
$\big{(}F_{\operatorname{Av}(\overline{a}_{i})}^{\varphi}\big{)}_{i\in\omega}$
converges pointwise to $F_{\mu}^{\varphi}$. Hence, $F_{\mu}^{\varphi}$ is
Baire-1 (and therefore Borel).

  3. ($iii$)

This follows from an encoding argument. Let
$(\varphi_{n}(x,y_{n}))_{n\in\omega}$ be an enumeration of the partitioned
$\mathcal{L}$-formulas. For each $n\in\mathbb{N}$, consider the partitioned
formula $\theta_{n}(x;y_{0},...,y_{n},z_{*},z_{0},...,z_{n})$ where
$|z_{*}|=|z_{i}|=1$ and

$\theta_{n}(x;\bar{y},\bar{z}):=\bigwedge_{i\leq
n}\left(\left(z_{*}=z_{i}\wedge\bigwedge_{\begin{subarray}{c}j\leq n\\\ j\neq
i\end{subarray}}z_{j}\neq z_{*}\right)\to\varphi_{i}(x,y_{i})\right).$

Since $\mu$ is finitely approximated over $M$, for $\epsilon=\frac{1}{n}$,
there exists some $\overline{a}_{n}$ in $(M^{x})^{<\omega}$ such that for
every $(\bar{b},\bar{c})\in\mathcal{U}^{\bar{y}\bar{z}}$,

$|\operatorname{Av}(\overline{a}_{n})(\theta_{n}(x,\bar{b},\bar{c}))-\mu((\theta_{n}(x,\bar{b},\bar{c}))|<\epsilon.$

Notice that $\theta_{n}(x;\bar{y},\bar{z})$ encodes the definable sets which
are obtained by the formulas $\varphi_{0}(x,y_{0}),...,\varphi_{n}(x,y_{n})$.
In particular, for every $b\in\mathcal{U}^{y_{j}}$ where $j\leq n$, consider
then tuple
$(\bar{d}_{b},\bar{c}_{j})=(d_{0},...d_{j-1},b,d_{j+1}...,d_{n},c_{*},c_{0},...,c_{n})$
where the $d_{i}$’s are arbitrary and $c_{*}=c_{l}$ if and only if $l=j$. Then

$|\operatorname{Av}(\overline{a}_{n})(\varphi_{j}(x,b))-\mu(\varphi_{j}(x,b))|=|\operatorname{Av}(\theta(x,\bar{d}_{b},\bar{c}_{j}))-\mu((\theta(x,\bar{d}_{b},\bar{c}_{j}))|.$

So for any $j\leq n$ and $b\in\mathcal{U}^{y_{j}}$,

$|\operatorname{Av}(\overline{a}_{n})(\varphi_{j}(x,b))-\mu(\varphi_{j}(x,b))|<\frac{1}{n}.$

It is clear that $\lim_{n\to\infty}\operatorname{Av}(\overline{a}_{n})=\mu$ in
$\mathfrak{M}_{x}(\mathcal{U})$.

  4. ($iv$)

The forward direction is Observation 3.2. We consider the converse. If
$\delta_{p}$ is sequentially approximated over $M$ then $\delta_{p}$ is
finitely satisfiable over a countable submodel $M_{0}$ by $(i)$ above. Then
$p$ is finitely satisfiable over $M_{0}$ and so by Theorem 1.1, $p$ is
sequentially approximated over $M_{0}$ (and also over $M$).

  5. ($v$)

Simply consider the approximating sequence restricted to the appropriate
coordinates. ∎

######  Proposition 3.4.

A measure $\mu$ is sequentially approximated and definable over $M$ if and
only if $\mu$ is finitely approximated over $M$.

###### Proof.

We first prove the forward direction. The proof is similar to the proof of [8,
Theorem 4.8]. Fix $\epsilon>0$. For any partitioned $\mathcal{L}$-formula
$\varphi(x,y)$, consider the map $F_{\mu}^{\varphi}:S_{y}(M)\to[0,1]$. Let
$(\overline{a}_{i})_{i\in\omega}$ be a sequence of points in
$(M^{x})^{<\omega}$ such that
$\lim_{i\to\infty}\operatorname{Av}(\overline{a}_{i})=\mu$ in
$\mathfrak{M}_{x}(\mathcal{U})$. Observe that each map
$F_{\operatorname{Av}(\overline{a})}^{\varphi}:S_{y}(M)\to[0,1]$ is continuous
and the sequence
$\big{(}F_{\operatorname{Av}(\overline{a}_{i})}^{\varphi}\big{)}_{i\in\omega}$
converge pointwise to $F_{\mu}^{\varphi}$. Since $\mu$ is definable, the map
$F_{\mu}^{\varphi}$ is continuous. By the Riesz representation theorem and
dominated convergence theorem, we have that
$\big{(}F_{\operatorname{Av}(\overline{a}_{i})}^{\varphi}\big{)}_{i\in\omega}$
converges weakly to $F_{\mu}^{\varphi}$ in $C(S_{y}(M))$. By a standard
application of Mazur’s lemma, there exists a sequence of functions
$(g_{j})_{j\in\omega}$ such that each $g_{j}$ is a rational convex combination
of $\\{F_{\operatorname{Av}(\overline{a}_{i})}^{\varphi}:i\leq n_{j}\\}$ for
some natural number $n_{j}$ and the sequence $(g_{j})_{j\in\omega}$ converges
uniformly to $F_{\mu}^{\varphi}$. Choose $m\in\mathbb{N}$ so that

$\sup_{p\in S_{y}(M)}|F_{\mu}^{\varphi}(p)-g_{m}(p)|<\epsilon.$

By construction, $g_{m}=F_{\operatorname{Av}(\overline{c})}^{\varphi}$ for
some $\overline{c}\in(M^{x})^{<\omega}$. Notice that

$\sup_{b\in\mathcal{U}^{y}}|\mu(\varphi(x,b))-\operatorname{Av}(\overline{c})(\varphi(x,b))|<\epsilon.$

For the converse, $\mu$ is definable over $M$ by $(iii)$ of Fact 2.8.
Moreover, $\mu$ is sequentially approximated over $M$ by $(iii)$ of
Proposition 3.3. ∎

We now show that sequentially approximated measures commute with definable
measures. It is well-known that in the context of NIP theories definable
measures commute with measures which are finitely satisfiable over a small
model (see [10, Lemma 3.1] or [16, Proposition 7.22]). Recently, it was shown
that in general, measures which are finitely satisfiable over a small model
(even $\operatorname{dfs}$ measures) do not always commute with definable
measures (see [7, Proposition 7.14]). We first present a topological proof (in
NIP theories) which shows that measures which are finitely satisfiable over a
small model commute with definable measures. We will then modify this proof
(by replacing an instance of continuity by the dominated convergence theorem)
to show that sequentially approximated measures commute with definable ones in
any theory. Recall the following facts.

######  Fact 3.5.

Let $\nu\in\mathfrak{M}_{y}(\mathcal{U})$, $N\prec\mathcal{U}$, and
$\varphi(x,y)$ be an $\mathcal{L}_{x,y}(N)$ formula. Let
$\mathfrak{M}_{x}(\mathcal{U},N)$ denote the collection of measures in
$\mathfrak{M}_{x}(\mathcal{U})$ which are finitely satisfiable over $N$.

  1. ($i$)

If $\nu$ is definable over $N$, then the map from
$\mathfrak{M}_{x}(\mathcal{U})$ to $[0,1]$ defined via
$\mu\to\nu\otimes\mu(\varphi(x,y))$ is continuous $($[7, Lemma 5.4]$)$.

  2. ($ii$)

$($T is NIP$)$ If $\nu$ is any measure, then the map from
$\mathfrak{M}_{x}(\mathcal{U},N)$ to $[0,1]$ defined via
$\mu\to\mu\otimes\nu(\varphi(x,y))$ is well-defined and continuous $($[4,
Proposition 6.3]$)$.

We remark that statement $(ii)$ of Fact 3.5 requires NIP for two reasons.
First, it is not true in general that measures which are finitely satisfiable
over a small model are Borel definable. In NIP theories, this is true ($(vi)$
of Fact 2.8). Secondly, the proof that this map is continuous relies on the
existence of a smooth extension of $\nu|_{N}$. Without NIP, this map need not
be continuous. The first proof of the following proposition can be found in
[10].

######  Proposition 3.6 (T is NIP).

Assume that $\mu\in\mathfrak{M}_{x}(\mathcal{U})$ and
$\nu\in\mathfrak{M}_{y}(\mathcal{U})$. If $\mu$ is finitely satisfiable over a
small model and $\nu$ is definable, then $\mu\otimes\nu=\nu\otimes\mu$.

###### Proof.

Fix a formula $\varphi(x,y)\in\mathcal{L}_{x,y}(\mathcal{U})$. Choose $N$ such
that $\mu$ is finitely satisfiable over $N$, $\nu$ is definable over $N$, and
$N$ contains all the parameters from $\varphi$. Since $\mu$ is finitely
satisfiable over $N$, there exists a net of measures
$(\operatorname{Av}(\overline{a}_{i}))_{i\in I}$ such that each
$\overline{a}_{i}\in(N^{x})^{<\omega}$ and $\lim_{i\in
I}\operatorname{Av}(\overline{a}_{i})=\mu$ in $\mathfrak{M}_{x}(\mathcal{U})$
($(iv)$ of Fact 2.2). By Fact 3.5

$\displaystyle\mu\otimes\nu(\varphi(x,y))=\int_{S_{y}(N)}F_{\mu}^{\varphi}d(\nu|_{N})$
$\displaystyle\overset{(a)}{=}\ \lim_{i\in
I}\int_{S_{y}(N)}F_{\operatorname{Av}(\overline{a}_{i})}^{\varphi}d(\nu|_{N})$
$\displaystyle\overset{(b)}{=}\ \lim_{i\in
I}\int_{S_{x}(N)}F_{\nu}^{\varphi^{*}}d(\operatorname{Av}(\overline{a}_{i})|_{N})$
$\displaystyle\overset{(c)}{=}\
\int_{S_{x}(N)}F_{\nu}^{\varphi^{*}}d(\mu|_{N})=\nu\otimes\mu(\varphi(x,y)).$

Where the equalities $(a)$ and $(c)$ follow from the fact that continuous
functions commute with nets. The equality $(b)$ is simple to check and is also
justified by statement $(i)$ of Fact 2.12. ∎

######  Proposition 3.7.

Sequentially approximated and definable measures commute. Assume that
$\mu\in\mathfrak{M}_{x}(\mathcal{U})$ and
$\nu\in\mathfrak{M}_{y}(\mathcal{U})$. If $\mu$ is sequentially approximated
and $\nu$ is definable, then $\mu\otimes\nu=\nu\otimes\mu$.

###### Proof.

Fix a formula $\varphi(x,y)\in\mathcal{L}_{x,y}(\mathcal{U})$. Choose $N$ such
that $\mu$ is sequentially approximated over $N$, $\nu$ is definable over $N$,
and $N$ contains all the parameters from $\varphi$. Let
$(\overline{a}_{i})_{i\in\omega}$ be a sequence of points in
$(N^{x})^{<\omega}$ such that
$\lim_{i\to\infty}\operatorname{Av}(\overline{a}_{i})=\mu$ in
$\mathfrak{M}_{x}(\mathcal{U})$. Now we consider the following computation.

$\displaystyle\mu\otimes\nu(\varphi(x,y))=\int_{S_{y}(N)}F_{\mu}^{\varphi}d(\nu|_{N})$
$\displaystyle\overset{(a)}{=}\
\lim_{i\to\infty}\int_{S_{y}(N)}F_{\operatorname{Av}(\overline{a}_{i})}^{\varphi}d(\nu|_{N})$
$\displaystyle\overset{(b)}{=}\
\lim_{i\to\infty}\int_{S_{x}(N)}F_{\nu}^{\varphi^{*}}d(\operatorname{Av}(\overline{a}_{i})|_{N})$
$\displaystyle\overset{(c)}{=}\
\int_{S_{x}(N)}F_{\nu}^{\varphi^{*}}d(\mu|_{N})=\nu\otimes\mu(\varphi(x,y)).$

Where the equality $(a)$ now holds from the dominated convergence theorem,
equality $(c)$ holds from $(i)$ of Fact 3.5 and the observation that
continuous functions commute with nets, and equality $(b)$ is easy to check
(also $(i)$ of Fact 2.12). ∎

######  Corollary 3.8.

Let $\mu\in\mathfrak{M}_{x}(\mathcal{U})$ and
$\nu\in\mathfrak{M}_{y}(\mathcal{U})$. If $\mu$ is finitely approximated and
$\nu$ is definable, then $\mu\otimes\nu=\nu\otimes\mu$.

###### Proof.

By (iii) of Proposition 3.3, $\mu$ is sequentially approximated. Apply
Proposition 3.7. ∎

###  3.2. Egorov’s theorem

It is interesting to note that sequentially approximated measures are not too
far away from finitely approximated measures. In particular, if we fix some
measure on the parameter space, any sequentially approximated measure is
almost finitely approximated. This result is in a similar vein as Khanaki’s
almost definable coheirs in the local setting ([12]). A direct application of
Egorov’s theorem gives our result.

######  Theorem 3.9 (Egorov’s Theorem).

Let $(X,B,\mu)$ be a finite measure space. Assume that $(f_{i})_{i\in\omega}$
is a sequence of measurable functions from $X\to\mathbb{R}$ such that
$(f_{i})_{i\in\omega}$ converges to a function $f$ pointwise. Then for every
$\epsilon>0$ there exists a $Y_{\epsilon}\in B$ such that
$f_{i}|_{Y_{\epsilon}}$ converges to $f|_{Y_{\epsilon}}$ uniformly on
$Y_{\epsilon}$ and $\mu(X\backslash Y_{\epsilon})<\epsilon$.

A proof of Egorov’s theorem can be found in [19, Theorem 3.2.4.1]. Restating
this theorem in our context gives the following result.

######  Corollary 3.10.

Assume that $p$ and $\mu$ are sequentially approximated over $M$. Let
$\nu\in\mathfrak{M}_{y}(M)$. Then, for every $\epsilon>0$, there exists a
Borel set $Y_{\epsilon}\subset S_{y}(M)$ such that

  1. (1)

$\nu(Y_{\epsilon})>1-\epsilon$.

  2. (2)

For every $\delta>0$ and every partitioned $\mathcal{L}$-formula
$\varphi(x,y)$, there exists $\overline{a}_{\delta}$ in $(M^{x})^{<\omega}$
such that for every $b\in\mathcal{U}^{y}$ so that $\operatorname{tp}(b/M)\in
Y_{\epsilon}$, we have

$|\mu(\varphi(x,b))-\operatorname{Av}(\overline{a}_{\delta})(\varphi(x,b))|<\delta.$

  3. (3)

For every partitioned $\mathcal{L}$-formula $\varphi(x,y)$, there exists $a$
in $M^{x}$ such that for every $b\in\mathcal{U}^{y}$ so that
$\operatorname{tp}(b/M)\in Y_{\epsilon}$, we have

$\varphi(x,b)\in p\iff\models\varphi(a,b).$

##  4\. Generically stable types

Throughout this section, we let $T$ be a countable theory and $\mathcal{U}$ be
a monster model of $T$. We show that if a type $p$ is generically stable over
a small submodel $M$ of $\mathcal{U}$, then $p$ is sequentially approximated
over $M$. Toward proving this result, we actually prove a slightly stronger
lemma than what is necessary. Namely, let $p$ be a $\operatorname{dfs}$ type
and let $M$ be a countable model such that $p$ is $\operatorname{dfs}$ over
$M$ (for any $\operatorname{dfs}$ type, these models always exist by (iv) of
Fact 2.6). We show that there exists a special sequence of points in $M$ such
that the limiting behavior of this sequence resembles a Morley sequence in $p$
over $M$. In the case where $p$ is generically stable over $M$, we show that
this special sequence converges to $p$. This is enough to show the result
since every generically stable type is generically stable over some countable
model. We now begin with a discussion on eventually indiscernible sequences,
which were introduced in [17].

######  Definition 4.1.

Let $(c_{i})_{i\in\omega}$ be a sequence of points in $\mathcal{U}^{x}$ and
$A\subset\mathcal{U}$. We say that $(c_{i})_{i\in\omega}$ is an eventually
indiscernible sequence over $A$ if for any formula $\varphi(x_{0},...,x_{k})$
in $\mathcal{L}_{(x_{i})_{i\in\omega}}(A)$, there exists some natural number
$N_{\varphi}$ such that for any indices $n_{k}>....>n_{0}>N_{\varphi}$ and
$m_{k}>...>m_{0}>N_{\varphi}$, we have that

$\mathcal{U}\models\varphi(c_{n_{0}},...,c_{n_{k}})\leftrightarrow\varphi(c_{m_{0}},...,c_{m_{k}}).$

######  Fact 4.2.

Let $(b_{i})_{i\in\omega}$ be a sequence of points in $\mathcal{U}^{x}$ and
$A\subset\mathcal{U}$ such that $|A|=\aleph_{0}$. Then there exists a
subsequence $(c_{i})_{i\in\omega}$ of $(b_{i})_{i\in\omega}$ such that
$(c_{i})_{i\in\omega}$ is eventually indiscernible over $A$.

The proof is a standard application of Ramsey’s theorem and taking the
diagonal (as mentioned in [17]). We prove a “continuous” version of this fact
in the next section and the proof is analogous (see Proposition 5.3 for
details).

For any eventually indiscernible sequence $(c_{i})_{i\in\omega}$ over a set of
parameters $A$, we can associate to this sequence a unique type in
$S_{(x_{i})_{i\in\omega}}(A)$. We call this the eventual Ehrenfeucht-Mostowski
type (or $\operatorname{EEM}$-type) of $(c_{i})_{i\in\omega}$ over $A$. We now
give the formal definition.

######  Definition 4.3.

Let $(b_{i})_{i\in\omega}$ be a sequence of points in $\mathcal{U}^{x}$ and
$A\subset\mathcal{U}$. Then the eventual Ehrenfeucht-Mostowski type (or EEM-
type) of $(b_{i})_{i\in\omega}$ over $A$, which is written as
$\operatorname{EEM}((b_{i})_{i\in\omega})/A)$, is a subset of
$\mathcal{L}_{(x_{i})_{i\in\omega}}(A)$ defined as follows:

Let $\varphi(x_{i_{0}},...,x_{i_{k}})$ be a formula in
$\mathcal{L}_{(x_{i})_{i\in\omega}}(A)$ where the indices are ordered
$i_{0}<...<i_{k}$. Then
$\varphi(x_{i_{0}},...x_{i_{k}})\in\operatorname{EEM}((b_{i})_{i\in\omega})/A)$
if and only if there exists an $N_{\varphi}$ such that for any
$n_{k}>...>n_{0}>N_{\varphi}$, we have that
$\mathcal{U}\models\varphi(b_{n_{0}},...,b_{n_{k}})$.

Notice that an $\operatorname{EEM}$-type of a sequence is always indiscernible
in the following sense: If we have indices $i_{0},...,i_{k}$ and
$j_{0},...,j_{k}$ where $i_{0}<...<i_{k}$ and $j_{0}<...<j_{k}$, then
$\varphi(x_{i_{0}},...,x_{i_{k}})$ is in the $\operatorname{EEM}$-type of
$(b_{i})_{i\in\omega}$ over $A$ if and only if
$\varphi(x_{j_{0}},...,x_{j_{k}})$ is. This follows directly from the
definition. We have some basic observations.

######  Observation 4.4.

Let be $(c_{i})_{i\in\omega}$ an eventually indiscernible sequence over $A$.

  1. (1)

Then $\operatorname{EEM}((c_{i})_{i\in\omega}/A)$ is a complete type in
$S_{(x_{i})_{i\in\omega}}(A)$.

  2. (2)

If $(c_{i})_{i\in\omega}$ is $A$-indiscernible, then
$\operatorname{EEM}((c_{i})_{i\in\omega}/A)=\operatorname{EM}((c_{i})_{i\in\omega}/A)$.

  3. (3)

If
$\operatorname{tp}((b_{i})_{i\in\omega}/A)=\operatorname{EEM}((c_{i})_{i\in\omega}/A)$,
then $(b_{i})_{i\in\omega}$ is $A$-indiscernible.

###### Proof.

Clear from the definitions and discussion above. ∎

We warn the reader that an eventually indiscernible sequence need not
“realize” its own $\operatorname{EEM}$-type. Consider the following example:

######  Example 4.5.

Let $T_{<}$ be the theory of $(\mathbb{R};<)$. Let $\mathcal{U}$ be a monster
model of $T_{real}$ and $\mathbb{R}\prec\mathcal{U}$. Then the sequence
$(a_{i})_{i\in\omega}$ where $a_{i}=i$ is eventually indiscernible over
$\mathbb{R}$ while the sequence $(b_{i})_{i\in\omega}$ where $b_{i}=i(-1)^{i}$
is not. Clearly, $(a_{i})_{i\in\omega}$ is not $\mathbb{R}$-indiscernible.
Moreover, for each $r\in\mathbb{R}$, the formula $x_{0}>r$ is in
$\operatorname{EEM}((a_{i})_{i\in\omega}/\mathbb{R})$ while $a_{1}>2$ clearly
does not hold. So if
$\operatorname{tp}((c_{i})_{i\in\omega}/\mathbb{R}))=\operatorname{EEM}((a_{i})_{i\in\omega}/\mathbb{R})$,
then $c_{i}>\mathbb{R}$ for each $i\in\omega$.

The next two lemmas prove the bulk of this section’s main theorem and their
proofs are similar to the proof of Theorem 1.1. The proof strategy for this
theorem is the following: If $p$ is in $S_{x}(\mathcal{U})$ and $p$ is
$\operatorname{dfs}$, then we can find a countable model $M$ such that $p$ is
$\operatorname{dfs}$ over $M$. Let $I$ be a Morley sequence in $p$ over $M$.
Using the fact that $p$ is finitely satisfiable over $M$, we can find a
sequence of points in $M^{x}$ which converge to $p|_{MI}$ in $S_{x}(MI)$.
After moving to an eventually indiscernible subsequence, we show that the
$\operatorname{EEM}$-type of this eventually indiscernible sequence is
precisely $p^{\omega}|_{M}$. With the stronger assumption that our type $p$ is
generically stable (instead of just $\operatorname{dfs}$), we show that this
eventually indiscernible subsequence must converge to $p$ in
$S_{x}(\mathcal{U})$.

######  Lemma 4.6.

Suppose $p$ is in $S_{x}(\mathcal{U})$ and $p$ is $\operatorname{dfs}$ over
$M$ where $|M|=\aleph_{0}$. Then there exists a sequence
$(c_{i})_{i\in\omega}$ in $M^{x}$ such that
$\operatorname{EEM}((c_{i})_{i\in\omega}/M)=p^{\omega}|_{M}$.

###### Proof.

Let $I=(a_{i})_{i\in\omega}$ be a Morley sequence in $p$ over $M$. Since $T$,
$M$, and $I$ are countable, $\mathcal{L}_{x}(MI)$ is countable. It follows
that $p|_{MI}$ is countable and we may enumerate this collection of formulas
as $(\varphi_{i}(x))_{i\in\omega}$. Since $p$ is $\operatorname{dfs}$ over
$M$, in particular $p$ is finitely satisfiable over $M$. For each natural
number $n$, we choose $b_{n}$ in $M^{x}$ such that
$\mathcal{U}\models\bigwedge_{j\leq n}\varphi_{j}(b_{n})$. By construction, we
have that $\lim_{i\to\infty}\operatorname{tp}(b_{i}/MI)=p|_{MI}$ in
$S_{x}(MI)$. By Fact 4.2, we may choose a subsequence $(c_{i})_{i\in\omega}$
of $(b_{i})_{i\in\omega}$ such that $(c_{i})_{i\in\omega}$ is eventually
indiscernible over $MI$. For ease of notation, we write $(c_{i})_{i\in\omega}$
as $J$.

We now show that
$\operatorname{EEM}(J/M)=\operatorname{EM}(I/M)=p^{\omega}|_{M}$. We remind
the reader that $\operatorname{EM}(I/M)=p^{\omega}|_{M}$ follows directly from
the definition of a Morley sequence. We prove the first equality by induction
on the number of free variables occurring in a formula. We begin with the base
case. It suffices to show that for every
$\varphi(x_{0})\in\mathcal{L}_{x_{0}}(M)$, if
$\varphi(x_{0})\in\operatorname{EM}(I/M)$, then
$\varphi(x_{0})\in\operatorname{EEM}(J/M)$. Notice that
$\lim_{n\to\infty}\operatorname{tp}(b_{n}/MI)=p|_{MI}$, and
$(c_{i})_{i\in\omega}$ is a subsequence of $(b_{n})_{n\in\omega}$,
$\lim_{i\to\infty}\operatorname{tp}(c_{i}/MI)=p|_{MI}$. This clearly implies
the base case.

Fix $k$ and suppose that for any formula $\theta(x_{0},...,x_{k})$ in
$\mathcal{L}_{x_{0},...,x_{k}}(M)$, we have that
$\theta(x_{0},...,x_{k})\in\operatorname{EM}(I/M)$ if and only if
$\theta(x_{0},...,x_{k})\in\operatorname{EEM}(J/M)$.

Towards a contradiction, we assume that
$\neg\theta(x_{0},...,x_{k+1})\in\operatorname{EEM}(J/M)$ and
$\theta(x_{0},...,x_{k+1})\in\operatorname{EM}(I/M)$. Since
$\neg\theta(\overline{x})\in\operatorname{EEM}(J/M)$, there exists some
natural number $N_{\theta_{1}}$ such that for any
$n_{k+1}>...>n_{0}>N_{\theta_{1}}$, we have that
$\mathcal{U}\models\neg\theta(c_{n_{0}},...,c_{n_{k+1}})$. Since
$\theta(\overline{x})\in\operatorname{EM}(I/M)$, we conclude that
$\mathcal{U}\models\theta(a_{0},...,a_{k+1})$. Since $p$ is
$\operatorname{dfs}$ over $M$, $I$ is totally indiscernible over $M$ by Fact
2.6. Therefore, $\mathcal{U}\models\theta(a_{k+1},a_{0}...,a_{k})$ and so
$\theta(x,a_{0},...,a_{k})\in p|_{Ma_{0},...,a_{k}}$. Since
$\lim_{i\to\infty}\operatorname{tp}(c_{i}/MI)=p|_{MI}$, there exists some
$N_{\theta_{2}}$ such that for every $n>N_{\theta_{2}}$, we have that
$\mathcal{U}\models\theta(c_{n},a_{0},...,a_{k})$. Choose
$n_{*}>\max\\{N_{\theta_{1}},N_{\theta_{2}}\\}$. Then the formula
$\theta(c_{n_{*}},x_{0},...,x_{k})\in\operatorname{tp}(a_{0},...,a_{k}/M)$. By
our induction hypothesis, we have that
$\theta(c_{n_{*}},\overline{x})\in\operatorname{EEM}(J/M)$ and so there exists
$N_{\theta_{3}}$ such that for any $m_{k}>...>m_{0}>N_{\theta_{3}}$, we have
that $\mathcal{U}\models\theta(c_{n_{*}},c_{m_{0}},...,c_{m_{k}})$. Now
consider what happens when $m_{0}>\max\\{N_{\theta_{3}},n_{*}\\}$. Then
$m_{k}>...>m_{0}>n_{*}>N_{\theta_{1}}$ and so
$\mathcal{U}\models\neg\theta(c_{n_{*}},c_{m_{0}},...,c_{m_{k}})$ by our
assumption. However, $m_{k}>...>m_{0}>N_{\theta_{3}}$ and therefore
$\mathcal{U}\models\theta(c_{n_{*}},c_{m_{0}},...,c_{m_{k}})$. This is a
contradiction. ∎

######  Lemma 4.7.

Suppose $p$ is in $S_{x}(\mathcal{U})$ and $M\prec\mathcal{U}$. Assume that
$p$ is generically stable over $M$. If $(c_{i})_{i\in\omega}$ is a sequence in
$M^{x}$ such that
$\operatorname{EEM}((c_{i})_{i\in\omega}/M)=p^{\omega}|_{M}$, then
$\lim_{i\to\infty}tp(c_{i}/\mathcal{U})=p$.

###### Proof.

Let $p$, $(c_{i})_{i\in\omega}$ and $M$ be as in the statement of the lemma.
Let $J=(c_{i})_{i\in\omega}$. We first argue that the sequence of global types
$(\operatorname{tp}(c_{i}/\mathcal{U}))_{i\in\omega}$ converges and then argue
that this sequence converges to $p$.

Claim 1: The sequence $(\operatorname{tp}(c_{i}/\mathcal{U}))_{i\in\omega}$
converges to a some type in $S_{x}(\mathcal{U})$.

It suffices to argue that for any formula
$\psi(x)\in\mathcal{L}_{x}(\mathcal{U})$,
$\lim_{i\to\infty}\mathbf{1}_{\psi}(c_{i})$ exists (recall that
$\mathbf{1}_{\psi(x)}$ is the characteristic function of the definable set
$\psi(x)$). Assume not. Then we may choose a subsequence
$(c_{i}^{\prime})_{i\in\omega}$ of $(c_{i})_{i\in\omega}$ such that
$\mathcal{U}\models\psi(c_{i}^{\prime})\leftrightarrow\neg\psi(c_{i+1}^{\prime})$.
For notational purposes, we also denote $(c^{\prime}_{i})_{i\in\omega}$ as
$J^{\prime}$. It is clear that $(c_{i}^{\prime})_{i\in\omega}$ is also
eventually indiscernible over $M$ and
$\operatorname{EEM}((c_{i}^{\prime})_{i\in\omega}/M)=\operatorname{EEM}((c_{i})_{i\in\omega}/M)$.
By using $J^{\prime}$, one can show that the following type is finitely
consistent:

$\Theta_{1}=\operatorname{EEM}(J^{\prime}/M)\cup\bigcup_{\textit{$i$ is
even}}\\{\psi(x_{i})\wedge\neg\psi(x_{i+1})\\}.$

Let $(d_{i})_{i\in\omega}$ realize this type. Then $(d_{i})_{i\in\omega}$ is a
Morley sequence in $p$ over $M$ because

$\operatorname{EM}((d_{i})_{i\in\omega}/M)=\operatorname{EEM}(J^{\prime}/M)=\operatorname{EEM}(J/M)=p^{\omega}|_{M}.$

Then $\mathcal{U}\models\psi(d_{i})$ if and only if $i$ is even. This
contradicts generic stability since
$\lim_{i\to\infty}\operatorname{tp}(d_{i}/M)$ does not converge.

Claim 2: The sequence $(\operatorname{tp}(c_{i}/\mathcal{U}))_{i\in\omega}$
converges to $p$.

Again, assume not. By claim 1,
$\lim_{i\to\infty}\operatorname{tp}(c_{i}/\mathcal{U})=q$ for some $q\in
S_{x}(\mathcal{U})$. By assumption, $q\neq p$ and so there exists a formula
$\psi(x)$ such that $\psi(x)\in p$ and $\neg\psi(x)\in q$. Since
$(\operatorname{tp}(c_{i}/\mathcal{U}))_{i\in\omega}$ converges to $q$, there
is an $N$ such that for every $n>N$, we have that
$\mathcal{U}\models\neg\theta(c_{n})$. By a similar argument as the previous
claim, one can show the following type is finitely consistent:

$\Theta_{2}=\operatorname{EEM}(J/M)\cup\bigcup_{i\in\omega}\neg\theta(x_{i}).$

Again, we let $(d_{i})_{i\in\omega}$ realize this type. Then
$(d_{i})_{i\in\omega}$ is a Morley sequence in $p$ over $M$ and we have that
$\lim_{i\to\infty}\operatorname{tp}(d_{i}/\mathcal{U})\neq p$ in
$S_{x}(\mathcal{U})$. This again contradicts the definition of generic
stability. ∎

######  Theorem 4.8.

Suppose $p$ is in $S_{x}(\mathcal{U})$ and $p$ is generically stable (over
$M$). Then $p$ is sequentially approximated (over $M$).

###### Proof.

If $p$ is generically stable, then $p$ is generically stable over a countable
submodel model $M_{0}$ contained in $M$ by Fact 2.6. Then $p$ is
$\operatorname{dfs}$ over $M_{0}$ and so by Lemma 4.6, one can choose
$(c_{i})_{i\in\omega}$ where each $c_{i}\in M_{0}^{x}$ and
$\operatorname{EEM}((c_{i})_{i\in\omega}/M_{0})=p^{\omega}|_{M_{0}}$. By Lemma
4.7, $\lim_{i\to\infty}\operatorname{tp}(c_{i}/\mathcal{U})=p$. ∎

######  Corollary 4.9.

Assume that $T^{\prime}$ is countable or uncountable in the language
$\mathcal{L^{\prime}}$, $\mathcal{U}^{\prime}\models T^{\prime}$, and
$M^{\prime}$ a submodel of $\mathcal{U}^{\prime}$. Assume that $p$ is
generically stable over $M^{\prime}$. Then for any countable collection of
formulas $\Delta=\\{\psi_{i}(x,y_{i})\\}_{i\in\omega}$ in
$\mathcal{L^{\prime}}$, there exists a sequence of points
$(c_{i})_{i\in\omega}$ each in $(M^{\prime})^{x}$ such that
$\lim_{i\to\infty}\operatorname{tp}_{\Delta}(c_{i}/\mathcal{U})=p|_{\Delta}$.

###### Proof.

Let $\mathcal{L}$ be a countable sublanguage of $\mathcal{L^{\prime}}$
containing all the formulas in $\Delta$. The corresponding type
$p|_{\mathcal{L}}$ is generically stable over the model $M$ where
$M=M^{\prime}|_{\mathcal{L}}$ (see [6, Remark 3.3]). Hence we may apply
Theorem 4.8. ∎

###  4.1. Examples and non-examples

We begin this subsection by collecting the known examples of sequentially
approximated types. We then go on to give two examples of types which are not
sequentially approximated (over any model).

######  Observation 4.10.

Assume that $p\in S_{x}(\mathcal{U})$ and let $M$ be a small elementary
submodel. Then, $p$ is sequentially approximated over $M$ if

  1. ($i$)

$T$ is stable, and $p$ is invariant over $M$,

  2. ($ii$)

$T$ is NIP, $|M|=\aleph_{0}$, and $p$ is finitely satisfiable over $M$, or

  3. ($iii$)

$p$ is generically stable over $M$.

We just proved $(iii)$. Clearly, $(i)$ follows from $(iii)$ (we remark that it
also follows from $(ii)$). As noted previously, the proof of $(ii)$ is
precisely [17, Lemma 2.8].

We now exhibit some concrete examples of types which are not sequentially
approximated. We begin by describing a type in an NIP theory which is finitely
satisfiable over a small model but not sequentially approximated (and its
associated Keisler measure is not sequentially approximated either). We then
discuss a finitely approximated type which is not sequentially approximated.

######  Proposition 4.11.

Let $\omega_{1}$ be the first uncountable ordinal, $M=(\omega_{1};<)$ with the
usual ordering, and let $T_{<}$ be the theory of $M$ in the language
$\\{<\\}$. Recall that $T_{<}$ is NIP. Let $p\in S_{x}(\omega_{1})$ be the
complete type extending $\\{\alpha<x:\alpha<\omega_{1}\\}$. Let $\mathcal{U}$
be a monster model of $T_{<}$ such that $M\prec\mathcal{U}$ and let $p_{*}\in
S_{x}(\mathcal{U})$ be the unique global coheir of $p$. Then, $p_{*}$ is not
sequentially approximated over any model.

###### Proof.

Assume for the sake of contradiction that $p_{*}$ is sequentially approximated
over some model $N$. Then there exists a sequence of points
$(b_{i})_{i\in\omega}$ in $N$ such that
$\lim_{i\to\infty}\operatorname{tp}(b_{i}/\mathcal{U})=p_{*}$ in
$S_{x}(\mathcal{U})$. There is either an infinite subsequence which is
strictly increasing or strictly decreasing and so without loss of generality,
$(b_{i})_{i\in\omega}$ has one of these two properties. First assume that
$(b_{i})_{i\in\omega}$ is strictly increasing. Notice that $b_{i}<x\in p_{*}$.
Since $p_{*}$ is a coheir of $p$, $p_{*}$ is finitely satisfiable over
$\omega_{1}$. So, for each $b_{i}$ there exists $\alpha$ in $\omega_{1}$ such
that $b_{i}<\alpha$. Now, for each $b_{i}$, we define
$\alpha_{i}:=\min\\{\alpha\in\omega_{1}:\mathcal{U}\models b_{i}<\alpha\\}$.
Since $\omega_{1}$ is well-ordered, $\alpha_{i}$ is well-defined. We let
$\beta$ be the supremum (in $\omega_{1}$) of $\\{\alpha_{i}:i\in\omega\\}$.
Then $\mathcal{U}\models b_{i}<\beta$ for each $i\in\omega$, and so but
$x<\beta\in p_{*}$, contradiction.

Now we assume that $(b_{i})_{i\in\omega}$ is a strictly decreasing
subsequence. Notice that for each $i\in\omega$, $b_{i}>x\in p_{*}$. Let
$\Theta(x)=\\{\alpha<x:\alpha\in\omega_{1}\\}\ \cup\\{x<b_{i}:i\in\omega\\}$.
By compactness, choose $c_{\infty}$ in $\mathcal{U}$ satisfying $\Theta(x)$.
Since $p_{*}$ is finitely satisfiable over $\omega_{1}$, we have
$c_{\infty}>x\in p_{*}$. But since $\mathcal{U}\models b_{i}>c_{\infty}$ for
each $i\in\omega$, we have that $x>c_{\infty}\in p$, contradiction. ∎

######  Remark 4.12.

The type $p_{*}$ in Proposition 4.11 is finitely satisfiable over a small
model, but not finitely satisfiable over any countable submodel by Theorem
1.1.

######  Proposition 4.13.

Let $p_{*}$ be as in Proposition 4.11. Then the associated Keisler measure
$\delta_{p_{*}}$ is not sequentially approximated.

###### Proof.

Clear from $(iv)$ of Proposition 3.3. ∎

######  Proposition 4.14.

Let $T^{2}_{s}$ be the theory of the random $K_{s}$-free graph in the language
$\mathcal{L}=\\{E(x,y)\\}$. Let $p_{*}$ be the unique global complete type
extending the formulas $\\{\neg E(x,b):b\in\mathcal{U}\\}$. Then,
$\delta_{p_{*}}$ is sequentially approximated (even finitely approximated over
any submodel) but $p_{*}$ is not sequentially approximated. Moreover,
$T^{2}_{s}$ admits no (non-realized) sequentially approximated types.

###### Proof.

The proof that $\delta_{p_{*}}$ is finitely approximated can be found in [6,
Theorem 5.8]. By $(iii)$ of Proposition 3.3, $\delta_{p_{*}}$ is sequentially
approximated. By $(v)$ of Proposition 3.3, it suffices to show that there are
no non-realized types in one variable which are sequentially approximated. Let
$p$ be any non-realized type in $S_{1}(\mathcal{U})$ and assume that
$(b_{i})_{i\in\omega}$ is a sequence of points in $\mathcal{U}^{x}$ such that
$\lim_{i\to\infty}\operatorname{tp}(b_{i}/\mathcal{U})=p$. Since $p$ is non-
realized, we may assume that the points in $(b_{i})_{i\in\omega}$ are
distinct. Then, by Ramsey’s theorem, there is a subsequence which is either
independent or complete. It cannot be complete, because that would violate
$K_{s}$-freeness. Therefore, $(b_{i})_{i\in\omega}$ contains an independent
subsequence, call it $(c_{i})_{i\in\omega}$. By compactness, there exists an
$a$ in $\mathcal{U}$ such that $\mathcal{U}\models E(c_{i},a)$ if and only if
$i$ is even. Then, $(\operatorname{tp}(c_{i}/\mathcal{U}))_{i\in\omega}$ does
not converge in $S_{x}(\mathcal{U})$ and so
$(\operatorname{tp}(b_{i}/\mathcal{U}))_{i\in\omega}$ does not converge in
$S_{x}(\mathcal{U})$. ∎

######  Question 4.15.

We say a global type $p$ in $S_{x}(\mathcal{U})$ is sad222Credit to James
Hanson for the terminology. if it is both sequentially approximated and
definable. Does there exist a global type $p$ which is sad over a model $M$
but is not generically stable over $M$? It is clear that if $T$ is NIP, then
all sad types are generically stable. Therefore an example of such a type must
come from the wild.

##  5\. Sequential approximations of measures in NIP theories

Throughout this section, we assume that $T$ is a countable NIP theory and
$\mathcal{U}$ is a monster model of $T$. We show that measures which are
finitely satisfiable over a countable model of $T$ are sequentially
approximated (Theorem T2). To do this, we introduce the notion of a smooth
sequence. These are sequences of global measures which are intended to play
the role of a Morley sequence for a measure. Unfortunately, these sequences
only exist (a priori) in the NIP context and it is currently not known how to
expand this idea to IP theories. At the end of this section, we give a
characterization of generic stability using smooth sequences (again, only in
the NIP context).

To motivate the machinery introduced in this section, we explain why Theorem
T2 does not follow directly from some approximation results currently in the
literature. One might assume that one could prove Theorem T2 from Theorem 1.1
in tandem with the following fact [16, Proposition 7.11],

######  Fact 5.1 (T is NIP).

Suppose that $\mu\in\mathfrak{M}_{x}(\mathcal{U})$ and $\mu$ is finitely
satisfiable over $M$. Then, for any formula $\varphi(x,y)\in\mathcal{L}$ and
every $\epsilon>0$, there exists types $p_{1},...,p_{n}\in
S_{x}(\mathcal{U})$, where for each $i\leq n$ the type $p_{i}$ is finitely
satisfiable over $M$, and

$\sup_{b\in\mathcal{U}^{y}}|\mu(\varphi(x,b))-\operatorname{Av}(\overline{p})(\varphi(x,b))|<\epsilon.$

If $\mu$ is in $\mathfrak{M}_{x}(\mathcal{U})$ and is finitely satisfiable
over a countable model $M$, then one can use Theorem 1.1 and Fact 5.1 together
to produce:

  1. (1)

a sequence of global measures
$(\operatorname{Av}(\overline{p}_{i}))_{i\in\mathbb{N}}$ such that each
$\overline{p}_{i}=(p_{i_{1}},...,p_{i_{k}})$, each $p_{i_{k}}\in
S_{x}(\mathcal{U})$ is finitely satisfiable over $M$, and
$\lim_{i\to\infty}\operatorname{Av}(\overline{p}_{i})=\mu$ in
$\mathfrak{M}_{x}(\mathcal{U})$,

  2. (2)

for each $i\in\mathbb{N}$, a sequence of points
$(\overline{a}_{i_{j}})_{j\in\mathbb{N}}$ each in $(M^{x})^{<\omega}$ so that
$\lim_{j\to\infty}\operatorname{Av}(\overline{a}_{i_{j}})=\operatorname{Av}(\overline{p}_{i})$.

This construction gives an array of points
$(\overline{a}_{i_{j}})_{(i,j)\in\mathbb{N}\times\mathbb{N}}$ in
$(M^{x})^{<\omega}$ so that

$\lim_{i\to\infty}\lim_{j\to\infty}\Big{(}\operatorname{Av}(\overline{a}_{i_{j}})\Big{)}=\mu\text{
in $\mathfrak{M}_{x}(\mathcal{U})$}.$

A priori, the convergence of an array does not imply that there exists a
subsequence of that array which converges to the array’s limit333For example,
any Baire-2 function which is not Baire-1 can be written as the limit of an
array of continuous functions, but cannot be written as the sequential limit
of continuous functions.. A similar situation arises by trying to iterate
Theorem 5.13. So, we must work slightly harder. As previously stated, our
proof essentially mimics the proof of Theorem 1.1 but with Morley sequences
replaced by smooth sequences. Finally we remark that if there were an
elementary proof using an array to show this result, then we would have a
moderately simple proof that $\operatorname{dfs}$ measures are finitely
approximated in NIP theories. In particular, this proof would bypass the
implicit use of randomizations (i.e. $(i)$ of Fact 5.5).

We formally begin this section by discussing a “continuous” analogue of
eventually indiscernible sequences.

###  5.1. Eventually indiscernible sequences revisited

We fix some notation. Fix distinct tuples of variables $x$ and
$x_{0},...,x_{n}$ such that $|x|=|x_{i}|$ for $i\leq n$. If
$\varphi(x_{0},...,x_{n})$ is a formula in
$\mathcal{L}_{x_{0},...,x_{n}}(\mathcal{U})$ and
$\overline{a}_{0},...,\overline{a}_{n}$ is a finite sequence of elements where
each $\overline{a}_{i}\in(\mathcal{U}^{x})^{<\omega}$ and
$\overline{a}_{i}=(a_{i,0},...,a_{i,m_{i}})$ for $i\leq n$, then we write
$\varphi_{c}(\overline{a}_{0},...,\overline{a}_{n})$ to mean,

$\bigotimes_{i=0}^{n}\operatorname{Av}(\overline{a}_{i})_{x_{i}}(\varphi(x_{0},...,x_{n})).$

Notice that $\varphi_{c}(\overline{a}_{0},...,\overline{a}_{n})$ is a real
number. We observe that by unpacking the definition of the product measure,
our formula can be computed as follows:

$\varphi_{c}(\overline{a}_{0},...,\overline{a}_{n})=\frac{1}{\prod_{i=0}^{n}(m_{i}+1)}\sum_{j_{0}=0}^{m_{0}}...\sum_{j_{n}=0}^{m_{n}}\mathbf{1}_{\varphi}(a_{0,j_{0}},...,a_{n,j_{n}}).$

######  Definition 5.2.

Let $(\overline{a}_{i})_{i\in\omega}$ be a sequence of elements in
$(\mathcal{U}^{x})^{<\omega}$ and let $A\subset\mathcal{U}$ be a collection of
parameters. Then we say that the sequence $(\overline{a}_{i})_{i\in\omega}$ is
eventually indiscernible over $A$ if for any formula
$\varphi(x_{0},...,x_{n})$ in $\mathcal{L}_{(x_{i})_{i\in\omega}}(A)$ and any
$\epsilon>0$, there exists $N_{\epsilon,\varphi}$ such that for any
$n_{k}>...>n_{0}>N_{\epsilon,\varphi}$ and
$m_{k}>....>m_{0}>N_{\epsilon,\varphi}$,

$|\varphi_{c}(\overline{a}_{n_{0}},...,\overline{a}_{n_{k}})-\varphi_{c}(\overline{a}_{m_{0}},...,\overline{a}_{m_{k}})|<\epsilon.$

######  Proposition 5.3.

Let $(\overline{a}_{i})_{i\in\omega}$ be a sequence of tuples in
$(\mathcal{U}^{x})^{<\omega}$. If $A$ is a countable set of parameters, then
there exists some subsequence $(\overline{c}_{i})_{i\in\omega}$ of
$(\overline{a})_{i\in\omega}$ such that $(\overline{c}_{i})_{i\in\omega}$ is
eventually indiscernible over $A$.

###### Proof.

This proof is a standard application of Ramsey’s theorem applied to the
“continuous” setting. Enumerate all pairs in
$\mathcal{L}_{(x_{i})_{i\in\omega}}(A)\times\mathbb{N}_{>0}$. Let
$(\overline{a}_{i}^{0})_{i\in\omega}:=(\overline{a}_{i})_{i\in\omega}$ and set
$B_{0}=\\{\overline{a}^{0}_{i}:i\in\omega\\}$. Now, assume we have constructed
the subsequence $(\overline{a}_{i}^{l})_{i\in\omega}$ and $B_{l}$ (where
$B_{l}=\\{\overline{a}_{i}^{l}:i\in\omega\\}$). We now construct
$(\overline{a}_{i}^{l+1})_{i\in\omega}$ and $B_{l+1}$. Assume that
$(\varphi(x_{0},...,x_{k}),n)$ is the $l+1$ indexed pair in our enumeration.
Then we define the coloring $r_{l+1}:[B_{l}]^{k+1}\to\\{0,...,n\\}$ via

$r(\\{\overline{a}^{l}_{i_{0}},...,\overline{a}^{l}_{i_{k}}\\})=\lfloor
n\cdot\varphi_{c}(\overline{a}^{l}_{i_{0}},...,\overline{a}^{l}_{i_{k}})\rfloor.$

where $i_{0}<i_{1}<...<i_{k}$. By Ramsey’s theorem, there is an infinite
monochromatic subset $B_{l}^{\prime}$ of $B_{l}$. Let
$(\overline{a}_{i}^{l+1})_{i\in\omega}$ be the obvious reindexed subsequence
of $(\overline{a}_{i}^{l})_{i\in\omega}$ with the elements only from the
monochromatic set $B_{l}^{{}^{\prime}}$. We let
$B_{l+1}=\\{\overline{a}^{l+1}_{i}:i\in\omega\\}$. By construction, the
sequence $(\overline{a}_{i}^{i})_{i\in\omega}$ is eventually indiscernible. ∎

We now present a collection of facts which will help us prove that the
associated average measures along eventually indiscernible sequences always
converge to a measure in $\mathfrak{M}_{x}(\mathcal{U})$ when the underlying
theory is NIP. The first fact is elementary and left to the reader as an
exercise.

######  Fact 5.4.

Assume that $(\mu_{i})_{i\in\omega}$ is a sequence of Keisler measures in
$\mathfrak{M}_{x}(\mathcal{U})$. If for every formula
$\varphi(x)\in\mathcal{L}_{x}(\mathcal{U})$,
$\lim_{i\to\infty}\mu_{i}(\varphi(x))$ converges, then
$(\mu_{i})_{i\in\omega}$ converges to a measure in
$\mathfrak{M}_{x}(\mathcal{U})$.

The next collection of facts can be found in [10]. In particular, $(i)$
follows immediately from Lemma 2.10 while $(ii)$ and $(iii)$ are from
Corollary 2.14. The proof of Lemma 2.10 is non-trivial and is an
interpretation of results in [2]. Implicitly, our proof uses the fact that the
randomization of an NIP theory is NIP.

######  Fact 5.5 (T is NIP).

Suppose that $\lambda\in\mathfrak{M}_{(x_{i})_{i\in\omega}}$ where
$|x_{i}|=|x_{j}|$ for each $i,j<\omega$. $\lambda$ is said to be
$M$-indiscernible if for every increasing sequence of indices
$i_{0},...,i_{n}$ and any formula $\varphi(x_{i_{0}},...,x_{i_{n}})$ in
$\mathcal{L}_{(x_{i})_{i\in\omega}}(M)$, we have that

$\lambda(\varphi(x_{i_{0}},...,x_{i_{n}}))=\lambda(\varphi(x_{0},...,x_{n})).$

Let $\mu,\nu\in\mathfrak{M}_{x}(\mathcal{U})$ such that $\mu,\nu$ are
invariant over $M$. The following statements are true.

  1. ($i$)

If $\lambda$ is $M$-indiscernible, then for any formula
$\varphi(x,b)\in\mathcal{L}_{x}(\mathcal{U})$, we have that
$\lim_{i\to\infty}\lambda(\varphi(x_{i},b))$ exists.

  2. ($ii$)

The measures $\mu^{(\omega)}$ and $\nu^{(\omega)}$ are $M$-indiscernible.

  3. ($iii$)

If $\mu^{(\omega)}|_{M}=\nu^{(\omega)}|_{M}$, then $\mu=\nu$.

We now establish a formal connection between eventually indiscernible
sequences of tuples and indiscernible measures. We use this connection to show
that the eventually indiscernible sequences converges to a measure in
$\mathfrak{M}_{x}(\mathcal{U})$.

######  Proposition 5.6.

Let $(\overline{c}_{i})_{i\in\omega}$ be a sequence of points in
$(\mathcal{U}^{x})^{<\omega}$. If $(\overline{c}_{i})_{i\in\omega}$ is an
eventually indiscernible sequence over some model $M$, then the sequence
$(\operatorname{Av}(\overline{c}_{i}))_{i\in\omega}$ converges in
$\mathfrak{M}_{x}(\mathcal{U})$.

###### Proof.

Assume not. Then there exists some formula $\psi(x,b)$ in
$\mathcal{L}_{x}(\mathcal{U})$, some $\epsilon_{0}>0$, and some subsequence
$(\overline{c}_{i}^{\prime})_{i\in\omega}$ of
$(\overline{c}_{i})_{i\in\omega}$ such that for each natural number $i$,

$|\operatorname{Av}(\overline{c}_{i}^{\prime})(\psi(x;b))-\operatorname{Av}(\overline{c}_{i+1}^{\prime})(\psi(x;b))|>\epsilon_{0}.$

It is clear that $(\overline{c}_{i}^{\prime})_{i\in\omega}$ is also eventually
indiscernible over $M$. We now aim to contradict $(i)$ of Fact 5.5 via
(topological) compactness of the space
$\mathfrak{M}_{\omega}(\mathcal{U}):=\mathfrak{M}_{(x_{i})_{i\in\omega}}(\mathcal{U})$.
For any formula
$\varphi(x_{i_{0}},...,x_{i_{k}})\in\mathcal{L}_{(x_{i})_{i\in\omega}}(M)$, we
let $r_{\varphi}$ be the unique real number such that for every $\epsilon>0$,
there exists an $N_{\epsilon,\varphi}$ so that for any
$n_{k}>...>n_{0}>N_{\epsilon,\varphi}$ we have

$|\varphi_{c}(\overline{c}^{\prime}_{n_{0}},...,\overline{c}^{\prime}_{n_{k}})-r_{\varphi}|<\epsilon.$

Since the sequence $(\overline{c}_{i}^{\prime})_{i\in\omega}$ is eventually
indiscernible over $M$, $r_{\varphi}$ exists for each
$\varphi(\overline{x})\in\mathcal{L}_{(x_{i})_{i\in\omega}}(M)$. Now, for
every $\varphi(\overline{x})\in\mathcal{L}_{(x_{i})_{i\in\omega}}(M)$ and
$\epsilon>0$, we define the following family of closed subsets of
$\mathfrak{M}_{\omega}(\mathcal{U})$;

$C_{\epsilon,\varphi}=\Big{\\{}\lambda\in\mathfrak{M}_{\omega}(\mathcal{U}):r_{\varphi}-\epsilon\leq\lambda(\varphi(\overline{x}))\leq
r_{\varphi}+\epsilon\Big{\\}}.$

We also define another family of sets and argue that they are closed; let

$D_{i}=\Big{\\{}\lambda\in\mathfrak{M}_{\omega}(\mathcal{U}):|\lambda(\psi(x_{i},b))-\lambda(\psi(x_{i+1},b))|\geq\frac{\epsilon_{0}}{2}\Big{\\}}.$

Notice that $D_{i}$ is closed since for every natural number $i$, the
evaluation map $E_{i}:\mathfrak{M}_{\omega}(\mathcal{U})\to[0,1]$ via
$E_{i}(\lambda)=\lambda(\varphi(x_{i},b))$ is continuous. Indeed, define
$F_{i}=E_{i}-E_{i+1}$ and $H_{i}=E_{i+1}-E_{i}$. Then we have
$D_{i}=F_{i}^{-1}([\frac{\epsilon_{0}}{2},1])\cup
H_{i}^{-1}([\frac{\epsilon_{0}}{2},1])$ and so $D_{i}$ is a union of two
closed sets and therefore closed. Using
$(\overline{c}_{i}^{\prime})_{i\in\omega}$, the collection
$\Phi=\\{C_{\epsilon,\varphi}:\epsilon>0,\varphi(\overline{x})\in\mathcal{L}_{\omega}(M)\\}\cup\\{D_{i}:i\in\omega\\}$
has the finite intersection property. Therefore, there exists some
$\lambda\in\mathfrak{M}_{\omega}(\mathcal{U})$ in the intersection of all the
sets in $\Phi$. Moreover, $\lambda$ is $M$-indiscernible by construction.
Since $\lambda$ is in $D_{i}$ for each $i$, its existence contradicts $(i)$ of
Fact 5.5. ∎

###  5.2. Smooth sequences

In this subsection, we define the notion of a smooth sequence and prove the
main theorem. If $\mu$ is a global $M$-invariant measure, then a smooth
sequence is a collection of models and measures meant to replicate a Morley
sequence. The ideology is the following: A Morley sequence in $p$ over $M$ is
to the infinite type $p^{\omega}|_{M}$ as a smooth sequence in $\mu$ over $M$
is to the measure $\mu^{(\omega)}|_{M}$. We now provide the formal definition.

######  Definition 5.7.

Let $\mu\in\mathfrak{M}_{x}(\mathcal{U})$ and assume that $\mu$ is invariant
over some small model $M$. Then, a smooth sequence in $\mu$ over $M$ is a
sequence of pairs of measures and small models,
$(\mu_{i},N_{i})_{i\in\omega}$, such that:

  1. $(i)$

$M\prec N_{0}$ and $N_{i}\prec N_{i+1}$ and each $N_{i}$ is small.

  2. $(ii)$

$\mu_{i}$ is smooth over $N_{i}$.

  3. $(iii)$

$\mu_{0}|_{M}=\mu|_{M}$ and for $i>0$, $\mu_{i}|_{N_{i-1}}=\mu|_{N_{i-1}}$.

Furthermore, we define
$\bigotimes_{i=0}^{\omega}\mu_{i}=\bigcup_{i=0}^{\omega}\bigotimes_{i=0}^{n}\mu_{i}$
which is an element of $\mathfrak{M}_{(x_{i})_{i\in\omega}}(\mathcal{U})$. We
let $N_{\omega}=\bigcup_{i\in\omega}N_{i}$. Notice that for each $i\in\omega$,
the measure $\mu_{i}$ is smooth over $N_{\omega}$.

######  Proposition 5.8.

If $T$ is a countable NIP theory, $\mu\in\mathfrak{M}_{x}(\mathcal{U})$, and
$\mu$ is invariant over $M$ where $|M|=\aleph_{0}$, then there exists a smooth
sequence $(\mu_{i},N_{i})_{i\in\omega}$ in $\mu$ over $M$ such that each
$N_{i}$ is countable.

###### Proof.

This follows directly from Proposition 2.9. ∎

######  Proposition 5.9 (T is NIP).

Assume that $\mu\in\mathfrak{M}_{x}(\mathcal{U})$ and $\mu$ is $M$-invariant.
Let $(\mu_{i},N_{i})_{i\in\omega}$ be a smooth sequence in $\mu$ over $M$.
Then, $\bigotimes_{i=0}^{\omega}\mu_{i}|_{M}=\mu^{(\omega)}|_{M}$. Hence,
$\bigotimes_{i=0}^{\omega}\mu_{i}$ is $M$-indiscernible.

###### Proof.

We prove this via induction on formulas in
$\mathcal{L}_{(x_{i})_{i\in\omega}}(\mathcal{U})$. For our base case, it is
true by construction that $\mu_{0}|_{M}=\mu|_{M}$. For our induction
hypothesis, we assume that
$\mu^{(k-1)}|_{M}=\bigotimes_{i=0}^{k-1}\mu_{i}|_{M}$. For ease of notation,
we set $\lambda=\bigotimes_{i=0}^{k-1}\mu_{i}$ and show the induction step:
Let $\varphi(x_{0},...,x_{k})$ be any formula in
$\mathcal{L}_{x_{0},...,x_{k}}(M)$. Since the product of smooth measures is
smooth (by $(iii)$ of Fact 2.12), we have that $\lambda$ is smooth over
$N_{k-1}$. In particular, $\lambda$ is invariant over $N_{k-1}$. We let
$\overline{x}=(x_{0},...,x_{k-1})$ and
$\theta(x_{k};\overline{x})=\varphi(x_{0},...,x_{k})$. We consider the
following computation followed by a list of justifications.

$\mu_{k}\otimes\lambda(\varphi(x_{0},...,x_{k}))=\int_{S_{\overline{x}}(N_{k})}F_{\mu_{k}}^{\theta}d(\lambda|_{N_{k}})\overset{(a)}{=}\int_{S_{x_{k}}(N_{k})}F_{\lambda}^{\theta^{*}}d(\mu_{k}|_{N_{k}})$
$\overset{(b)}{=}\int_{S_{x_{k}}(N_{k-1})}F_{\lambda}^{\theta^{*}}d(\mu_{k}|_{N_{k-1}})\overset{(c)}{=}\int_{S_{x_{k}}(N_{k-1})}F_{\lambda}^{\theta^{*}}d(\mu|_{N_{k-1}})\overset{(a)}{=}\int_{S_{\overline{x}}(N_{k-1})}F_{\mu}^{\theta}d(\lambda|_{N_{k-1}})$
$\overset{(d)}{=}\int_{S_{\overline{x}}(M)}F_{\mu}^{\theta}d(\lambda|_{M})\overset{(e)}{=}\int_{S_{\overline{x}}(M)}F_{\mu}^{\theta}d(\mu^{(k-1)}|_{M})=\mu\otimes\mu^{(k-1)}(\varphi(x_{0},...,x_{k})).$

We provide the following justifications:

  1. ($a$)

Smooth measures commute with invariant measures.

  2. ($b$)

Changing space of integration since $\lambda$ is invariant over $N_{k-1}$.

  3. ($c$)

By construction of smooth sequences, we have that
$\mu_{k}|_{N_{k-1}}=\mu|_{N_{k-1}}$.

  4. ($d$)

Changing space of integration since $\mu$ is invariant over $M$.

  5. ($e$)

By our induction hypothesis. ∎

We now begin the proof of our main theorem. Again, the proof is similar to
both the generically stable case in the previous section and even more so to
the proof of Lemma 2.8 in [17]. Here, the major difference is that we replace
the Morley sequence in that proof with a countable model, $N_{\omega}$, which
“contains” a smooth sequence in $\mu$ over $M$. Then we find a sequence of
elements in $(M^{x})^{<\omega}$ such that the associated average measures
converge to $\mu|_{N_{\omega}}$ in $\mathfrak{M}_{x}(N_{\omega})$. After
choosing an eventually indiscernible subsequence, we know from our NIP
assumption that this new sequence converges to a global measure $\nu$ in
$\mathfrak{M}_{x}(\mathcal{U})$. Finally, we demonstrate that
$\nu^{(\omega)}|_{M}=\mu^{(\omega)}|_{M}$ which completes the proof.

######  Theorem 5.10 ($T$ is NIP).

Let $\mu$ be finitely satisfiable over a countable model $M$. Then there
exists a sequence $(\overline{a})_{i\in\omega}$ of elements, each in
$(M^{x})^{<\omega}$, such that for any
$\theta(x)\in\mathcal{L}_{x}(\mathcal{U})$, we have that,

$\lim_{i\to\infty}\operatorname{Av}(\overline{a}_{i})(\theta(x))=\mu(\theta(x)).$

###### Proof.

Choose a smooth sequence $(\mu_{i},N_{i})_{i\in\omega}$ in $\mu$ over $M$. By
Proposition 5.8 we may choose this sequence so that for each $i\in\omega$,
$N_{i}$ is countable. In particular, this implies that $N_{\omega}$ is a
countable model. We begin by constructing a sequence of elements
$(\overline{a}_{i})_{i\in\omega}$ in $(M^{x})^{<\omega}$ such that
$(\operatorname{Av}(\overline{a}_{i})|_{N_{\omega}})_{i\in\omega}$ converges
to $\mu|_{N_{\omega}}$ in $\mathfrak{M}_{x}(N_{\omega})$. Since $N_{\omega}$
is countable, we let $(\theta_{i}(x))_{i\in\omega}$ be an enumeration of the
formulas in $\mathcal{L}_{x}(N_{\omega})$. Since $\mu$ is finitely satisfiable
over $M$, we can find we find $\overline{a}_{k}\in(M^{x})^{<\omega}$ such that
for any $j\leq k$, we have that,

$|\mu(\theta_{j}(x))-\operatorname{Av}(\overline{a}_{k})(\theta_{j}(x))|<\frac{1}{k}.$

By construction, it is clear that the sequence
$(\operatorname{Av}(\overline{a}_{i})|_{N_{\omega}})_{i\in\omega}$ converges
to $\mu|_{N_{\omega}}$ in $\mathfrak{M}_{x}(N_{\omega}$). Now, we let
$(\overline{c}_{i})_{i\in\omega}$ be a subsequence of
$(\overline{a}_{i})_{i\in\omega}$ so that $(\overline{c}_{i})_{i\in\omega}$ is
eventually indiscernible over $N_{\omega}$. Then the sequence
$(\operatorname{Av}(\overline{c}_{i}))_{i\in\omega}$ converges in
$\mathfrak{M}_{x}(\mathcal{U})$ by Proposition 5.6. Assume that
$(\operatorname{Av}(\overline{c}_{i}))_{i\in\omega}$ converges to some measure
$\nu\in\mathfrak{M}_{x}(\mathcal{U})$. Hence, $\nu$ is finitely satisfiable
over $M$ by $(i)$ of Proposition 3.3 and therefore $\nu$ is invariant over
$M$. We show that $\nu^{(\omega)}|_{M}=\mu^{(\omega)}|_{M}$. This will
conclude the proof by $(iii)$ of Fact 5.5.

Since $(\overline{c}_{i})_{i\in\omega}$ is a subsequence of
$(\overline{a}_{i})_{i\in\omega}$, it follows that
$\nu|_{N_{\omega}}=\mu|_{N_{\omega}}$ and therefore $\nu|_{M}=\mu|_{M}$. We
now proceed by induction. Assume that $\nu^{(k-1)}|_{M}=\mu^{(k-1)}|_{M}$. Fix
$\varphi(x_{0},...,x_{k})$ in $\mathcal{L}_{x_{0},...,x_{k}}(M)$. For ease of
notation, set $\lambda=\bigotimes_{i=0}^{k-1}\mu_{i}$. We recall that
$\lambda$ is smooth over $N_{\omega}$ (see Fact 2.12). By Proposition 5.9,
$\mu^{(k-1)}|_{M}=\lambda|_{M}$. We let $\overline{x}=(x_{0},...,x_{k-1})$ and
let $\theta(x_{k};\overline{x})=\varphi(x_{0},...,x_{k})$. We now consider the
critical computation followed a small glossary of justifications.

$\nu^{(k)}(\varphi(x_{0},...,x_{k}))=\int_{S_{\overline{x}}(M)}F_{\nu}^{\theta}d(\nu^{(k-1)}|_{M})\overset{(a)}{=}\int_{S_{\overline{x}}(M)}F_{\nu}^{\theta}d(\mu^{(k-1)}|_{M})$
$\overset{(b)}{=}\int_{S_{\overline{x}}(M)}F_{\nu}^{\theta}d(\lambda|_{M})\overset{(c)}{=}\int_{S_{\overline{x}}(N_{\omega})}F_{\nu}^{\theta}d(\lambda|_{N_{\omega}})\overset{(d)}{=}\int_{S_{x_{k}}(N_{\omega})}F_{\lambda}^{\theta^{*}}d(\nu|_{N_{\omega}})$
$\overset{(e)}{=}\int_{S_{x_{k}}(N_{\omega})}F_{\lambda}^{\theta^{*}}d(\mu|_{N_{\omega}})\overset{(d)}{=}\int_{S_{\overline{x}}(N_{\omega})}F_{\mu}^{\theta}d(\lambda|_{N_{\omega}})\overset{(c)}{=}\int_{S_{\overline{x}}(M)}F_{\mu}^{\theta}d(\lambda|_{M})$
$\overset{(b)}{=}\int_{S_{\overline{x}}(M)}F_{\mu}^{\theta}d(\mu^{(k-1)}|_{M})=\mu^{(k)}(\varphi(x_{0},...,x_{k})).$

We provide the following justifications:

  1. (a)

Induction hypothesis.

  2. (b)

$\mu^{(k-1)}|_{M}=\lambda|_{M}$.

  3. (c)

Changing the space of integration.

  4. (d)

Smooth measures commute with invariant measures.

  5. (e)

$\nu|_{N_{\omega}}=\mu|_{N_{\omega}}$ ∎

We now observe that we have another proof of the theorem that global measures
in NIP theories which are definable and finitely satisfiable are also finitely
approximated.

######  Corollary 5.11.

If $T^{\prime}$ is a countable or uncountable NIP theory and $\mu$ is
$\operatorname{dfs}$ over $M$, then $\mu$ is finitely approximated over $M$.

###### Proof.

After restricting to a countable language, we still have a
$\operatorname{dfs}$ measures (by [6, Proposition 2.9]). By Proposition 2.9,
$\mu$ restricted to this language is $\operatorname{dfs}$ over a countable
model, $M_{0}$. By the previous result, $\mu$ is sequentially approximated
over $M_{0}$. Since $\mu$ is also definable, an application of Proposition 3.4
yields the result. ∎

######  Observation 5.12.

Assume that $\mu\in\mathfrak{M}_{x}(\mathcal{U})$ and let $M$ be a small
elementary submodel. Then, $\mu$ is sequentially approximated over $M$ if

  1. (1)

$T$ is stable, and $\mu$ is invariant over $M$,

  2. (2)

$T$ is NIP, $|M|=\aleph_{0}$, and $\mu$ is finitely satisfiable over $M$, or

  3. (3)

$\mu$ is finitely approximated over $M$.

Finally, one may ask what happens in the local context. We remark that there
exists two proofs for a local version of Theorem T2 which both rely on an
important result of Bourgain, Fremlin, and Talagrand whose connection to model
theory is (by now) well-known (e.g. [11, 18, 12, 8]). Chronologically, the
first proof of the following theorem is implicit in the work of Khanaki (see
[12, Remark 3.21, Theorem 3.26]) (through the observation that measures are
types over models of the randomization in continuous model theory and [1,
Proposition 1.1]),

######  Theorem 5.13.

Suppose $\mu$ is a Keisler measure in $\mathfrak{M}_{\varphi}(\mathcal{U})$,
$\mu$ is finitely satisfiable over $M$ where $|M|=\aleph_{0}$, and
$\varphi(x,y)$ is an NIP formula. Then there exists a sequence of points
$(\overline{a}_{i})_{i\in\omega}$ in $(M^{x})^{<\omega}$ such that for each
$b\in\mathcal{U}^{y}$,

$\lim_{i\to\infty}\operatorname{Av}(\overline{a}_{i})(\varphi(x,b))=\mu(\varphi(x,b)).$

There is another proof for the case of just Keisler measures via the VC
theorem (see [8, Lemma 4.7]) which came later.

###  5.3. Smooth sequences and generically stable measures in NIP theories

We now give an equivalent characterization for generically stable measures in
NIP theories. We invite the reader to review the definition of a generically
stable type prior to reading this section. Recall the following theorem due to
Hrushovski, Pillay, and Simon [10, Theorem 3.2].

######  Theorem 5.14 (T is NIP).

Assume that $\mu\in\mathfrak{M}_{x}(\mathcal{U})$. Then the following are
equivalent.

  1. ($i$)

$\mu$ is dfs.

  2. ($ii$)

$\mu$ is finitely approximated.

  3. ($iii$)

$\mu$ is fim (see [10, Definition 2.7]).

  4. ($iv$)

$\mu$ is invariant and $\mu_{x}\otimes\mu_{y}=\mu_{y}\otimes\mu_{x}$.

Moreover, a Keisler measure (in an NIP theory) is called generically stable if
it satisfies any/all of $(i)-(iv)$.

We will now show that smooth sequences can also give a characterization of
generically stable measures in NIP theories.

######  Lemma 5.15 (T is NIP).

Let $\mu\in\mathfrak{M}_{x}(\mathcal{U})$. Suppose that $\mu$ is generically
stable over $M$. For any smooth sequence $(\mu_{i},N_{i})_{i\in\omega}$ in
$\mu$ over $M$, we have that $\lim_{i\to\infty}\mu_{i}=\mu$ in
$\mathfrak{M}_{x}(\mathcal{U})$.

###### Proof.

Since $(\mu_{i},N_{i})_{i\in\omega}$ is a smooth sequence in $\mu$ over $M$,
the measure $\bigotimes_{i=0}^{\omega}\mu_{i}$ is indiscernible over $M$ by
Proposition 5.9. By $(i)$ of Fact 5.5, we know that
$\lim_{i\to\infty}\mu_{i}=\nu$ for some $\nu\in\mathfrak{M}_{x}(\mathcal{U})$.
Since each $\mu_{i}$ is finitely satisfiable over $N_{i}$, it follows that
$\nu$ is finitely satisfiable over $N_{\omega}$. By $(iii)$ of Fact 5.5, it is
enough to show that
$\nu^{(\omega)}|_{N_{\omega}}=\mu^{(\omega)}|_{N_{\omega}}$. The base case is
trivial. Assume that $\nu^{(k-1)}|_{N_{\omega}}=\mu^{(k-1)}|_{N_{\omega}}$.
Fix $\varphi(x_{0},...,x_{k})\in\mathcal{L}_{x_{0},...,x_{k}}(N_{\omega})$ and
$\epsilon>0$. Let $\overline{x}=(x_{0},...,x_{k-1})$ and
$\theta(x_{k};\overline{x})=\varphi(x_{0},...,x_{k})$. Since $\mu$ is
generically stable over $M$, $\mu^{(k-1)}$ is generically stable over $M$
($(v)$ of Fact 2.12) and so also definable over $N_{\omega}$. Therefore by
$(v)$ of Fact 2.8, there exists formulas
$\psi_{1}(x_{k}),...,\psi_{n}(x_{k})\in\mathcal{L}_{x_{k}}({N_{\omega}})$ and
real numbers $r_{1},...,r_{n}\in[0,1]$ so that

$\sup_{q\in
S_{x_{k}}(N_{\omega})}|F_{\mu^{(k-1)}}^{\theta^{*}}(q)-\sum_{i=1}^{n}r_{i}\mathbf{1}_{\psi_{i}(x_{k})}(q)|<\epsilon.$

Consider the following sequence of equations followed by a short list of
justifications.

$\nu^{(k)}(\varphi(x_{0},...,x_{k}))=\int_{S_{\bar{x}}(N_{\omega})}F_{\nu}^{\theta}d(\nu^{(k-1)}|_{N_{\omega}})\overset{(a)}{=}\int_{S_{\bar{x}}(N_{\omega})}F_{\nu}^{\theta}d(\mu^{(k-1)}|_{N_{\omega}})$
$\overset{(b)}{=}\int_{S_{x_{k}}(N_{\omega})}F_{\mu^{(k-1)}}^{\theta^{*}}d(\nu|_{N_{\omega}})\approx_{\epsilon}\int_{S_{x_{k}}(N_{\omega})}\sum_{i=1}^{n}r_{i}\mathbf{1}_{\psi_{i}(x_{k})}d(\nu|_{N_{\omega}})$
$=\sum_{i=1}^{n}r_{i}\nu(\psi_{i}(x_{k}))\overset{(c)}{=}\sum_{i=1}^{n}r_{i}\mu(\psi_{i}(x_{k}))=\int_{S_{x_{k}}(N_{\omega})}\sum_{i=1}^{n}r_{i}\mathbf{1}_{\psi_{i}(x_{k})}d(\mu|_{N_{\omega}})$
$\approx_{\epsilon}\int_{S_{x_{k}}(N_{\omega})}F_{\mu^{(k-1)}}^{\theta^{*}}d(\mu|_{N_{\omega}})\overset{(b)}{=}\int_{S_{x_{k}}(N_{\omega})}F_{\mu}^{\theta}d(\mu^{(k-1)}|_{N_{\omega}})=\mu^{(k)}(\varphi(x_{0},...,x_{k})).$

  1. (a)

Induction hypothesis.

  2. (b)

(T is NIP) Generically stable measures commute with invariant measures (see
$(b)$ of Fact 2.12).

  3. (c)

Base case.

As $\epsilon$ was arbitrary, this proves the result. ∎

######  Lemma 5.16 (T is NIP).

Assume that $\mu$ is $M$-invariant. If for every smooth sequence
$(\mu_{i},N_{i})_{i\in\mathbb{N}}$ in $\mu$ over $M$, we have that
$\lim_{i\to\infty}\mu_{i}=\mu$, then $\mu$ is generically stable over $M$.

###### Proof.

Since $T$ is NIP, all invariant measures are Borel definable. By Theorem 5.14,
it suffices to show that $\mu$ commutes with itself, i.e.
$\mu_{x}\otimes\mu_{y}=\mu_{y}\otimes\mu_{x}$. Fix
$\varphi(x,y)\in\mathcal{L}_{x,y}(\mathcal{U})$. Let $M_{1}$ be a small model
such that $M\prec M_{1}$ and $M_{1}$ contains all the parameters from
$\varphi(x,y)$. We choose a smooth sequence $(\mu_{i,x};N_{i})_{i\in\omega}$
in $\mu_{x}$ over $M_{1}$ and let $N_{\omega}=\bigcup_{i\in\omega}N_{i}$. By
construction, the sequence $(\mu_{i,x},N_{i})_{i\in\omega}$ is a smooth
sequence in $\mu_{x}$ over $M$. Consider the following computation.

$\mu_{x}\otimes\mu_{y}(\varphi(x,y))=\int_{S_{y}(M_{1})}F_{\mu_{x}}^{\varphi}d(\mu_{y}|_{M_{1}})\overset{(a)}{=}\int_{S_{y}(N_{\omega})}F_{\mu_{x}}^{\varphi}d(\mu_{y}|_{N_{\omega}})$
$\overset{(b)}{=}\lim_{i\to\infty}\int_{S_{y}(N_{\omega})}F_{\mu_{i,x}}^{\varphi}d(\mu_{y}|_{N_{\omega}})\overset{(c)}{=}\lim_{i\to\infty}\int_{S_{x}(N_{\omega})}F_{\mu_{y}}^{\varphi^{*}}d(\mu_{i,x}|_{N_{\omega}})$
$\overset{(d)}{=}\lim_{i\to\infty}\int_{S_{x}(M_{1})}F_{\mu_{y}}^{\varphi^{*}}d(\mu_{i,x}|_{M_{1}})\overset{(e)}{=}\lim_{i\to\infty}\int_{S_{x}(M_{1})}F_{\mu_{y}}^{\varphi^{*}}d(\mu_{x}|_{M_{1}})$
$=\int_{S_{x}(M_{1})}F_{\mu_{y}}^{\varphi^{*}}d(\mu_{x}|_{M_{1}})=\mu_{y}\otimes\mu_{x}(\varphi(x,y)).$

We provide a list of the following justifications:

  1. $(a)$

Changing the space of integration.

  2. $(b)$

Dominated convergence theorem.

  3. $(c)$

Smooth measures commute with Borel definable measures.

  4. $(d)$

Since $\mu_{y}$ is $M_{1}$ invariant.

  5. $(e)$

Since $\mu_{i,x}|_{M_{1}}=\mu_{x}|_{M_{1}}$ for any $i\in\omega$. ∎

######  Theorem 5.17 (T is NIP).

Let $\mu\in\mathfrak{M}_{x}(\mathcal{U})$. Then the following are equivalent:

  1. (1)

$\mu$ is generically stable over $M$.

  2. (2)

For any smooth sequence $(\mu_{i},N_{i})_{i\in\omega}$ in $\mu$ over $M$,

$\lim_{i\to\infty}\mu_{i}=\mu\text{ in $\mathfrak{M}_{x}(\mathcal{U})$.}$

###### Proof.

Follows directly from the previous two lemmas. ∎

##  6\. Local Measures revisited

We generalize the main theorem of [8]. Fix a partitioned NIP formula
$\varphi(x,y)$ and let $\mu$ be a $\varphi$-measure. In [8], we proved two
main theorems. We showed that if $\varphi(x,y)$ is an NIP formula and $\mu$ is
$\varphi$-definable and finitely satisfiable over a countable model $M$, then
$\mu$ is $\varphi$-finitely approximated. We then proved that if $\mu$ is
definable and finitely satisfiable over any small model $M$, then $\mu$ is
finitely approximated in $M$ by reducing to the previous theorem. But this was
somewhat unsatisfactory and the following question was left open: if $\mu$ is
$\varphi$-definable and finitely satisfiable over a small model, then is $\mu$
$\varphi$-finitely approximated? We give a positive answer to this question by
modifying one of the important technical lemmas in the proof. Let us first
recall some definitions.

######  Definition 6.1.

Fix $\mathcal{U}$ and a formula $\varphi(x,y)$ in $\mathcal{L}(\mathcal{U})$.

  1. (1)

$\mathcal{L}_{\varphi}(\mathcal{U})$ denotes the Boolean algebra of definable
sets of $\mathcal{U}^{x}$ generated by the collection
$\\{\varphi(x,b):b\in\mathcal{U}\\}$.

  2. (2)

A $\varphi$-measure is a finitely additive measure on the Boolean algebra
$\mathcal{L}_{\varphi}(\mathcal{U})$.

  3. (3)

The collection of all $\varphi$-measures is denoted
$\mathfrak{M}_{\varphi}(\mathcal{U})$.

  4. (4)

Let $M\prec\mathcal{U}$ and assume that $M$ contains all the parameters from
$\varphi(x,y)$. For any $\mu\in\mathfrak{M}_{\varphi}(\mathcal{U})$, we say
that $\mu$ is $(M,\varphi)$-invariant if for any $b,c\in\mathcal{U}^{y}$ such
that $\operatorname{tp}(b/M)=\operatorname{tp}(c/M)$, we have that
$\mu(\varphi(x,b))=\mu(\varphi(x,c))$.

  5. (5)

Let $\mu\in\mathfrak{M}_{\varphi}(M)$. If $\mu$ is $(M,\varphi)$-invariant,
then we define can the fiber map $F_{\mu}^{\varphi}:S_{y}(M)\to[0,1]$ via
$F_{\mu,M}^{\varphi}(q)=\mu(\varphi(x,b))$ where $b\models q|_{M}$. When $M$
is clear from context, we write $F_{\mu,M}^{\varphi}$ simply as
$F_{\mu}^{\varphi}$.

  6. (6)

Let $\mu\in\mathfrak{M}_{\varphi}(\mathcal{U})$. Then $\mu$ is said to be
$\varphi$-definable if the map $F_{\mu,M}^{\varphi}:S_{y}(M)\to[0,1]$ is
continuous.

  7. (7)

Let $\mu\in\mathfrak{M}_{\varphi}(\mathcal{U})$. Then $\mu$ is said to be
definable if for any formula $\theta(x,\overline{y})$ in the algebra generated
by $\\{\varphi(x,y_{i}):i\in\mathbb{N}\\}$, $\mu$ is $(M,\theta)$-invariant
and the map $F_{\mu}^{\varphi}:S_{y}(M)\to[0,1]$ is continuous.

  8. (8)

For any $\mu\in\mathfrak{M}_{\varphi}(\mathcal{U})$, $\mu$ is said to be
finitely satisfiable in $M$ if for every
$\theta(x)\in\mathcal{L}_{\varphi}(\mathcal{U})$ such that $\mu(\theta(x))>0$,
there exists some $a\in M$ so that $\mathcal{U}\models\theta(a)$.

  9. (9)

For each $a\in M$ we let $F_{a}^{\varphi}:S_{y}(M)\to[0,1]$ via
$F_{a}^{\varphi}=\mathbf{1}_{\varphi(a,y)}$. We denote the collection of such
functions as $\mathbb{F}_{M}$. We let $\operatorname{conv}(\mathbb{F}_{M})$ be
the collection of convex combinations of elements in $\mathbb{F}_{M}$. We let
$F=[0,1]^{S_{y}(M)}$ endowed with the Tychonoff topology and if $A\subset F$,
we let $\operatorname{cl}(A)$ denote its closure in this space and so the set
$\operatorname{cl}(\operatorname{conv}(A))$ is well-defined.

Recall the following facts about $\varphi$-measures which can be found in [8].

######  Fact 6.2.

Let $\mu\in\mathfrak{M}_{\varphi}(\mathcal{U})$ and $M\prec\mathcal{U}$.

  1. $(i)$

If $\mu$ is finitely satisfiable or $\varphi$-definable over $M$ then $\mu$ is
$(M,\varphi)$-invariant.

  2. $(ii)$

If $\mu$ is $\varphi$-definable over $M$ then $\mu$ is
$(M_{0},\varphi)$-invariant for some $M_{0}\prec M$ such that
$|M_{0}|=\aleph_{0}$.

  3. $(iii)$

If $\mu$ is finitely satisfiable over $M$ then $F_{\mu,M}^{\varphi}$ is in
$\operatorname{cl}(\operatorname{conv}(\mathbb{F}_{M}))$.

  4. $(iv)$

If $|M|=\aleph_{0}$ and $\varphi(x,y)$ is NIP, there exists a sequence of
elements $(g_{i})_{i\in\omega}$ with each
$g_{i}\in\operatorname{conv}(\mathbb{F}_{M})$ so that
$\lim_{i\to\infty}g_{i}=F_{\mu,M}^{\varphi}$.

The following lemma is essentially the missing lemma from [8]. The missed
observation is that one can consider finitely many parameters at once (instead
of a single parameter).

######  Lemma 6.3.

Suppose that $\mu\in\mathfrak{M}_{\varphi}(\mathcal{U})$ and $\mu$ is finitely
satisfiable in a small submodel $N$ and $(M,\varphi)$-invariant. Then the map
$F_{\mu,M}^{\varphi}\in\operatorname{cl}(\operatorname{conv}(\mathbb{F}_{M}))$.

###### Proof.

The proof is similar to the proof for types [16, Lemma 2.18] as well as the
proof for measures [8, Proposition 4.13] (which has both a stronger assumption
and conclusion). It suffices to show that for any finite collection of types
$p_{1},...,p_{n}\in S_{y}(M)$ and $\epsilon>0$ there exists
$\overline{a}\in(M^{x})^{<\omega}$ such that
$F_{\operatorname{Av}(\overline{a}),M}^{\varphi}(p_{i})\approx_{\epsilon}F_{\mu,M}^{\varphi}(p_{i})$
for each $i\leq n$.

Fix $p_{1},...,p_{n}\in S_{y}(M)$ and $\epsilon>0$. Choose $b_{i}\models
p_{i}$ for $i\leq n$. Let $q=\operatorname{tp}(N/M)\in S_{|N|}(M)$. Let
$\hat{q}\in S_{|N|}(\mathcal{U})$ such that $\hat{q}\supset q$ and $\hat{q}$
is finitely satisfiable in $M$, i.e. $\hat{q}$ is a global coheir of $q$. Let
$N_{1}\models\hat{q}|_{Mb_{1},...,b_{n}}$.

By compactness, there exists elements
$b_{1}^{\prime},...,b_{n}^{\prime}\in\mathcal{U}$ such that
$\operatorname{tp}(N_{1}b_{1},...,b_{n}/M)=tp(Nb_{1}^{\prime},...,b_{n}^{\prime}/M)$.
Since $\mu$ is $(M,\varphi)$-invariant, we have that

$F_{\mu,M}^{\varphi}(p_{i})=\mu(\varphi(x,b_{i}))=\mu(\varphi(x,b^{\prime}_{i})),$

for each $i\leq n$. Since $\mu$ is finitely satisfiable in $N$, there exists
some $m$ and $\overline{c}\in(N^{x})^{m}$ such that
$\operatorname{Av}(\overline{c})(\varphi(x,b^{\prime}_{i}))\approx_{\epsilon}\mu(\varphi(x,b_{i}^{\prime}))$
for $i\leq n$. Let $B_{i}=\\{j\leq m:\models\varphi(c_{j},b^{\prime}_{i})\\}$.
Now consider the formula

$\theta(x_{1},...,x_{m},y_{1},...y_{n})=\bigwedge_{i\leq
n}\Big{(}\bigwedge_{j\in B_{i}}\varphi(x_{j},y_{i})\wedge\bigwedge_{j\not\in
B_{i}}\neg\varphi(x_{j},y_{i})\Big{)}.$

By construction
$\theta(\overline{x},\overline{y})\in\operatorname{tp}(\overline{c},\overline{b^{\prime}}/M)$
and so for an appropriate choice of indices,
$\theta(\overline{x},\overline{y})\in\operatorname{tp}(Nb_{1}^{\prime},...,b_{n}^{\prime}/M)$.
Hence
$\theta(\overline{x},\overline{y})\in\operatorname{tp}(N_{1}b_{1},...,b_{n}/M)$
and so
$\theta(\overline{x},\overline{b})\in\operatorname{tp}(N_{1}/Mb_{1},...,b_{n})\subset\hat{q}$.
Since $\hat{q}$ is finitely satisfiable in $M$, there exists
$\overline{a}\in(M^{x})^{m}$ such that $\models\theta(\bar{a},\bar{b})$. By
construction, we have that for any $i\leq n$,

$F_{\operatorname{Av}(\overline{a}),M}^{\varphi}(p_{i})=\operatorname{Av}(\overline{a})(\varphi(x,b_{i}))=\operatorname{Av}(\overline{c})(\varphi(x,b^{\prime}_{i}))\approx_{\epsilon}\mu(\varphi(x,b^{\prime}_{i}))=F_{\mu,M}^{\varphi}(p_{i}).$

This concludes the proof. ∎

######  Theorem 6.4.

Fix a formula $\varphi(x,y)$ and a small model $M$ containing all the
parameters from $\varphi(x,y)$. Assume that
$\mu\in\mathfrak{M}_{\varphi}(\mathcal{U})$. If

  1. (1)

$\varphi(x;y)$ is NIP,

  2. (2)

$\mu$ is $\varphi$-definable over $M$,

  3. (3)

and $\mu$ is finitely satisfiable in $M$,

Then for every $\epsilon>0$, there exists $a_{1},...,a_{n}\in M^{x}$ such
that,

$\sup_{b\in\mathcal{U}^{y}}|\mu(\varphi(x,b))-\operatorname{Av}(\overline{a})(\varphi(x,b))|<\epsilon.$

###### Proof.

We remark that the proof is similar to that of Proposition 3.4. Since $\mu$ is
$\varphi$-definable over $M$, $\mu$ is $(M_{0},\varphi)$-invariant where
$M_{0}$ is a countable submodel of $M$. By Lemma 6.3, the map
$F_{\mu,M_{0}}^{\varphi}\in\operatorname{cl}(\operatorname{conv}(\mathbb{F}_{M_{0}}))$.
By Fact 6.2, there exists a sequence $(g_{i})_{i\in I}$ so that
$\lim_{i\to\infty}g_{i}=F_{\mu,M_{0}}^{\varphi}$. By Mazur’s lemma, for every
$\epsilon>0$, there exists a finite set $I\subset\mathbb{N}$ and positive real
numbers $\\{r_{i}:i\in I\\}$ such that $\sum_{i\in I}r_{i}=1$ and

$\sup_{q\in S_{y}(M_{0})}|F_{\mu,M_{0}}^{\varphi}(q)-\sum_{i\in
I}r_{i}g_{i}(q)|<\epsilon.$

The map $\sum_{i\in I}r_{i}g_{i}$ can clearly be uniformly approximated by an
average function. More explicitly, there exists
$\overline{d}\in(M^{x})^{<\omega}$ such that

$\sup_{q\in S_{y}(M)}|\sum_{i\in
I}r_{i}g_{i}(q)-F^{\varphi}_{\operatorname{Av}(\overline{d}),M}(q)|<\epsilon.$

Hence

$\sup_{b\in\mathcal{U}^{y}}|\mu(\varphi(x,b))-\operatorname{Av}(\overline{d})(\varphi(x,b))|=\sup_{q\in
S_{y}(M)}|F_{\mu,M}^{\varphi}(q)-F_{\operatorname{Av}(\overline{d}),M}^{\varphi}(q)|<2\epsilon.$

which completes the proof. ∎

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