Werner Porod

# Predictions for Composite Higgs models from gauge/gravity dualities

###### Abstract

Gauge/gravity dualities provide a very useful approach into solving strongly
coupled systems. We apply this to Composite Higgs models and determine the
mass hierarchies of the corresponding bound states. As a cross check we apply
this to QCD and compare the results to existing lattice calculations for which
we find good agreement. We then focus on a particular example whose
phenomenology has recently been studied in the literature in a generic way and
outline first phenomenological implications of our findings for the spectrum.

##  1 Introduction

The extension of the the AdS/CFT correspondence [1, 2, 3] to less symmetric
gauge/gravity dualities has proven to be a powerful tool in describing group
by adding probe branes [4]. This allows to study the related meson operators
[5, 6]. These methods were successfully used to obtain gravity duals of chiral
symmetry breaking ($\chi SB$) in confining non-abelian gauge theories [7, 8].
It is natural to apply this approach to other strongly coupled theories. We
focus here on composite Higgs models, see e.g. [9, 10] for reviews. These
models are characterised by strongly coupled gauge theory and an underlying
set of fermions dubbed hyperquarks in the following.

In composite Higgs models $\chi SB$ in the fermion sector is caused by a by a
strongly coupled gauge theory, similar to QCD. In this way at least four
Nambu-Goldstone bosons are generated [11]. By weakly gauging part of the
global chiral symmetries, four then pseudo-Nambu Goldstone bosons (pNGBs) can
be placed in the fundamental representation of SU(2)L to become the complex
Higgs field. The composite nature of the Higgs removes the huge level of fine
tuning in the Standard Model (SM) hierarchy problem. This strong dynamics
would occur at a scale of a few TeV, the expected mass scale for the bound
states.

We use non-conformal gauge/gravity models that explicitly include the gauge
theories’ dynamics through the running of the anomalous dimension $\gamma$ of
the hyperquark mass [12, 13]. The models are inspired by top-down models
involving probe D7-branes embedded into ten-dimensional supergravity. However,
this is combined with a phenomenological approach and sensible guesses for the
running of $\gamma$ which are based on perturbation theory. With this we
obtain prediction for part of the mesonic and baryonic spectrum of the theory.
Here we will demonstrate that power of this methods by comparing our results
with data for QCD bound states. Then we will focus on two models: (i) an Sp(4)
theory with 4 fundamental and 6 sextet Weyl fermions [14] for which also
lattice studies exist [15, 16]; and an Sp(4) theory with five sextet Weyl and
6 fundamental fermions [17]. Further examples can be found in ref. [13].

##  2 Gauge/gravity duality, basic idea and some technical aspects

We introduce here briefly the model based on gauge/gravity duality, which was
first suggested in ref. [18]. In [13] it has been dubbed Dynamic AdS/YM (Anti-
de Sitter/Yang-Mills) to emphasise that it can be used to holographically
describe the chiral symmetry breaking dynamics of any gauge theory. These
models include hyperquarks in several, potentially inequivalent,
representations.

We summarize here key elements and refer to [13] for further details. In this
modelling, the renormalization group (RG) scale emerges as an extra
holographic direction. In a five dimensional AdS5 space one has the metric

$\displaystyle ds^{2}=\frac{dr^{2}}{r^{2}}+r^{2}dx_{(1,3)}\,.$ (1)

The radial direction $r$ is interpreted as the RG scale and slices at some
fixed $r$ describe the gauge theory on the corresponding $1+3$ dimensional
sub-space. Fields in the AdS bulk have solutions which describe gauge
invariant observables, i.e. operators $\mathcal{O}$ and sources $\mathcal{J}$.

In QCD with two light flavours the chiral flavour symmetry
SU(2)${}_{L}\times$SU(2)R is broken to SU(2)V by the formation of a vacuum
expectation value for the operator $\bar{q}_{L}q_{R}+$ h.c. Holographically
the operator is described as a dimension one scalar field $L$ in AdS5
satisfying the Klein-Gordon equation [6]

$\displaystyle\frac{\partial}{\partial r}\left(r^{3}\frac{\partial L}{\partial
r}\right)-r\,\Delta m^{2}\,L=0\,,\quad
L=\frac{m}{r^{\gamma}}+\frac{\langle\bar{q}q\rangle}{r^{2-\gamma}}\,,\quad\gamma(\gamma-2)=\Delta
m^{2}\,.$ (2)

The solutions describe a dimension one mass $m$ and the dimension three quark
condensate $\langle\bar{q}q\rangle$ if $\Delta^{2}m=0$. For non-zero $\Delta
m^{2}$ these operators develop an anomalous dimension $\gamma$.

In AdS5 the instability bound for a scalar is given by the Breitenlohner-
Freedman (BF) bound [19] corresponding to $\Delta m^{2}=-1$. The massless
$L=0$ solution becomes unstable if $\Delta m^{2}$ passes through this bound.
This corresponds to $\gamma=1$ or in other words if the dimension of
$\bar{q}q$ falls to two then there is an instability due to the generation of
a quark condensate and $\chi SB$ takes place. Note, that this criteria matches
that from gap equation analysis in ref. [20]. Therefore, one can make
holographic models of $\chi SB$ which are essentially Higgs-like theories with
the scalar field $L$ but whose potential $\Delta m^{2}$ changes with the RG
scale $r$. The potential is determined by the running of the anomalous
dimension $\gamma$ which in turn is determined by the gauge dynamics. Chiral
symmetry breaking occurs at the scale where $\gamma=1$.

Holography allows one to determine the low energy mesonic theory as follows.
For a given background vacuum, e.g. a solution for the field $L$, one allows
small fluctuations of the form $\delta(r)e^{ik\cdot x}$ with $k^{2}=-M^{2}$,
which describe fluctuations of the $\bar{q}q$ operator, for example the
$f_{0}$ state in QCD. Generically one obtains a Sturm-Louville system which
only has normalizable solutions for particular values of $M^{2}$. Other states
such as spin-1 states like the $\rho$ meson can be included via a gauge field
dual to $\bar{q}\gamma^{\mu}q$ and so forth. In this way one obtains the meson
spectrum. Substituting these solutions back into the action and integrating
over the radial direction $r$ yields a $1+3$ dimensional theory of the mesons
and their interaction couplings. In a similar way one can also include baryons
and their couplings with the mesons [13] which eventually also include the
Yukawa coupling of the top-quark [12, 13].

Last be not least, we note, that holography is a strong weak duality and so
the gravitational dual should only describe the infrared (IR) region of an
asymptotically free gauge theory with a large coupling. Models based on
gauge/gravity duality show some remnant of the N=4 super Yang-Mills theory
left in the description that enforces conformality in the UV. This matches the
weak coupling physics of these models and one might be tempted to let the dual
extend to the far UV. However, one should cut off the description, e.g. around
a few GeV in QCD, where the coupling becomes weak. Here, one can simply impose
a cut off in the bulk but then there is a matching problem: one should align
the dual to meet at QCD the intermediate coupling. This means the inclusion of
some higher dimension operators (HDO) at the UV boundary $\Lambda_{UV}$. This
can be implemented using Witten’s multi-trace prescription [21] as has been
detailed in ref. [13] to which refer for further details. The basic idea
behind this is, that one catches in this way the first corrections describing
the stringy nature of excited states. In ref. [13] the following operators
have been considered

${g^{2}_{S}\over\Lambda_{UV}^{2}}|\bar{q}q|^{2}\,,\quad{g^{2}_{V}\over\Lambda_{UV}^{2}}|\bar{q}\gamma^{\mu}q|^{2}\,,\quad{g^{2}_{A}\over\Lambda_{UV}^{2}}|\bar{q}\gamma^{\mu}\gamma_{5}q|^{2}\,,\quad{g^{2}_{\rm
B}\over\Lambda_{UV}^{5}}|qqq|^{2}\,,$ (3)

with $g_{i}$ being dimensionless couplings.

We first demonstrate the power of the gauge/gravity duality by applying it to
QCD as a test case. The corresponding model is described by the action [18,
13]

$\displaystyle S=\int
d^{4}xd\rho\,\text{Tr}\rho^{3}\left[\frac{1}{\rho^{2}+|X|^{2}}|DX|^{2}+\frac{\Delta^{2}m}{\rho^{2}}+\frac{1}{2g^{2}_{5}}(F_{V}^{2}+F_{A}^{2})\overline{\Psi}(\not{D}_{\text{AAdS}}-m)\Psi\right]$
(4)

where $X=Le^{2i\pi T}$ describes the quark condensate/$\sigma$ and pion
fields, $F_{V}$ the $\rho$ meson and $F_{A}$ the $a$ meson. $\Psi$ is a Dirac
field corresponding to the nucleon. The factors with $r^{2}=\rho^{2}+|X|^{2}$
of $X$ in the metric are deduced from top down (probe D7 brane) models [6] and
are the simplest ansatz to communicate the background quark condensate to the
fluctuation fields.

The starting point is the perturbative results for the running of $\gamma$.
Expanding $\Delta m^{2}=\gamma(\gamma-2)$ at small $\gamma$ gives

$\displaystyle\Delta
m^{2}=-2\gamma=-\frac{3({N_{c}}^{2}-1)}{2N_{c}\pi}\alpha\,$ (5)

with $\alpha$ being the QCD coupling constant. Here we allow ourselves to
extend the perturbative results as a function of renormalization RG scale
$\mu=r=\sqrt{\rho^{2}+L^{2}}$ to the non-perturbative regime. The resulting
running of $\Delta m$ in the Dynamic AdS/QCD model is shown in fig. 1 on the
left – the BF bound is violated close to the scale $r=\mu=1$.

Figure 1:  The $N_{c}=3,N_{f}=2$ QCD model: on the left we display the running
of the AdS scalar mass $\Delta m^{2}$ against log RG scale (we use
$\mu=\sqrt{\rho^{2}+L^{2}}$ in the holographic model). On the right we show
the the vacuum solution for $|X|=L(\rho)$ against $\rho$. The 45$\degree$ line
is where we apply the on mass shell infrared boundary condition. The $L(\rho)$
with a massless UV quark has $L_{IR}=0.43$. The quark masses from top to
bottom are 1, 0.75, 0.5, 0.25, 0.05, 0. Here units are set by
$\alpha(\rho=1)=0.65$. See ref. [13] for further details.

In table 1 we display our results for the spectrum. Note, that we use here and
below the following notation: we label the model as AdS/SU(3) to indicate the
gauge group and 2$F$ 2$\bar{F}$ to show that there are 2 Weyl fermions in the
fundamental and two in the anti-fundamental representation which is equivalent
to 2 Dirac fermions in the fundamental representation. The second column gives
the measured masses of the QCD bound states whereas the third one gives the
results for AdS/SU(3) setting all quark masses to zero. Moreover, we have used
the $\rho$-meson mass to set the scale. We see, that the ground states are
reasonably well described but the pion decay constant is somewhat low which is
partially caused by neglecting the UV quark mass(es). The $\sigma$ (S) mass is
high, but possibly should be compared to the f0(980) if the f0(500) is a pion
bound state. The proton mass is clearly too high. The first excited states are
clearly more off but this had to be expected as no string like structures are
taking into account. The situation clearly improves once HDOs are taken into
account as shown in the fourth column. However, this comes at the cost of
fewer predictions at this stage as we have fitted the coefficients of the HDOs
to data because a calculation from first principles is quite difficult.

| QCD | AdS/SU(3) |  | AdS/SU(3) | with
---|---|---|---|---|---
|  | 2$F$ 2$\bar{F}$ |  | 2 F 2 $\bar{F}$ | HDO
$M_{\rho}$ | 775 | 775∗ |  | 775∗ |
$M_{A}$ | 1230 | 1183 | \- 4% | 1230∗ | $g^{2}_{A}=5.76149$
$M_{S}$ | 500/990 | 973 | +64%/-2% | 597 | $+20\%/-40\%$
$M_{B}$ | 938 | 1451 | +43% | 938∗ | $g^{2}_{B}=25.1558$
$f_{\pi}$ | 93 | 55.6 | -50% | 93∗ | $g^{2}_{S}=4.58981$
$f_{\rho}$ | 345 | 321 | \- 7% | 345∗ | $g^{2}_{V}=4.64807$
$f_{A}$ | 433 | 368 | -16% | 444 | $+2.5\%$
$M_{\rho,n=1}$ | 1465 | 1678 | +14% | 1532 | $+4.5\%$
$M_{A,n=1}$ | 1655 | 1922 | +19% | 1789 | $+8\%$
$M_{S,n=1}$ | 990 /1200-1500 | 2009 | +64%/+35% | 1449 | $+46\%/0\%$
$M_{B,n=1}$ | 1440 | 2406 | +50% | 1529 | $+6\%$
Table 1:  Predictions for masses in MeV for $N_{f}=2$ SU(3) gauge theory. The columns show the QCD value [22], the holographic prediction for the theory with massless quarks and the improved holographic QCD theory with HDOs. The $\rho$-meson mass has been used to set the scale and other quantities that are used to fix the HDO couplings are indicated by *s. | QCD | AdS/SU(3) |
---|---|---|---
|  | 3 F 3 $\bar{F}$ |
$\rho(770)$, $\omega(782)$ | $775.26\pm 0.23$ | 775* |
$K^{*}(892)$ | $891.67\pm 0.26$ | 966 | 8%
$\phi(1020)$ | $1019.461\pm 0.016$ | 1120 | 9%
$a_{1}(1260)$, $f_{1}(1260)$ | $1230\pm 40$ | 1103 | 11%
$K_{1}(1400)$ | $1403\pm 7$ | 1432 | 2%
$f_{1}(1420)$ | $1426.3\pm 0.9$ | 1847 | 26%
$a_{0}(980)$, $f_{0}(980)$ | $980\pm 20$ | 930 | 5%
$K_{0}^{*}(700)$ | $845\pm 17$ | 987 | 16%
$f_{0}(1370)$ | 1370 | 1031 | 28%
$\pi^{0,\pm}$ | $139.57039\pm 0.00017$ | 128 | 9%
$K^{0,\pm}$ | $497.611\pm 0.013$ | 497 | O(0.1) %
Table 2:  Meson masses in MeV in the three flavour case compared with the
experimental data [22]. We have fixed the masses for the vector bosons $\rho$
and $\omega$ as indicated by the * and calculated the masses for the axial
vectors, the scalars and the pNGBs. The UV quark masses used are
$m_{u}=m_{d}=3.1$ MeV and $m_{s}=95.7$ MeV.

We note here for completeness, that strictly speaking these results do not
fully take into account the flavour structure of the theory, e.g. they do not
take into account for example mass splittings between different isospin
multiplets nor the fact that the quarks have different UV masses. This can be
taken into account by starting with a non-abelian Dirac-Born-Infeld action as
has been shown in ref. [23]. Table 2 gives a corresponding results from ref.
[23] assuming $m_{u}=m_{d}\neq m_{s}$ and one finds a reasonable agreement
between data and our model.

##  3 Example Sp(4) gauge theories

We focus here on Sp(4) gauge theories which have been proposed in refs. [14,
24, 17, 25] as candidates for CH models. Moreover, phenomenological aspects of
a particular example of such models, dubbed M5 in ref. [25], have been
recently worked out in a series of papers, see [26, 27, 28, 29, 30]. We will
briefly comment on the impact of our findings to phenomenological aspects in
the subsequent section. We discuss first a realisation that allows for a
comparison with lattice data. It contains 4 Weyl fermions in the fundamental
representation $F$. Thus, the global group containing eventually the
electroweak sector of the SM is SU(4) which gets broken to Sp(4) via the
condensate. One can introduce top partners [14] into the Sp(4) model by
including of six additional fermions in the sextet, two index anti-symmetric
representation $A_{2}$ of the gauge group. In the nomenclature of ref. [25]
this is model M8. The condensate breaks the SU(6) global group of this sector
to SO(6) which contains SU(3) as subgroup. The full symmetry breaking pattern
is given by

$\displaystyle\text{SU(4)}\times\text{SU(6)}\times\text{U(1)}$
$\displaystyle\to\underbrace{\text{Sp(4)}}_{\text{SU(2)}_{L}\times\text{U(1)}}\times\underbrace{\text{SO(6)}}_{\text{SU(3)}\times\text{U(1)}}\times\text{U(1)}$
(6)

where the U(1) factors give eventually the hypercharge.

For the holographic model one needs the running of the gauge coupling as
outlined in the previous section, see e.g. [13] for the corresponding
formulae. In this model one has two condensates $\langle FF\rangle$ and
$\langle A_{2}A_{2}\rangle$ and the relevant one-loop anomalous dimensions are

$\displaystyle\gamma_{F}=\frac{15}{8\pi}\alpha\quad\text{ and
}\quad\gamma_{A_{2}}=\frac{6}{2\pi}\alpha\,.$ (7)

We note, that one would expect the $A_{2}$ fermions to condense ahead of the
fundamental fields since the corresponding critical value for $\alpha$ is
smaller. Once the $A_{2}$s condense, SU(6) is broken to SO(6). At this point
the $A_{2}$s become massive but it is unclear how quickly they decouple from
the running of $\alpha$. This is still an open question and we refer to ref.
[13] for a first discussion.

We give in table 3 the spectrum for various approximations which are needed on
the one hand to estimate on the on hand the uncertainty due to the decoupling
of the $A_{2}$ fermions and on the other hand for comparison with lattice
data. The first column gives our results without approximation for the various
bound states: scalar $S$, axial spin-1 $A$, vector spin-1 $V$ and baryons $B$.
Mesonic bound states which are either $A_{2}$ or $F$ bound states carry a
corresponding index. There difference in the masses arises as the different
vacuum solutions of their respective condensate enter their equations of
motions [13]. The baryons are $FA_{2}F$ bound states. It is still an open
question how to precisely model such states within gauge/gravity duality. We
compute their masses using either the vacuum of the $F$ condensate or the one
of the $A_{2}$ condensate as indicated by the corresponding subscript.

For the results in the first/second column we do not/do decouple the $A_{2}$
fermions at the scale where the $A_{2}$ condense. This clearly does not affect
the masses of the $A_{2}$ mesons which is the reason for us to fix the overall
scale by the mass of the vector spin-1 state $VA_{2}$. The decoupling of the
$A_{2}$ implies an increase in the running of $\alpha$ which in turn leads to
the observed increase of the masses with $F$ index in the second column.
Independent of this uncertainty we see that the $A_{2}$ mesons are heavier
than the $F$ mesons. Note, that in particular the mass of the scalar $SV$
depends strongly on the decoupling behaviour of the $A_{2}$ fermions. In case
of the baryons, the variation gives a first measure of the uncertainty on the
underlying mass calculation. Last but not least we have added a NJL term to
enforce an equal scale of condensation and the corresponding results are given
in the last column of this table.

| AdS/$Sp(4)$ | AdS/Sp(4) | AdS/Sp(4) | lattice [15] | lattice [16] | AdS/Sp(4)
---|---|---|---|---|---|---
| no decouple | A2 decouple | quench | quench | unquench | \+ NJL
$M_{VA_{2}}$ | 1* | 1* | 1* | 1.000(32) |  | 1*
$M_{VF}$ | 0.61 | 0.814 | 0.962 | 0.83(19) | 0.83(27) | 1.03
$M_{AA_{2}}$ | 1.35 | 1.35 | 1.28 | 1.75 (13) |  | 1.35
$M_{AF}$ | 0.938 | 1.19 | 1.36 | 1.32(18) | 1.34(14) | 1.70
$M_{SA_{2}}$ | 0.375 | 0.375 | 1.14 | 1.65(15) |  | 0.375
$M_{SF}$ | 0.325 | 0.902 | 1.25 | 1.52 (11) | 1.40(19) | 0.375
$M_{BA_{2}}$ | 1.85 | 1.85 | 1.86 |  |  | 1.85
$M_{BF}$ | 1.13 | 1.53 | 1.79 |  |  | 1.88
Table 3:  AdS/$Sp(4)~{}4F~{}6A_{2}$. Ground state spectra for various
gauge/gravity models and comparison to lattice results. The subscript $A_{2}$
and $F$ indicate the quantity in each of the two different representation
sectors. Note here for the unquenched lattice results, which do not include
the $A_{2}$ fields, we have normalized the $F$ vector meson mass to that of
the quenched computation.

Lattice studies of this model have been made in ref. [15] in the quenched
approximation. Within gauge/gravity duality the quenching corresponds to
neglecting the hyperquark contributions to the running of the gauge coupling.
This implies a steeper running of the coupling resulting in a more compressed
spectrum as can be seen in the third column of table 3. In [16] the group
followed up that work of [15] by unquenching the fundamental hyperquark sector
using Wilson fermions. We show the results of both lattice studies in column 4
and 5 of this table for direct comparison to the holographic results. The
quenched results from both, the lattice and the holographic model, show
considerable correlation. This provides confidence that trends as the fields
are unquenched may be trustworthy. Consequently we would expect that if the
$A_{2}$ fermions were included as unquenched fields, the $F$ sector would
decrease in mass by about 20-40$\%$. Moreover, we also expect the scalar meson
masses to be considerably lower than predicted by the quenched lattice
computation.

A huge advantage of calculations in the gauge/gravity framework is, that one
can easily incorporate changes in the number of the underlying fermions which
is much more difficult in case of lattice calculations. We take as an example
the M5 model of ref. [25] which contains 5 $A_{2}$ and 6 $F$ corresponding to
the following breaking pattern of the global group

$\displaystyle\text{SU(5)}\times\text{SU(6)}\times\text{U(1)}$
$\displaystyle\to\underbrace{\text{SO(5)}}_{\text{SU(2)}_{L}\times\text{U(1)}}\times\underbrace{\text{Sp(6)}}_{\text{SU(3)}\times\text{U(1)}}\times\text{U(1)}$
(8)

where again the U(1) factors give eventually the hypercharge.

| $M_{VA_{2}}$ | $M_{VF}$ | $M_{AA_{2}}$ | $M_{AF}$ | $M_{SA_{2}}$ | $M_{SF}$ | $M_{BA_{2}}$ | $M_{BF}$
---|---|---|---|---|---|---|---|---
$4F~{}6A_{2}$ | 1* | 0.618 | 1.4 | 0.862 | 0.376 | 0.348 | 1.85 | 1.15
$5A_{2}~{}6F$ | 1* | 0.61 | 1.35 | 0.938 | 0.38 | 0.33 | 1.85 | 1.13
Table 4:  For the masses in Sp(4) theories with two different matter
representations that can trigger chiral symmetry breaking: $4F~{}6A_{2}$ (M8
model) and $5A_{2}~{}6F$ (M5 model).

The resulting masses are given in table 4. As one would naively expect, the
results are quite similar as we have changed the fermion content only
slightly. However, note that the embedding of the SM in the corresponding
global groups differ in both models and, thus, consequently the
phenomenological implications. We take as an example the spin-1 vectors
states: in case of $4F~{}6A_{2}$ ($5A_{2}~{}6F$) the ones carrying electroweak
quantum numbers will be significantly lighter (heavier) compared to the ones
carrying SU(3) quantum numbers.

##  4 Phenomenological aspects

We would like to emphasis that the implications of our results for
phenomenological investigations discussed below have to be taken with care.
The results presented here so far take are based originally on an abelian
Born-Dirac-Infeld action and, thus, they do not take into account possible
multiplet structures within a given type of bound states. Taking for example
the M5 model, there is actually not one spin-1/2 state but several of them
[26] which transform as follows under the SM-gauge group
SU(3)$\times$SU(2)${}_{L}\times$U(1)Y: $(3,2,\pm 7/6)$, $(3,2,\pm 1/6)$,
$(3,1,\pm 2/3)$, $(8,2,\pm 1/2)$, $(8,1,0)$, $(1,2,\pm 1/2)$ and $(1,1,0)$.
The colour triplets are the usual top-partners. For a proper calculation of
the corresponding masses one has to a consider a non-abelian Born-Dirac-Infeld
action as starting which complicates the calculation quite a bit. However, we
have recently started this for some simple cases, SU(3)$\times$SU(3)$/$SU(3)
[23] and SU(4)$/$Sp(4) [31]. While these are not applicable for the M5 model
directly, one can nevertheless infer some requirements needed for a successful
description.

In [26] it has been shown that the M5 model contains an accidental ‘baryon
number’ implying that either one of the baryons or the coloured pNGB
transforming as $(3,1,2/3)$, called $\pi_{3}$ in [26], has to be stable.
Obviously this should be the $(1,1,0)$ baryon, called $\tilde{B}$ in [26],
which could then serve as a DM candidate. It might surprise at first glance
that a baryon should be lighter than a pNGB. Such a behaviour can in principle
occur if the underlying hyperquarks have mass terms which break the global
groups explicitly as has been shown in [13, 31]. In such a scenario $\pi_{3}$
would decay into $\tilde{B}t$ like the right-handed scalar top in
supersymmetric models if there is sufficient phase space. In case of smaller
mass differences, three-body decays via an off-shell top-quark would become
important, again similar to supersymmetric models [32, 33, 34]. Actually, this
scenario could easily be confused with supersymmetric models at first glance
as several signatures are very similar.

Assuming, that the results in table 4 still give the same rough patterns, the
coloured baryons would be significantly heavier than the coloured spin-1
resonances represented by $VF$ and $AF$, which actually represents different
SU(3) states including $V_{8}^{0}$ and $A_{3}^{-2/3}$ [30]. Here the
subscripts give the SU(3) representation and the superscripts the electric
charge. These would yield completely new decay channels of the baryons such as

$\displaystyle\tilde{g}$ $\displaystyle\to
A_{6}^{-2/3}t\to\pi_{8}\pi_{8}\pi^{*}_{3}t\to 4t+2\bar{t}+\tilde{B}$ (9)
$\displaystyle T_{R}$ $\displaystyle\to V_{8}^{0}t\to\pi_{3}\pi^{*}_{3}t\to
2t+\bar{t}+2\tilde{B}\,.$ (10)

which to our knowledge have not yet been considered in the literature up to
now. We denoted here the $(8,1,0)$ and $(3,1,2/3)$ baryons by $\tilde{g}$ and
$T_{R}$, respectively. Moreover, $\pi_{8}$ are strongly interacting pNGBs
decaying dominantly into $t\bar{t}$. Note also, that $\tilde{B}$ would lead to
missing transverse momentum in the signal. The model M5 contains also an
extended pNGB sector in the electroweak sector containing for example a doubly
charged scalar $S^{++}$ which has similar quantum numbers as the one from type
II seesaw models [35, 36] and thus the same production cross section at the
LHC [28]. However, it has a quite different dominant decay mode [28], namely

$\displaystyle S^{++}\to t\bar{b}W^{+}\,.$ (11)

The resulting increased hadronic activity implies that bounds on its mass are
quite weak [28]. This scalar pNGB can also be produced in the decays of top-
partners. For example, is leads to an additional decay channel for the
$X_{5/3}$ baryon from the $(3,2,7/6)$ multiplet:

$\displaystyle X_{5/3}$ $\displaystyle\to W^{+}t$ (standard channel) (12)
$\displaystyle X_{5/3}$ $\displaystyle\to S^{++}b\to tb\bar{b}W^{+}$ (new
channel). (13)

The importance of the new channel depends of course what is the mass of the
$S^{++}$ compared to the baryons. We are investigating currently how important
this and related decay modes are in scenarios in which the required mass
splitting of the baryons can be achieved by giving different masses to the
underlying fermions. We note for completeness, that exotic signatures of top-
partners haven been discussed in [37, 38, 27, 39].

##  5 Conclusions and outlook

We have shown in this contribution that gauge/gravity duality methods are a
powerful tool for obtaining sensible estimates for masses of the bound states,
both, in QCD as well as in Composite Higgs models. We have seen that
reasonable agreement with data in case of QCD and with lattice data in case of
a specific Composite Higgs model are obtained. The advantage of this framework
is, that it allows for a relatively fast computations of the spectrum, more
precisely ratio of masses. Most of the results have been obtained neglecting
the fact the bound states form multiplets of the unbroken global sub-group
which can have somewhat different masses. We have started recently to take
this into account, see ref. [31], for the case of SU(4)/Sp(4). This will be
extended in different directions in the future: (i) we will take into account
fermionic bound states. (ii) We will systematically explore the models
proposed in [25]. (iii) We will study the impact of our results on the
phenomenology of these models in a coherent global picture.

## Acknowledgments

I thank A. Banerjee, G. Cacciapaglia, J. Erdmenger, N. Evans, G. Ferretti, Y.
Liu, T. Flacke, J. H. Kim, M. Kunkel, P. Ko, K. S. Rigatos and R. Ströhmer for
interesting and fruitful discussions. I also would like to thank the
organizers of the Corfu workshops for creating such a stimulating environment.
This work has been supported by DFG projects nr PO/1337-11/1 and nr
PO/1337-12/1.

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# Concerning Keisler Measures over ultraproducts

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

As consequence of the VC theorem, any pseudo-finite measure over an NIP
ultraproduct is generically stable. We demonstrate a converse of this theorem
and prove that any finitely approximable measure over an ultraproduct is
itself pseudo-finite (even without the NIP assumption). We also analyze the
connection between the Morley product and the pseudo-finite product. In
particular, we show that if $\mu$ is definable and both $\mu$ and $\nu$ are
pseudo-finite, then the Morley product of $\mu$ and $\nu$ agrees with the
pseudo-finite product of $\mu$ and $\nu$. Using this observation, we construct
generically stable idempotent measures on pseudo-finite NIP groups.

##  1\. Introduction

First, we remind the reader that Keisler measures over ultraproducts have been
studied by many people in many contexts (e.g. [1, 2, 5, 8, 9, 10, 13, 15, 16,
18, 19]).

In the NIP setting, there are several methods for constructing generically
stable measures. One such method is via ultralimits. The following theorem is
a consequence of the VC theorem (e.g. [17, Example 7.32] or [18, Corollary
1.3] for similar statements):

######  Theorem 1.1.

Let $(M_{i})_{i\in I}$ be an indexed family of $\mathcal{L}$-structures, $D$
be an ultrafilter on $I$, and $\mathcal{M}:=\prod_{D}M_{i}$ the be the
ultraproduct. Suppose that $\mathcal{M}$ is NIP. For each $i\in I$, let
$\mu_{i}$ be a Keisler measure on $M_{i}$ which concentrates on finitely many
realized types. If $\mu:=\lim_{D}\mu_{i}$ is the ultralimit (a Keisler measure
on $\mathcal{M}$) then $\mu$ is finitely approximated, i.e. for every formula
$\varphi(x,y)\in\mathcal{L}$ and $\epsilon>0$, there exists
$\overline{\mathbf{a}}:=\mathbf{a}_{1},...,\mathbf{a}_{n}$ in
$\mathcal{M}^{x}$ such that

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

In other words, suppose that $\mathcal{M}$ is NIP and
$\mu\in\mathfrak{M}_{x}(\mathcal{M})$. If $\mu$ is pseudo-finite then $\mu$ is
finitely approximated.

The first goal of this paper is to prove a converse to Theorem 1.1 (answering
a question of Chernikov and Starchenko). We show that if $\mathcal{L}$ is
countable and our ultrafilter $D$ is countably incomplete then any finitely
approximated measure over our ultraproduct is actually pseudo-finite (Theorem
3.5). We remark that the second assumption, i.e. the ultrafilter being
countably incomplete, is quite tame since one needs to assume the existence of
inaccessible cardinals to have a countably complete ultrafilter. We also
remark that this result does not require an NIP assumption.

Secondly, we write down a bit of folklore. When working over an ultraproduct
$\mathcal{M}$, there are two notions of product for measures. There is the
Morley product (denoted $\otimes$) and the pseudo-finite product (denoted
$\boxtimes$). We remark that one runs into an immediate problem when trying to
compare these two products since these operations are often defined over
different collections of measures. The pseudo-finite product is defined on
pairs of pseudo-finite measures [_duh_] while the Morley product is usually
defined on pairs of global Keisler measures which are Borel-definable over a
small submodel. However, if one adds a definability assumption, then one can
recover a version of the Morley product over arbitrary models. To be explicit,
we prove that if $\mu$ and $\nu$ are pseudo-finite measures over an
ultraproduct and $\mu$ is definable, then the Morley product and pseudo-finite
product agree (Theorem 4.8). As a consequence, one can show that if $\nu$ is
also definable, then $\mu\otimes\nu=\nu\otimes\mu$ (Corollary 4.9). We then
apply local versions of these results to demonstrate that certain measures are
not pseudo-finite over ultraproducts of the Paley graphs (Theorem 4.14).
Finally, we return to the classical Keisler measure setting (measures over a
monster model) and use Theorems 1.1 and 4.8 to build generically stable
idempotent measures in the context of pseudo-finite NIP groups (Theorem 5.5).

We begin with a preliminaries section. The rest of the paper follows the
outline presented in the preceding paragraphs.

### Acknowledgements

We thank Artem Chernikov for both specific and general comments as well as
discussion.

##  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 $|r-s|<\epsilon$ and
$r\approxeq_{\epsilon}s$ to mean $|r-s|\leq\epsilon$.

For the most part, our notation is standard. The letter $\mathcal{L}$ denotes
a first order language. The letters $x,y,z$ denote finite tuples of variables.
We use the letter $M$ to denote an arbitrary $\mathcal{L}$-structure. If
$A\subseteq M$, then 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”. An
$\mathcal{L}$-formula is a $\mathcal{L}(\emptyset)$-formula. We use
$\mathcal{L}_{x}(A)$ to denote the $\mathcal{L}(A)$-formulas with free
variable(s) $x$. We write partitioned formulas as $\varphi(x,y)$ with variable
variables $x$ and parameter variables $y$. We let $\varphi^{*}(y,x)$ denote
the exact same formula as $\varphi(x,y)$ but with the variables and parameters
swapped.

Unlike similar papers about Keisler measures, we do not identify a type and
its corresponding Keisler measure. Let $A$ be a subset of a model $M$. We let
$S_{x}(A)$ denote the space of types over $A$ (in variable(s) $x$). We let
$\mathfrak{M}_{x}(A)$ denote the space of Keisler measures over $A$ (in
variable(s) $x$). For any (tuple of) variable(s) $x$, and any subset
$A\subseteq M$, we have a map $\delta:S_{x}(A)\to\mathfrak{M}_{x}(M)$ 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 $M^{x}$,
then we let $\operatorname{Av}(\overline{a})$ be the associated average
measure in $\mathfrak{M}_{x}(M)$. Explicitly, for any
$\psi(x)\in\mathcal{L}_{x}(M)$, we define

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

Moreover we let

$\operatorname{conv}_{x}(M):=\Big{\\{}\sum_{i=1}^{n}r_{i}\delta_{\operatorname{tp}(a_{i}/M)}:a_{i}\in
M^{x},\sum_{i=1}^{n}r_{i}=1,r_{i}>0\Big{\\}}\subseteq\mathfrak{M}_{x}(M).$

###  2.1. Ultraproducts, ultralimits and pseudo-finite measures

We first discuss our conventions on ultraproducts. Let $I$ be an (infinite)
indexing set, $(M_{i})_{i\in I}$ be an indexed family of
$\mathcal{L}$-structures, and $D$ be an ultrafilter on $I$. $D$ will always
denote a non-principal ultrafilter. We denote the ultraproduct of
$(M_{i})_{i\in I}$ relative to $D$ as $\prod_{D}M_{i}$. In practice, we will
usually write ultraproducts as $\mathcal{M}$ when the indexing set, sequence
of models, and ultrafilter are unambiguous. We will write elements of
$\mathcal{M}$ as $\mathbf{a},\mathbf{b},\mathbf{c}$, etc. If $(c_{i})_{i\in
I}$ is an indexed family of elements such that $c_{i}\in M_{i}$, then we let
$[(c_{i})]_{D}$ denote the corresponding element in $\mathcal{M}$, i.e. the
equivalence class of $(c_{i})_{i\in I}$ modulo $D$. If
$\mathbf{b}\in\mathcal{M}$, then we write $(\mathbf{b}(i))_{i\in I}$ or simply
$\mathbf{b}(i)$ to denote a (choice of) sequence such that each
$\mathbf{b}(i)\in M_{i}$ and $[\mathbf{b}(i)]_{D}=\mathbf{b}$.

We recall the definition of an ultralimit. Since this paper is primarily
concerned with finitely additive measures, we will restrict our definition to
the case of ultralimits of indexed families of real numbers.

######  Definition 2.1.

Let $(r_{i})_{i\in I}$ be a bounded indexed family of real numbers and let $D$
be an ultrafilter on $I$. Then the ultralimit of $(r_{i})_{i\in I}$ (with
respect to $D$) is the unique real number $r$ such that for every
$\epsilon>0$, $\\{i\in I:|r_{i}-r|\leq\epsilon\\}\in D$. We denote this
ultralimit as $\lim_{D}r_{i}$.

We remind the reader that ultralimits of bounded indexed families of real
numbers always exist. The following facts are elementary.

######  Fact 2.2.

Let $r_{1},...,r_{n}$ be real numbers, let $(s_{i}^{1})_{i\in
I},...,(s_{i}^{n})_{i\in I}$ be sequence of real numbers, and $D$ be an
ultrafilter on $I$. Then the following hold:

  1. (1)

$|\lim_{D}s^{1}_{i}|=\lim_{D}|s^{1}_{i}|$.

  2. (2)

$\sum_{j=1}^{n}r_{i}\lim_{D}s_{i}^{j}=\lim_{D}\sum_{j=1}^{n}r_{i}\cdot
s_{i}^{j}.$

  3. (3)

If there exists $A\in D$ such that for any $i\in A$, $s_{i}^{1}\geq r_{1}$,
then $\lim_{D}s_{i}^{1}\geq r_{1}$.

  4. (4)

If there exists $A\in D$ such that for any $i\in A$, $s_{i}^{1}\leq r_{1}$,
then $\lim_{D}s_{i}^{1}\leq r_{1}$.

  5. (5)

If there exists $A\in D$ such that for any $i\in A$,
$|s_{i}^{1}-s_{i}^{2}|\leq\epsilon$, then

$\lim_{D}s_{i}^{1}\approxeq_{\epsilon}\lim_{D}s_{i}^{2}.$

We now recall the definition of an ultralimit of a family of measures as well
as the definition of a pseudo-finite measure/type. We remark that the
ultralimit can be defined on the sigma-algebra generated by internal subsets
of the ultraproduct (Loeb measure construction). However, since we only care
about Keisler measures, we restrict our discussion to definable subsets of the
ultraproduct.

######  Definition 2.3.

Let $(M_{i})_{i\in I}$ be a family of $\mathcal{L}$-structures and $D$ be an
ultrafilter on $I$. Let $\mathcal{M}:=\prod_{D}M_{i}$ and for each $i$, we let
$\mu_{i}\in\mathfrak{M}_{x}(M_{i})$. The ultralimit of $(\mu_{i})_{i\in I}$
(relative to $D$) is the unique measure $\mu\in\mathfrak{M}_{x}(\mathcal{M})$
such that for every formula
$\varphi(x,\mathbf{b})\in\mathcal{L}_{x}(\mathcal{M})$ we have that

$\mu(\varphi(x,\mathbf{b}))=\lim_{D}\mu_{i}(\varphi(x,\mathbf{b}(i))).$

Moreover we say that $\mu$ is pseudo-finite (with respect to the indexed
family $(M_{i})_{i\in I}$ and ultrafilter $D$) if there exists an indexed
family of measures $(\mu_{i})_{i\in I}$ such that
$\mu_{i}\in\operatorname{conv}_{x}(M_{i})$ and $\lim_{D}\mu_{i}=\mu$. Finally,
we say that the type $p$ is pseudo-finite if the corresponding Keisler measure
$\delta_{p}$ is pseudo-finite.

The following fact is straightforward and left to the reader as an exercise.

######  Fact 2.4.

The ultralimit $\lim_{D}\mu_{i}$ in the above definition is well-defined.

###  2.2. Working over models.

A lot of literature dealing with Keisler measures focuses on global measures,
i.e. those which are defined over a monster model. On the other hand this
paper is concerned with ultraproducts which are generally not monster models.
Therefore we need to set some conventions.

######  Definition 2.5.

Let $M$ be any model of $T$, $p\in S_{x}(M)$ and $\mu\in\mathfrak{M}_{x}(M)$.

  1. (1)

We say that $p$ is $M$-definable if for every $\mathcal{L}$-formula
$\varphi(x,y)$, there exists a formula
$d^{\theta}_{p}(y)\in\mathcal{L}_{y}(\mathcal{M})$ such that for any $b\in M$,

$\theta(x,b)\in p\iff M\models d^{\theta}_{p}(b).$

When $M$ is obvious, we sometimes simply write that $p$ is definable.

  2. (2)

We say that $\mu$ is $M$-definable if for every $\mathcal{L}$-formula
$\varphi(x,y)$, there exists a continuous function
$F^{\varphi}_{\mu,M}:S_{y}(M)\to[0,1]$ such that for every $b\in M^{y}$,
$F^{\varphi}_{\mu,M}(\operatorname{tp}(b/M^{y}))=\mu(\varphi(x,b))$. We remark
that if such a continuous function exists, then it is the unique continuous
function with this property. When $M$ is obvious, we sometimes simply write
that $\mu$ is definable.

  3. (3)

We say that $\mu$ is finitely approximated if for every partitioned
$\mathcal{L}$-formula $\varphi(x,y)$ and every $\epsilon>0$, there exists a
sequence $a_{1},...,{a}_{n}\in M^{x}$ such that

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

  4. (4)

We say that $p$ is finitely approximated if the corresponding Keisler measure
$\delta_{p}$ is finitely approximated.

One might (and should) wonder why we are restricting ourselves to
$M$-definable measures and not working with “$M$-Borel-definable" measures.
This is because being Borel-definable does not have such a good “over-a-model"
analogue. We discuss why in the warning below.

######  Warning 2.6.

One may be tempted to define Borel-definable over a model in a similar way to
definability. A plausible definition might be the following: We say that a
measure $\mu\in\mathfrak{M}_{x}(M)$ is M-Borel-definable∗ if for every formula
$\varphi(x,y)\in\mathcal{L}(M)$, there exists a unique Borel function
$f_{\mu}^{\varphi}:S_{y}(M)\to[0,1]$ such that for any $b\in M^{y}$,
$f_{\mu}^{\varphi}(\operatorname{tp}(b/M))=\mu(\varphi(x,b))$. The reason why
we require uniqueness is so that we can construct a unique Morley product with
any measure in $\mathfrak{M}_{y}(M)$ (see Definition 4.1).

However, a measure $\mu$ can have _multiple Borel-definable extensions_. In
particular, there exists a model $M$ and a measure $\mu\in\mathfrak{M}_{x}(M)$
such that $\mu$ is $M$-definable but not $M$-Borel-definable∗. Consider
$M:=(\mathbb{R},<)$ and $p$ is the unique complete type extending
$\\{x>a:a\in\mathbb{R}\\}$. The measure $\mu=\delta_{p}$ is
$\mathbb{R}$-definable but not $\mathbb{R}$-Borel-definable∗ essentially
because both the global heir and global coheir are Borel-definable extensions
of $p$ and so uniqueness fails.

The following facts are straightforward and left to the reader as an exercise.

######  Fact 2.7.

Let $M$ be any model of $T$, $p\in S_{x}(M)$, $\mu\in\mathfrak{M}_{x}(M)$, and
$M\prec\mathcal{U}$.

  1. (1)

The type $p$ is definable if and only if the measure $\delta_{p}$ is
definable.

  2. (2)

If $\mu$ is finitely approximated, then $\mu$ is definable.

  3. (3)

If $p$ is finitely approximated, then $p$ is definable.

######  Proposition 2.8.

Suppose that $\mu$ is finitely approximated. Then for any finite collection of
$\mathcal{L}$-formulas $\theta_{1}(x,y_{1}),...,\theta_{n}(x,y_{n})$, there
exists $\overline{a}:=a_{1},...,a_{n}\in M^{x}$ such that

$\sup_{c\in
M^{y_{i}}}|\mu(\theta(x,c))-\operatorname{Av}(\overline{a})(\theta_{i}(x,c))|<\frac{1}{n}.$

###### Proof.

Follows from a standard encoding argument. Consider the new variables
$z_{*},z_{1},...,z_{n}$ where $|z_{*}|=|z_{i}|=1$. Now define the formula

$\gamma(x,y_{0},y_{1},...,y_{n},z_{*},z_{1},...,z_{n}):=\bigwedge_{i\leq
n}\left(\left(z_{*}=z_{i}\wedge\bigwedge_{j\leq n;j\neq i}^{n}z_{j}\neq
z_{*}\right)\to\theta_{i}(x,y_{i})\right).$

By finite approximability, there exists $\overline{a}\in(M^{x})^{<\omega}$
such that for any $\bar{b}\in\mathcal{U}^{\bar{y}}$,

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

We claim $\operatorname{Av}(\overline{a})$ satisfies the condition. ∎

##  3\. Finitely Approximated implies pseudo-finite

The purpose of this section is to prove a (partial) converse to Theorem 1.1.
We will show that if a measure $\mu$ on an ultraproduct $\mathcal{M}$ is
finitely approximated, then $\mu$ is pseudo-finite (modulo some minimal
assumptions). Our result is partial since we require our language to be
countable and our ultrafilter to be countably incomplete. However this result
does not require an NIP assumption. For clarity, we first provide a version of
the proof for types over a countable index set before proving the general case
for measures. The measure case is structurally the same as the type case,
however we feel that the type case is pedagogically helpful.

######  Theorem 3.1.

Let $\mathcal{L}$ be a countable language, $I=\mathbb{N}$, $D$ be a non-
principal ultrafilter on $I$, and $(M_{i})_{i\in I}$ be an indexed family of
$\mathcal{L}$-structures. Let $\mathcal{M}=\prod_{D}M_{i}$ and $p\in
S_{x}(\mathcal{M})$. If $\delta_{p}$ is finitely approximated then
$\delta_{p}$ is pseudo-finite.

###### Proof.

Let $(\varphi_{i}(x,y_{i}))_{i\in\mathbb{N}}$ be an enumeration of partitioned
$\mathcal{L}$-formulas. Since $\delta_{p}$ is finitely approximated,
$\delta_{p}$ is definable (by Proposition 2.7). Hence for each partitioned
$\mathcal{L}$-formula $\varphi_{i}(x,y_{i})$, there exists an
$\mathcal{L}(\mathcal{M})$-formula $d_{p}^{\varphi_{i}}(y_{i},\mathbf{c}_{i})$
such that for any $\mathbf{b}\in\mathcal{M}^{|y_{i}|}$, $\mathcal{M}\models
d_{p}^{\varphi_{i}}(\mathbf{b},\mathbf{c}_{i})$ if and only if
$\varphi_{i}(x,\mathbf{b})\in p$. Since $\delta_{p}$ is finitely approximated,
it follows from Proposition 2.8 that for every $n\in\mathbb{N}$, there exists
some $\overline{\mathbf{a}}\in(\mathcal{M})^{<\omega}$ such that for every
$i\leq n$,

$\sup_{\mathbf{b}\in\mathcal{M}^{|y_{i}|}}|\delta_{p}(\varphi_{i}(x,\mathbf{b}))-\operatorname{Av}(\overline{\mathbf{a}})(\varphi_{i}(x,\mathbf{b}))|<\frac{1}{n}.$

Therefore for each $n\in\mathbb{N}$ there exists some number $r(n)$ such that

$\displaystyle\mathcal{M}\models\exists x_{1},...,x_{r(n)}$
$\displaystyle\bigwedge_{i\leq n}\Big{(}\forall y_{i}\Big{(}$ $\displaystyle
d_{p}^{\varphi_{i}}(y_{i},\mathbf{c}_{i})\leftrightarrow\bigvee_{A\in\Gamma(n)}\Big{(}\bigwedge_{s\in
A}\varphi_{i}(x_{s},y_{i})\wedge\bigwedge_{t\not\in
A}\neg\varphi_{i}(x_{t},y_{i})\Big{)}\Big{)}\Big{)},$

where
$\Gamma(n)=\\{A\subset\\{1,...,r(n)\\}:\frac{|A|}{r(n)}>1-\frac{1}{n}\\}$.
This sentence states that there exists $r(n)$ elements in $\mathcal{M}$ whose
average uniformly approximates the measure $\delta_{p}$ on the formulas
$\varphi_{1}(x,y_{1}),...,\varphi_{n}(x,y_{n})$ up to an error of
$\frac{1}{n}$. For each $i\in\mathbb{N}$ choose a sequence
$(c_{i_{j}})_{j\in\mathbb{N}}$ such that $[(c_{i_{j}})]_{D}=\mathbf{c}_{i}$.
By Łoś’s theorem, it follow that

$\displaystyle E_{n}=\Big{\\{}k\in$
$\displaystyle\mathbb{N}:M_{k}\models\exists x_{1},...,x_{r(n)}$
$\displaystyle\bigwedge_{i\leq n}$ $\displaystyle\Big{(}\forall
y_{i}\Big{(}d_{p}^{\varphi_{i}}(y_{i},c_{i_{k}})\leftrightarrow\bigvee_{A\in\Gamma(n)}\Big{(}\bigwedge_{s\in
A}\varphi_{i}(x_{s},y_{i})\wedge\bigwedge_{t\not\in
A}\neg\varphi_{i}(x_{t},y_{i})\Big{)}\Big{)}\Big{)}\Big{\\}}\in D.$

Let $X_{n}=\\{i\in\mathbb{N}:i\geq n\\}$ and let $Y_{n}=X_{n}\cap E_{n}$.
Notice that $\bigcap Y_{n}=\emptyset$ and $Y_{n}\in D$ for each
$n\in\mathbb{N}$. We now construct a sequence of Keisler measures
$(\mu_{j})_{j\in\mathbb{N}}$ such that each $\mu_{j}$ is in
$\operatorname{conv}_{x}(M_{j})$. We then argue that
$\lim_{D}\mu_{j}=\delta_{p}$. Consider the function
$g:\mathbb{N}\to\mathbb{N}$ where

$g(j)=\begin{cases}\begin{array}[]{cc}\max\\{n\in\mathbb{N}:j\in
Y_{n}\\}&\text{if exists,}\\\ 0&\text{otherwise.}\end{array}\end{cases}$

If $g(j)=0$, set $\mu_{j}$ equal to any measure in
$\operatorname{conv}_{x}(M_{j})$. If $g(j)=m>0$, then

$\displaystyle M_{j}\models\exists x_{1},...,x_{r(m)}$
$\displaystyle\bigwedge_{i\leq m}\Big{(}\forall y_{i}$
$\displaystyle\Big{(}d_{p}^{\varphi_{i}}(y_{i},c_{i_{j}})\leftrightarrow\bigvee_{A\in\Gamma(m)}\Big{(}\bigwedge_{s\in
A}\varphi_{i}(x_{s},y_{i})\wedge\bigwedge_{t\not\in
A}\neg\varphi_{i}(x_{t},y_{i})\Big{)}\Big{)}.$

Choose $\overline{a}_{j}=(a_{1}^{j},...,a_{r(n)}^{j})$ in $M_{j}^{r(n)}$ which
satisfy the formula above (after removing the existential quantifiers $\exists
x_{1},...,x_{r(m)}$). Let $\mu_{j}=\operatorname{Av}(\overline{a}_{j})$.

We now demonstrate that the ultralimit of $(\mu_{j})_{j\in\mathbb{N}}$ is
equal to $\delta_{p}$. Fix
$\theta(x,\mathbf{e})\in\mathcal{L}_{x}(\mathcal{M})$. Without loss of
generality, assume that $\delta_{p}(\theta(x,\mathbf{e}))=1$. Fix $\epsilon>0$
and let $(e_{i})_{i\in\mathbb{N}}$ be a sequence such that
$[(e_{i})]_{D}=\mathbf{e}$. By Fact 2.2, it suffices to show that
$K=\\{j\in\mathbb{N}:\mu_{j}(\theta(x,e_{j}))\geq 1-\epsilon\\}\in D$. Notice
that $\theta(x,y)$ appears in our index, say
$\theta(x,y)=\varphi_{m}(x,y_{m})$. Since $\delta_{p}(\theta(x,\mathbf{e}))=1$
we have that $\mathcal{M}\models
d_{p}^{\varphi_{m}}(\mathbf{e},\mathbf{c}_{m})$ and so the set
$Z=\\{j\in\mathbb{N}:M_{j}\models d_{p}^{\varphi_{m}}(e_{j},c_{m_{j}})\\}$ is
in $D$. Let $m_{*}=\max\\{m,\left\lceil\frac{1}{\epsilon}+1\right\rceil\\}$.
By construction $Y_{m_{*}}\cap Z\in D$. We argue that $Y_{m_{*}}\cap
Z\subseteq K$ and so $K$ is in $D$ (which will complete our proof).

Let $j\in Y_{m_{*}}\cap Z$. Since $j\in Y_{m_{*}}$, it follows that

$M_{j}\models
d_{p}^{\varphi_{m}}(e_{j},c_{m_{j}})\leftrightarrow\bigvee_{A\in\Gamma(m_{*})}\Big{(}\bigwedge_{s\in
A}\varphi_{m}(a^{j}_{s},e_{j})\wedge\bigwedge_{t\not\in
A}\neg\varphi_{m}(a_{t}^{j},e_{j})\Big{)}.$

Since $j\in Z$, it follows that $M_{j}\models
d_{p}^{\varphi_{m}}(e_{j},c_{m_{j}})$. So, $M_{j}\models\bigwedge_{s\in
A_{*}}\varphi_{m}(a^{j}_{s},e_{j})$ for some $A_{*}\in\Gamma(m_{*})$. Hence

$\mu_{j}(\theta(x,e_{j}))=\frac{|\\{s\leq
r(m_{*}):M_{j}\models\varphi(a_{s}^{j},e_{j})\\}|}{r(m_{*})}\geq\frac{|A_{*}|}{r(m_{*})}>1-\frac{1}{m_{*}}>1-\epsilon.$

Thus $Y_{m_{*}}\cap Z\subset K$ and so $K\in D$. ∎

######  Remark 3.2.

In the proof above, we used $I=\mathbb{N}$. However, we only need our
ultrafilter to be countably incomplete for the result to go through. We will
see this in the general version of the proof.

The following fact is stated without proof for global definable measures in
[3, Fact 2.3]. We provide a short proof for completeness.

######  Proposition 3.3.

Let $\mu\in\mathfrak{M}_{x}(M)$ and suppose that $\mu$ is $M$-definable. Then
for every natural number $n$ (say greater than 4) and formula
$\varphi(x,y)\in\mathcal{L}$, there exists definable sets
$d^{\varphi}_{0,n}(y),...,d^{\varphi}_{n,n}(y)$ in $\mathcal{L}_{y}(M)$ such
that

  1. (1)

The collection $\\{d^{\varphi}_{i,n}(y)\\}_{i=0}^{n}$ cover $M^{y}$, i.e.
$M\models\forall y\left(\bigvee_{i=0}^{n}d_{i,n}^{\varphi}(y)\right)$.

  2. (2)

If $M\models d^{\varphi}_{i,n}(b)$, then
$|\mu(\varphi(x,b))-\frac{i}{n}|<\frac{1}{n}$.

  3. (3)

If $|\mu(\varphi(x,b))-\frac{k}{n}|<\frac{1}{n}$, then
$M\models\bigvee_{j=k-1}^{k+1}d_{j,n}^{\varphi}(b)$.

###### Proof.

Since $\mu$ is definable over $M$, the map
$F_{\mu,M}^{\varphi}:S_{y}(M)\to[0,1]$ is continuous. Consider the sets
$I_{n}=\\{0,\frac{1}{n},\frac{2}{n},...,\frac{n-1}{n},n\\}$. Let
$A_{0}=\left(F_{\mu}^{\varphi}\right)^{-1}([0,\frac{1}{n}))$,
$A_{n}=\left(F_{\mu}^{\varphi}\right)^{-1}((\frac{n-1}{n},1])$, and for
$0<i<n$, let
$A_{i}=\left(F_{\mu}^{\varphi}\right)^{-1}((\frac{i-1}{n},\frac{i+1}{n}))$.
Since $F_{\mu}^{\varphi}$ is continuous, we have that for each $0\leq i\leq
n$, $A_{i}=\bigvee_{j\in J_{i}}[\theta_{j}^{i}(y)]$. The collection
$\\{[\theta_{j}^{i}(y)]:j\in J_{i};i\in I_{n}\\}$ forms an open cover of
$S_{y}(M)$. So we have a finite subcover say $\\{[\theta_{j}^{i}(y)]:j\in
J_{i}^{*};i\in I_{n}\\}$ where $J_{i}^{*}$ is a finite subset of $J_{i}$. For
each $0\leq i\leq n$, we define

$d_{i,n}^{\varphi}(y)=\begin{cases}\begin{array}[]{cc}\bigvee_{j\in
J_{i}^{*}}\theta_{j}^{0}(y)&J_{i}^{*}\neq\emptyset,\\\ y\neq
y&\text{otherwise}.\end{array}\end{cases}$

We claim that $\\{d_{i,n}^{\varphi}(y)\\}_{i=0}^{n}$ as constructed satisfy
the theorem. ∎

We also need the following fact.

######  Fact 3.4.

Suppose $m\in\mathbb{N}_{>0}$ and $r,q\in\mathbb{R}$. If $|q-r/m|<1/m$ then

$\left|q-\frac{\left\lfloor r\right\rfloor}{m}\right|<\frac{1}{m}\text{ or
}\left|q-\frac{\left\lceil r\right\rceil}{m}\right|<\frac{1}{m}.$

######  Theorem 3.5.

Let $\mathcal{L}$ be a countable language, $I=\kappa$, $D$ be a countably
incomplete ultrafilter on $I$, and $(M_{i})_{i\in I}$ be an indexed family of
$\mathcal{L}$-structures. Let $\mathcal{M}=\prod_{D}M_{i}$ and
$\mu\in\mathfrak{M}_{x}(\mathcal{M})$. If $\mu$ is finitely approximated then
$\mu$ is pseudo-finite.

###### Proof.

The proof of this theorem is structurally similar to the proof of Proposition
3.1, but slightly more complicated since it is more difficult to encode the
measure of a formula into the language. Let
$(\varphi_{i}(x,y_{i}))_{i\in\mathbb{N}}$ be an enumeration of partitioned
$\mathcal{L}$-formulas. Since $\mu$ is finitely approximated, $\mu$ is
definable (Proposition 2.7). By Propositions 3.3, for each partitioned
$\mathcal{L}$-formula $\varphi_{i}(x,y_{i})$ and $n\in\mathbb{N}$, there
exists $\mathcal{L}_{y_{i}}(\mathcal{M})$-formulas (not necessarily a
partition)
$d_{0,n}^{\varphi_{i}}(y_{i},\mathbf{c}_{0,n}^{i}),...,d_{n,n}^{\varphi_{i}}(y_{i},\mathbf{c}_{n,n}^{i})$
such that

  1. (1)

$\\{d_{l,n}^{\varphi_{i}}(y_{i},\mathbf{c}_{l,n}^{i})\\}_{l=0}^{n}$ covers
$\mathcal{M}^{|y_{i}|}$,

  2. (2)

if $\mathcal{M}\models
d_{j,n}^{\varphi_{i}}(\mathbf{b},\mathbf{c}_{j,n}^{i})$, then
$|\mu(\varphi_{i}(x,\mathbf{b}))-\frac{j}{n}|<\frac{1}{n}$,

  3. (3)

if $|\mu(\varphi(x,b))-\frac{k}{n}|<\frac{1}{n}$, then
$\mathcal{M}\models\bigvee_{j=k-1}^{k+1}d_{j,n}^{\varphi_{i}}(\mathbf{b},\mathbf{c}_{j,n}^{i})$,

  4. (4)

and if $l<0$ or $l>n$, let
$d_{l,n}^{\varphi_{i}}(y_{i},\mathbf{c}_{l,n}^{i}):=y_{i}\neq y_{i}$.

For simplicity, we will suppress the parameters, the $\mathbf{c}$’s,
throughout the rest of the proof. Since $\mu$ is finitely approximated, it
follows from Proposition 2.8 that for every $n\in\mathbb{N}$, there exists
some $\overline{\mathbf{a}}\in(\mathcal{M})^{<\omega}$ such that for every
$i\leq n$,

($\dagger$)
$\sup_{\mathbf{b}\in\mathcal{M}^{|y_{i}|}}|\mu(\varphi_{i}(x,\mathbf{b}))-\operatorname{Av}(\overline{\mathbf{a}})(\varphi_{i}(x,\mathbf{b}))|<\frac{1}{n}.$

We now construct a sentence similar to the one found in Proposition 3.1. We
claim that for every $n\in\mathbb{N}$, there exists a number $r(n)$ such that
the following sentence is true in $\mathcal{M}$.

$\displaystyle\mathcal{M}\models$ $\displaystyle\exists x_{1},...,x_{r(n)}$
$\displaystyle\bigwedge_{i\leq n}\Big{(}\forall y_{i}\Big{(}\bigwedge_{j\leq
n}\Big{(}d_{j,n}^{\varphi_{i}}(y_{i})\to\bigvee_{A\in\Gamma(n,j)}\Big{(}\Big{(}\bigwedge_{s\in
A}\varphi_{i}(x_{s},y_{i})\wedge\bigwedge_{t\not\in
A}\neg\varphi_{i}(x_{t},y_{i})\Big{)}\Big{)}\Big{)}$
$\displaystyle\wedge\Big{(}\bigwedge_{A\subseteq\mathcal{P}([r(n)])}\Big{(}\Big{(}\bigwedge_{l\in
A}\varphi(x_{l},y_{i})\wedge\bigwedge_{l\not\in
A}\neg\varphi(x_{l},y_{i})\Big{)}\to\bigvee_{t=-1}^{2}d^{\varphi_{i}}_{\gamma(A)+t,n}(y_{i})\Big{)}\Big{)}\Big{)}\Big{)},$

where $\Gamma(n,j)=\\{A\subseteq
r(n):|\frac{|A|}{r(n)}-\frac{j}{n}|<\frac{2}{n}\\}$ and
$\gamma(A)=\left\lfloor\frac{|A|}{r(n)}\cdot n\right\rfloor\cdot\frac{1}{n}$.
For each $n$, we let $\psi_{n}=\exists\overline{x}\sigma_{n}(\overline{x})$ be
the sentence described above. This sentence essentially states that given some
$n\in\mathbb{N}$, there exists points
$\mathbf{a}_{1},...,\mathbf{a}_{r(n)}\in\mathcal{M}$ such that if $\models
d_{j,n}^{\varphi_{i}}(\mathbf{b})$, then
$\operatorname{Av}(\overline{\mathbf{a}})(\varphi(x,\mathbf{b}))\in(\frac{j-1}{n},\frac{j+1}{n})$
and if $\operatorname{Av}(\overline{\mathbf{a}})(\varphi(x,b))=r$, then the
correct definition formula holds (modulo some wiggle room).

We now justify why $\mathcal{M}\models\psi_{n}$. Let
$\mathbf{a}_{1},...,\mathbf{a}_{r(n)}$ be points in $\mathcal{M}$ which
satisfy equation $(\dagger)$. Now assume that $\mathcal{M}\models
d_{j,n}^{\varphi_{i}}(\mathbf{b})$. Then
$|\mu(\varphi(x,\mathbf{b}))-\frac{j}{n}|<\frac{1}{n}$. Since
$\mathbf{a}_{1},...,\mathbf{a}_{r(n)}$ satisfy equation $(\dagger)$, we have
that
$|\mu(\varphi(x,\mathbf{b}))-\operatorname{Av}(\overline{\mathbf{a}})(\varphi(x,\mathbf{b}))|<\frac{1}{n}$
and hence
$|\operatorname{Av}(\mathbf{a})(\varphi(x,\mathbf{b}))-\frac{j}{n}|<\frac{2}{n}$
and so

$\Big{|}\frac{|\\{i\leq
r(n):\models\varphi(\mathbf{a}_{i},\mathbf{b})\\}|}{r(n)}-\frac{j}{n}\Big{|}<\frac{2}{n}.$

Now suppose that

$\bigwedge_{A\subseteq\mathcal{P}([r(n)])}\bigwedge_{l\in
A}\varphi(\mathbf{a}_{l},\mathbf{b})\wedge\bigwedge_{t\not\in
A}\neg\varphi(\mathbf{a}_{t},\mathbf{b}).$

Then
$\operatorname{Av}(\overline{\mathbf{a}})(\varphi(x,\mathbf{b}))=\frac{|A|}{r(n)}$.
By equation $(\dagger)$, notice
$|\mu(\varphi(x,\mathbf{b}))-\frac{|A|}{r(n)}|<\frac{1}{n}$, and so
$|\mu(\varphi(x,\mathbf{b}))-\frac{\frac{|A|}{r(n)}\cdot n}{n}|<\frac{1}{n}$.
Hence by Fact 3.4,

$|\mu(\varphi(x,\mathbf{b}))-\frac{\left\lfloor\frac{|A|}{r(n)}\cdot
n\right\rfloor}{n}|<\frac{1}{n}\text{ or
}|\mu(\varphi(x,\mathbf{b}))-\frac{\left\lceil\frac{A}{r(n)}\cdot
n\right\rceil}{n}|<\frac{1}{n}.$

By construction, this implies that

$\mathcal{M}\models\bigvee_{j=-1}^{1}d_{\gamma(A)+j,n}^{\varphi_{i}}(\mathbf{b})\vee\bigvee_{j=-1}^{1}d_{\gamma(A)+1+j,n}^{\varphi_{i}}(\mathbf{b}).$

which is precisely what we want.

As in the proof of Proposition 3.1, we construct the collection of sets
$E_{n}$ where $E_{n}=\\{k\in\mathbb{N}:M_{k}\models\psi_{n}\\}$ with the
parameters in $\psi_{n}$ replaced with the appropriate parameters in each
$M_{k}$. By Łoś’s theorem, $E_{n}\in D$. Since $D$ is countably incomplete, we
let $\\{X_{i}:i\in\mathbb{N}\\}\subseteq D$ such that
$\bigcap_{i\in\mathbb{N}}X_{i}=\emptyset$. Again, we set $Y_{n}=E_{n}\cap
X_{n}$ and we notice that $\bigcap Y_{n}=\emptyset$. In a similar way as to
Proposition 3.1, we now construct a sequence of Keisler measure
$(\mu_{i})_{i\in I}$ where each $\mu_{i}$ is in $\mathfrak{M}_{x}(M_{i})$. We
define the map $g:I\to\mathbb{N}$ where $g(i)=\max\\{n\in\mathbb{N}:i\in
Y_{n}\\}$ and $0$ otherwise. If $g(i)=0$, we let $\mu_{i}$ be any measure in
$\operatorname{conv}(M_{i})$. If $g(i)=n>0$, then we choose
$\overline{a}_{i}=(a_{1}^{i},...,a_{r(n)}^{i})$ in $M_{i}$ such that
$M_{i}\models\sigma_{n}(\overline{a}_{i})$ and let
$\mu_{i}=\operatorname{Av}(\overline{a}_{i})$.

We claim that the ultralimit of $(\mu)_{i\in I}$ is equal to $\mu$. Let
$\theta(x,y)$ be an $\mathcal{L}$-formula and assume that
$\mu(\theta(x,\mathbf{e}))=s$. Fix $\epsilon>0$ and let
$(e_{i})_{i\in\omega}/D=\mathbf{e}$. It suffices to show that $K=\\{i\in
I:|\mu_{i}(\theta(x,e_{i}))-s|<\epsilon\\}\in D$. Notice that $\theta(x,y)$
appears in our index, say $\theta(x,y)=\varphi_{m}(x,y_{m})$. Let
$m_{*}=\max\\{m+1,\left\lceil\frac{10}{\epsilon}\right\rceil\\}$. Since
$\mu(\theta(x,\mathbf{e}))=s$, notice that

$0=|\mu(\varphi_{m}(x,\mathbf{e}))-s|=|\mu(\varphi_{m}(x,\mathbf{e}))-\frac{s\cdot
m_{*}}{m_{*}}|<\frac{1}{m_{*}}$

and so by Fact 3.4,

$|\mu(\varphi_{m}(x,\mathbf{e}))-\frac{\left\lfloor s\cdot
m_{*}\right\rfloor}{m_{*}}|<\frac{1}{m_{*}}\text{ or
}|\mu(\varphi_{m}(x,\mathbf{e}))-\frac{\left\lceil s\cdot
m_{*}\right\rceil}{m_{*}}|<\frac{1}{m_{*}}.$

Let $j_{*}=\left\lfloor s\cdot m_{*}\right\rfloor$. Then by construction
$\mathcal{M}\models\bigvee_{\alpha=-1}^{2}d^{\varphi_{m}}_{j_{*}+\alpha,m_{*}}(\mathbf{e})$
and so for some $i_{*}\in\\{j_{*}-1,j_{*},j_{*}+1,j_{*}+2\\}$,
$\mathcal{M}\models d_{i_{*},m_{*}}^{\varphi_{m}}(\mathbf{e})$. Let $Z=\\{i\in
I:M_{i}\models d_{i_{*},m_{*}}^{\varphi_{m}}(e_{i})\\}$. By Łoś’s theorem,
$Z\in D$. We now argue that $Y_{m_{*}}\cap Z\subset K$. This which implies
$K\in D$ and completes the proof.

Let $k\in Z\cap Y_{m_{*}}$. Since $k\in Y_{m_{*}}$, it follows that

$M_{k}\models
d_{i_{*},m_{*}}^{m}(e_{k})\to\bigvee_{A\in\Gamma(m_{*},i_{*})}\Big{(}\bigwedge_{l\in
A}\varphi(a_{l}^{k},e_{k})\wedge\bigwedge_{t\not\in
A}\neg\varphi(a_{t}^{k},e_{k})\Big{)}.$

Since $k\in Z$, it follows that for some $A_{*}\in\Gamma(m_{*},i_{*})$

$M_{k}\models\bigwedge_{l\in
A_{*}}\varphi(a_{l}^{k},e_{k})\wedge\bigwedge_{t\not\in
A_{*}}\neg\varphi(a_{t}^{k},e_{k}).$

Now since $A_{*}\in\Gamma(m_{*},i_{*})$, it follows that
$|\frac{|A_{*}|}{r(m_{*})}-\frac{i_{*}}{m_{*}}|<\frac{2}{m_{*}}$. By
definition, this implies that

$|\mu_{k}(\varphi(x,e_{k}))-\frac{i_{*}}{m_{*}}|<\frac{2}{m_{*}}.$

Now notice

$|\mu_{k}(\varphi(x,e_{k}))-s|\leq|\mu_{k}(\varphi(x,e_{k}))-\frac{i_{*}}{m_{*}}|+|\frac{i_{*}}{m_{*}}-s|<\frac{2}{m*}+\frac{4}{m_{*}}<\frac{10}{m_{*}}<\epsilon.$

Hence $k\in K$ and we conclude $K\in D$. ∎

The following corollary follows tautologically from Theorems 1.1 and 3.5.

######  Corollary 3.6.

Suppose that $\mathcal{L}$ is countable, $I=\kappa$, $(M_{i})_{i\in I}$ is an
indexed family of $\mathcal{L}$-structures, $D$ is a countably incomplete
ultrafilter on $I$, $\mathcal{M}=\prod_{D}M_{i}$ is NIP, and
$\mu\in\mathfrak{M}_{x}(\mathcal{M})$. Then $\mu$ is generically stable if and
only if $\mu$ is pseudo-finite.

We now discuss a brief theoretical application of the previous theorem. Let
$T_{n}$ be the theory of the $n$-Henson graph [11]. More explicitly, $T_{n}$
is the theory of the Fraïssé limit of all finite $K_{n}$-free graphs. It is
open whether or not this theory is pseudo-finite.

######  Corollary 3.7.

Suppose that $T_{n}$ is pseudo-finite. Let $(G_{i})_{i\in\mathbb{N}}$ be a
countable sequence of graphs and $D$ an ultrafilter on $\mathbb{N}$ such that
$\mathcal{G}:=\prod_{D}G_{i}\models T_{n}$. Consider the measure $\delta_{p}$
where $p$ is the unique type over $\mathcal{G}$ such that $p\supset\\{\neg
R(x,\mathbf{b}):\mathbf{b}\in\mathcal{G}\\}$. It was shown in [6] that
$\delta_{p}$ is finitely approximated. Hence by the previous theorem, if
$T_{n}$ is pseudo-finite then $\delta_{p}$ is itself pseudo-finite.

##  4\. The Morley product and the pseudo-finite product

The main goal of this section is to show that the Morley product and the
pseudo-finite product agree for reasonable pairs of measures. Explicitly, we
prove that these products agree when the left-hand-side measure is
$\mathcal{M}$-definable and both are pseudo-finite. If both measures are
definable and pseudo-finite, then the Morley product commutes on this pair. We
then use these results to show that certain natural measures are not pseudo-
finite over non-trivial ultraproducts of the Paley graphs. We begin this
section by recalling the definitions of both the Morley and pseudo-finite
products.

######  Definition 4.1.

Let $\mu\in\mathfrak{M}_{x}(M)$, $\nu\in\mathfrak{M}_{y}(M)$, and suppose
$\mu$ is definable. Then the Morley product of $\mu$ and $\nu$, denoted
$\mu\otimes\nu$, is the unique measure in $\mathfrak{M}_{xy}(M)$ such that for
any $\varphi(x,y,c)\in\mathcal{L}_{xy}(M)$,

$\mu\otimes\nu(\varphi(x,y,c))=\int_{S_{y}(M)}F_{\mu,M}^{\varphi_{c}}d\tilde{\nu},$

where $F_{\mu,M}^{\varphi_{c}}:S_{y}(M)\to[0,1]$ is the unique continuous map
extending $\operatorname{tp}(b/M)\to\mu(\varphi(x,b,c))$ and $\tilde{\nu}$ is
the unique regular Borel probability measure on $S_{y}(M)$ such that for any
clopen set $[\psi(y)]\subseteq S_{y}(M)$,
$\tilde{\nu}([\psi(y)])=\nu(\psi(y))$. We will often drop the tilde and write
$\tilde{\nu}$ simply as $\nu$.

We remind the reader that in general, the Morley product is non-commutative
and can be a nuisance. We now recall/define the pseudo-finite product of
Keisler measures. In this paper, we want our ultralimits of Keisler measures
to be honest-to-goodness Keisler measures over the ultraproduct and so we do
not care about internal subsets of our ultraproduct (only definable ones).

######  Definition 4.2.

Suppose that $(M_{i})_{i\in I}$ is a sequence of $\mathcal{L}$-structures, $D$
is an ultrafiliter on $I$, and $\mathcal{M}:=\prod_{D}M_{i}$. Suppose that
$\mu\in\mathfrak{M}_{x}(\mathcal{M})$ and
$\nu\in\mathfrak{M}_{y}(\mathcal{M})$ and both $\mu$ and $\nu$ are pseudo-
finite. So there exists sequences of measures $(\mu_{i})_{i\in I}$ and
$(\nu_{i})_{i\in I}$ such that $\mu_{i}\in\operatorname{conv}_{x}(M_{i})$,
$\nu_{i}\in\operatorname{conv}_{y}(M_{i})$, $\lim_{D}\mu_{i}=\mu$, and
$\lim_{D}\nu_{i}=\nu$. Then the pseudo-finite product of $\mu$ and $\nu$,
denoted $\mu\boxtimes\nu$, is the unique measure in
$\mathfrak{M}_{xy}(\mathcal{M})$ such that for any
$\varphi(x,y,\mathbf{c})\in\mathcal{L}_{xy}(\mathcal{M})$,

$\mu\boxtimes\nu(\varphi(x,y,\mathbf{c}))=\lim_{D}\Big{(}\mu_{i}\otimes\nu_{i}(\varphi(x,y,\mathbf{c}(i))\Big{)}.$

######  Remark 4.3.

In the definition above, since $\mu_{i}\in\operatorname{conv}_{x}(M_{i})$ and
$\nu_{i}\in\operatorname{conv}_{y}(M_{i})$, we have that
$\mu_{i}=\sum_{k=1}^{n}r_{k}\delta_{a_{k}}$ and
$\nu_{i}=\sum_{j=1}^{m}s_{j}\delta_{b_{j}}$. One can check that for any
$\theta(x,y)\in\mathcal{L}_{xy}(M_{i})$,

$\mu_{i}\otimes\nu_{i}(\theta(x,y))=\sum_{k=1}^{n}\sum_{j=1}^{m}r_{k}s_{j}\delta_{(a_{k},b_{j})}(\theta(x,y))=\nu_{i}\otimes\mu_{i}(\theta(x,y)).$

One could be (and probably should be) worried about whether or not the pseudo-
finite product is well-defined. We claim that it is well-defined and work
towards proving this now. Towards this goal we start with a definition and
prove a general lemma.

######  Definition 4.4.

Fix a sequence of $\mathcal{L}$-structures $(M_{i})_{i\in I}$ and an
ultrafilter $D$ on $I$. Let $\mathcal{M}=\prod_{D}M_{i}$. For each $i\in I$,
let $g_{i}:S_{y}(M_{i})\to[0,1]$ be a function. Then we define the map
$g_{D}:\mathcal{M}\to[0,1]$ via
$g_{D}(\mathbf{a})=\lim_{D}(g_{i}(\operatorname{tp}(a_{i}/M_{i})))$.

It is easy to check that the above definition is well-defined. The following
lemma is the main lemma of this section. We assume it is more or less
folklore.

######  Lemma 4.5.

Fix a sequence of $\mathcal{L}$-structures $(M_{i})_{i\in I}$ and an
ultrafilter $D$ on $I$. Let $\mathcal{M}=\prod_{D}M_{i}$. For each $i\in I$,
let $g_{i},h_{i}:S_{y}(M_{i})\to[0,1]$ be continuous functions and suppose
that
$\sup_{\mathbf{a}\in\mathcal{M}}|g_{D}(\mathbf{a})-h_{D}(\mathbf{a})|<\epsilon$.
If $(\nu_{i})_{i\in I}$ is any sequence of Keisler measures, then

$\lim_{D}\left(\int_{S_{y}(M_{i})}g_{i}d\nu_{i}\right)\approxeq_{\epsilon}\lim_{D}\left(\int_{S_{y}(M_{i})}h_{i}d\nu_{i}\right).$

###### Proof.

Let $A=\\{i\in I:\sup_{q\in S_{y}(M_{i})}|g_{i}-h_{i}|(q)\leq\epsilon\\}$. We
claim $A\in D$. If not, then $I\backslash A\in D$. For each $i\in I\backslash
A$, there exists some $q_{i}^{*}\in S_{y}(M_{i})$ such that
$|g_{i}-h_{i}|(q_{i}^{*})>\epsilon$. Since the map
$|g_{i}-h_{i}|:S_{y}(M_{i})\to[0,1]$ is continuous and $M_{i}^{y}$ is a dense
subset of $S_{y}(M_{i})$, there exists some $e_{i}\in M_{i}^{y}$ such that
$|g_{i}-h_{i}|(\operatorname{tp}(e_{i}/M_{i}))>\epsilon$. We define
$(c_{i})_{i\in I}$ where

$c_{i}=\begin{cases}\begin{array}[]{cc}e_{i}&i\in I\backslash A,\\\
\text{anything in $M_{i}$}&\text{otherwise}.\end{array}\end{cases}$

and let $\mathbf{c}=[(c_{i})]_{D}$. Then by Fact 2.2,

$\epsilon\leq\lim_{D}|g_{i}-h_{i}|(\operatorname{tp}(c_{i}/M_{i}))=|\lim_{D}g_{i}(\operatorname{tp}(c_{i}/M_{i}))-\lim_{D}h_{i}(\operatorname{tp}(c_{i}/M_{i}))|$
$=|g_{D}(\mathbf{c})-h_{D}(\mathbf{c})|<\sup_{\mathbf{a}\in\mathcal{M}}|g_{D}(\mathbf{a})-h_{D}(\mathbf{a})|<\epsilon,$

which is a contradiction. Therefore we conclude that $A\in D$.

Now consider the set $\tilde{A}=\\{i\in
I:|\int_{S_{y}(M_{i})}g_{i}d\nu_{i}-\int_{S_{y}(M_{i})}h_{i}d\nu_{i}|\leq\epsilon\\}$.
We claim that $\tilde{A}\supseteq A$ and so $\tilde{A}\in D$. Observe that if
$i\in A$, then

$\displaystyle\left|\int_{S_{y}(M_{i})}g_{i}d\nu_{i}-\int_{S_{y}(M_{i})}h_{i}d\nu_{i}\right|$
$\displaystyle\leq\int_{S_{y}(M_{i})}|g_{i}-h_{i}|d\nu_{i}$
$\displaystyle\leq\int_{S_{y}(M_{i})}\epsilon d\nu_{i}$
$\displaystyle=\epsilon.$

Thus $i\in\tilde{A}$ and we conclude $\tilde{A}\in D$. Now by $(5)$ of Fact
2.2,

$\lim_{D}\left(\int_{S_{y}(M_{i})}g_{i}d\nu_{i}\right)\approxeq_{\epsilon}\lim_{D}\left(\int_{S_{y}(M_{i})}h_{i}d\nu_{i}\right).\qed$

######  Lemma 4.6.

Let $\mu\in\mathfrak{M}_{x}(\mathcal{M})$,
$\nu\in\mathfrak{M}_{y}(\mathcal{M})$ and suppose there exists
$\mu^{1}_{i},\mu^{2}_{i}\in\mathfrak{M}_{x}(M_{i})$,
$\nu^{1}_{i},\nu^{2}_{i}\in\mathfrak{M}_{y}(M_{i})$ such that

  1. (1)

$\lim_{D}\mu^{1}_{i}=\lim_{D}\mu^{2}_{i}=\mu$ and
$\lim_{D}\nu^{1}_{i}=\lim_{D}\nu^{2}_{i}=\nu$.

  2. (2)

For each $i\in I$, $\mu^{1}_{i},\mu^{2}_{i},\nu^{1}_{i},\nu^{2}_{i}$ are each
$M_{i}$-definable.

  3. (3)

For each $i\in I$,
$\mu^{2}_{i}\otimes\nu^{1}_{i}=\nu^{1}_{i}\otimes\mu^{2}_{i}$ and
$\mu^{2}_{i}\otimes\nu^{2}_{i}=\nu^{2}_{i}\otimes\mu^{2}_{i}$

Then
$\lim_{D}(\mu^{1}_{i}\otimes\nu^{1}_{i})=\lim_{D}(\mu^{2}_{i}\otimes\nu^{2}_{i})$.

###### Proof.

By Fact 2.4, it suffices to show that for any
$\varphi(x,y,\mathbf{b})\in\mathcal{L}_{xy}(\mathcal{M})$,

$\lim_{D}\mu_{i}^{1}\otimes\nu_{i}^{1}(\varphi(x,y,\mathbf{b}(i)))=\lim_{D}\mu_{i}^{2}\otimes\nu_{i}^{2}(\varphi(x,y,\mathbf{b}(i)))$

Since $\mu_{i}^{1},\mu_{i}^{2},\nu_{i}^{1},$ and $\nu_{i}^{2}$ are each
$M_{i}$-definable, the maps
$F_{\mu_{i}^{1}}^{\varphi_{\mathbf{b}(i)}},F_{\mu_{i}^{2}}^{\varphi_{\mathbf{b}(i)}}:S_{y}(M_{i})\to[0,1]$
and
$F_{\nu_{i}^{1}}^{\varphi^{*}_{\mathbf{b}(i)}},F_{\nu_{i}^{2}}^{\varphi^{*}_{\mathbf{b}(i)}}:S_{x}(M_{i})\to[0,1]$
are each continuous. Moreover, notice that for any $\mathbf{a}\in\mathcal{M}$,

$\left(F_{\mu_{i}^{1},M_{i}}^{\varphi_{\mathbf{b}(i)}}\right)_{D}(\mathbf{a})=\lim_{D}\mu_{i}^{1}(\varphi(x,\mathbf{a}(i),\mathbf{b}(i)))=\mu(\varphi(x,\mathbf{a},\mathbf{b}))$
$=\lim_{D}\mu_{i}^{2}(\varphi(x,\mathbf{a}(i),\mathbf{b}(i)))=\left(F_{\mu_{i}^{2},M_{i}}^{\varphi_{\mathbf{b}(i)}}\right)_{D}(\mathbf{a}).$

Likewise,
$\left(F_{\nu_{i}^{1},M_{i}}^{\varphi^{*}_{\mathbf{b}(i)}}\right)_{D}=\left(F_{\nu_{i}^{2},M_{i}}^{\varphi^{*}_{\mathbf{b}(i)}}\right)_{D}$.
Therefore, we can apply the Lemma 4.5 (to two different pairs of sequences of
functions) and with any $\epsilon>0$. We now have the following sequence of
equations:

$\lim_{D}\mu^{1}_{i}\otimes\nu^{1}_{i}(\varphi(x,y,\mathbf{b}(i)))=\lim_{D}\left(\int_{S_{y}(M_{i})}F_{\mu^{1}_{i},M_{i}}^{\varphi_{\mathbf{b}(i)}}d\nu^{1}_{i}\right)$
$\overset{(a)}{=}\lim_{D}\left(\int_{S_{y}(M_{i})}F_{\mu^{2}_{i},M_{i}}^{\varphi_{\mathbf{b}(i)}}d\nu^{1}_{i}\right)\overset{(b)}{=}\lim_{D}\left(\int_{S_{y}(M_{i})}F_{\nu^{1}_{i},M_{i}}^{\varphi^{*}_{\mathbf{b}(i)}}d\mu^{2}_{i}\right)$
$\overset{(a)}{=}\lim_{D}\left(\int_{S_{y}(M_{i})}F_{\nu^{2}_{i},M_{i}}^{\varphi^{*}_{\mathbf{b}(i)}}d\mu^{2}_{i}\right)\overset{(b)}{=}\lim_{D}\left(\int_{S_{y}(M_{i})}F_{\mu^{2}_{i},M_{i}}^{\varphi_{\mathbf{b}(i)}}d\nu^{2}_{i}\right)$
$=\lim_{D}\mu_{i}^{2}\otimes\nu_{i}^{2}(\varphi(x,y,\mathbf{b}(i))).$

These equations are justified by the following:

  1. $(a)$

Lemma 4.5 applied with arbitrarily small $\epsilon$.

  2. $(b)$

Remark 4.3. ∎

######  Proposition 4.7.

The pseudo-finite product $\boxtimes$ is well-defined.

###### Proof.

It suffices to argue that if $\mu\in\mathcal{M}_{x}(\mathcal{M})$,
$\nu\in\mathfrak{M}_{y}(\mathcal{M})$,
$\mu^{1}_{i},\mu^{2}_{i}\in\operatorname{conv}_{x}(M_{i})$,
$\nu^{1}_{i},\nu^{2}_{i}\in\operatorname{conv}_{y}(M_{i})$,
$\lim_{D}\mu^{1}_{i}=\lim_{D}\mu^{2}_{i}=\mu$, and
$\lim_{D}\nu_{i}^{1}=\lim_{D}\nu_{i}^{2}=\nu$ then
$\lim_{D}(\mu_{i}\otimes\nu_{i})=\lim_{D}(\mu_{i}\otimes\nu_{i})$. Notice that
every measure which concentrates on finitely many realized types is definable.
By Remark 4.3, all necessary products commute. Hence we can apply Lemma 4.6
and the proof is complete. ∎

We now connect the Morley product and the pseudo-finite product and derive the
main result of this section.

######  Theorem 4.8.

Let $(M_{i})_{i\in I}$ be a indexed family of $\mathcal{L}$-structures, $D$ be
an ultrafilter on $I$, and $\mathcal{M}:=\prod_{D}M_{i}$. Let
$\mu\in\mathfrak{M}_{x}(\mathcal{M})$ and
$\nu\in\mathfrak{M}_{y}(\mathcal{M})$. Suppose that both $\mu$ and $\nu$ are
pseudo-finite. If $\mu$ is definable, then $\mu\otimes\nu=\mu\boxtimes\nu$.

###### Proof.

Fix a formula $\varphi(x,y,\mathbf{d})\in\mathcal{L}_{xy}(\mathcal{M})$,
$\epsilon>0$, and let $[(d_{i})]_{D}=\mathbf{d}$. Choose $(\mu_{i})_{i\in I}$
and $(\nu_{i})_{i\in I}$ such that $\mu_{i}\in\operatorname{conv}_{x}(M_{i})$,
$\nu_{i}\in\operatorname{conv}_{y}(M_{i})$, $\lim_{D}\mu_{i}=\mu$, and
$\lim_{D}\nu_{i}=\nu$. Since $\mu$ is definable, there exists real numbers
$r_{1},...,r_{n}$ and $\mathcal{L}_{y}(\mathcal{M})$-formulas
$\psi_{1}(y,\mathbf{a}_{1}),...,\psi_{n}(y,\mathbf{a}_{n})$ which partition
$\mathcal{M}^{y}$ such that

$None$ $\sup_{q\in
S_{y}(\mathcal{M})}|F_{\mu,\mathcal{M}}^{\varphi}(q)-\sum_{j=1}^{n}r_{j}\mathbf{1}_{\psi_{j}(y,\mathbf{a}_{j})}(q)|<\epsilon.$

Let $f_{i}:=\sum_{j=1}^{n}r_{j}\mathbf{1}_{\psi_{j}(y,\mathbf{a}_{j}(i))}$ and
so $f_{i}:S_{y}(M_{i})\to[0,1]$. Then we have that

$\mu\otimes\nu(\varphi(x,y,\mathbf{d}))=\int_{S_{y}(\mathcal{M})}F_{\mu,\mathcal{M}}^{\varphi_{\mathbf{d}}}d\nu\approx_{\epsilon}\int_{S_{y}(\mathcal{M})}\sum_{j=1}^{n}r_{j}\mathbf{1}_{\psi_{j}(y,\mathbf{a}_{j})}d\nu=\sum_{i=1}^{n}r_{j}\nu(\psi_{j}(y,\mathbf{a}_{j}))$
$=\sum_{j=1}^{n}r_{j}\left(\lim_{D}\nu_{i}(\psi_{j}(y,\mathbf{a}_{j}(i)))\right)=\lim_{D}\left(\sum_{j=1}^{n}r_{j}\nu_{i}(\psi_{j}(y,\mathbf{a}_{j}(i))\right)$
$=\lim_{D}\left(\int_{S_{y}(M_{i})}\sum_{j=1}^{n}r_{j}\mathbf{1}_{\psi_{j}(y,\mathbf{a}_{j}(i))}d\nu_{i}\right)=\lim_{D}\left(\int_{S_{y}(M_{i})}f_{i}d\nu_{i}\right),$
$\overset{(*)}{\approxeq_{\epsilon}}\lim_{D}\left(\int_{S_{y}(M_{i})}F_{\mu_{i},M_{i}}^{\varphi_{\mathbf{d}(i)}}d\nu_{i}\right)=\mu\boxtimes\nu(\varphi(x,y,\mathbf{d})).$

We justify equation $(*)$ using Lemma 4.5. Clearly for each $i\in I$, we have
that $F_{\mu_{i},M_{i}}^{\varphi_{\mathbf{d}(i)}}$ and $f_{i}$ are continuous.
Moreover, for any $\mathbf{c}\in\mathcal{M}$,

$|(F_{\mu_{i},M_{i}}^{\varphi_{\mathbf{d}(i)}})_{D}(\mathbf{c})-f_{D}(\mathbf{c})|=|\mu(\varphi(x,\mathbf{c},\mathbf{d}))-\sum_{j=1}^{n}r_{j}\mathbf{1}_{\psi_{j}(y,\mathbf{a}_{j})}(\operatorname{tp}(\mathbf{c}/\mathcal{M}))|<\epsilon.$

The first equation above is a direct consequence of the definitions (left
term) and Łoś’s theorem (right term). Hence Lemma 4.5 applies and
approximation $(*)$ is justified. ∎

######  Corollary 4.9.

Let $(M_{i})_{i\in I}$ be a indexed family of $\mathcal{L}$-structures, $D$ be
an ultrafilter on $I$, and $\mathcal{M}:=\prod_{D}M_{i}$. Let
$\mu\in\mathfrak{M}_{x}(\mathcal{M})$ and
$\nu\in\mathfrak{M}_{y}(\mathcal{M})$. Suppose that both $\mu$ and $\nu$ are
pseudo-finite. If $\mu$ and $\nu$ are both definable, then
$\mu\otimes\nu=\nu\otimes\mu$.

###### Proof.

Fix a formula $\varphi(x,y,\mathbf{d})\in\mathcal{L}_{xy}(\mathcal{M})$. Let
$\mu_{i}\in\operatorname{conv}_{x}(M_{i})$ and
$\nu_{i}\in\operatorname{conv}_{y}(M_{i})$ such that $\lim_{D}\mu_{i}=\mu$ and
$\lim_{D}\nu_{i}=\nu$. Then Theorem 4.8 and Remark 4.3 yield the following,

$\mu\otimes\nu(\varphi(x,y,\mathbf{d}))=\mu\boxtimes\nu(\varphi(x,y,\mathbf{d}))=\lim_{D}\mu_{i}\otimes\nu_{i}(\varphi(x,y,\mathbf{d}(i)))$
$=\lim_{D}\nu_{i}\otimes\mu_{i}(\varphi(x,y,\mathbf{d}(i)))=\nu\boxtimes\mu(\varphi(x,y,\mathbf{d}))=\nu\otimes\mu(\varphi(x,y,\mathbf{d})).\qed$

######  Remark 4.10.

The previous results goes through locally. In particular, suppose that
$\varphi(x,y)$ is an $\mathcal{L}$-formula, $(M_{i})_{i\in I}$ is an indexed
family of $\mathcal{L}$-structures, $D$ is an ultrafilter on $I$, and
$\prod_{D}M_{i}=\mathcal{M}$. Suppose that $\mu$ and $\nu$ are pseudo-finite
and $\mu$ is $\varphi$-definable; i.e. we only assume that the map
$F_{\mu,\mathcal{M}}^{\varphi}:S_{y}(\mathcal{M})\to[0,1]$ is well-defined and
continuous.

  1. (1)

Then $\mu\otimes\nu(\varphi(x,y))=\mu\boxtimes\nu(\varphi(x,y))$.

  2. (2)

If $\nu$ is $\varphi^{*}$-definable, then
$\mu\otimes\nu(\varphi(x,y))=\nu\otimes\mu(\varphi(x,y))$.

###  4.1. An application

Remark 4.10 can be used to show that many measures are not pseudofinite (with
respect to a particular sequence of $\mathcal{L}$-structures and choice of
ultrafilter). Here we consider the Paley graphs and demonstrate that the
“unfair coin-flipping measures" are not pseudo-finite in this context.

######  Definition 4.11 (Paley Graphs).

Let $(q_{i})_{i\in\mathbb{N}}$ be the sequence of primes such that $q_{i}=1$
mod $4$. Let $\mathbb{F}_{q_{i}}$ be the finite field of size $q_{i}$ and
$G_{q_{i}}=(\mathbb{F}_{q_{i}};R)$ where $G_{q_{i}}\models R(a,b)$ if and only
if there exists some $z\in\mathbb{F}_{q_{i}}$ such that $z^{2}=a-b$. We remark
that $G_{q_{i}}$ is a graph.

######  Fact 4.12.

The following statements are true.

  1. (1)

For any non-principal ultrafilter $D$ on $I$, $\prod_{D}G_{q_{i}}$ is a model
of the random graph [15, Example 3.4].

  2. (2)

For each $q_{i}$ and $a\in G_{q_{i}}$, the degree of $a$ is
$\frac{q_{i}-1}{2}$ [21].

######  Definition 4.13.

Let $M$ be a model of the random graph. We say that a measure $\nu$ is an
unfairly weighted coin-flipping measure if $\nu=\nu_{p}$ where
$p\in[0,1]\backslash\\{1/2\\}$ and for any finite sequence of distinct
singletons $a_{1},...,a_{n},b_{1},...,b_{m}$,

$\nu_{p}\left(\bigwedge_{i=1}^{n}R(x,a_{i})\wedge\bigwedge_{j=1}^{m}\neg
R(x,b_{j})\right)=\left(\frac{1}{p}\right)^{n}\left(\frac{1}{1-p}\right)^{m}.$

By quantifier elimination, $\nu_{p}$ is unique.

######  Theorem 4.14.

Suppose that $\mathcal{L}=\\{R\\}$, $I=\mathbb{N}$, $M_{i}=G_{q_{i}}$, $D$ is
a non-principal ultrafilter on $I$, and $\mathcal{M}:=\prod_{D}M_{i}$. We let
$\mu$ be the pseudo-finite counting measure, i.e. $\mu=\lim_{D}\mu_{i}$ where
for each $\varphi(x)\in\mathcal{L}_{x}(M_{i})$,
$\mu_{i}(\varphi(x))=\frac{|\\{a\in
M_{i}:M_{i}\models\varphi(a)\\}|}{|M_{i}|}$.

  1. (1)

The measure $\mu$ is $R$-definable and in particular $F_{\mu,\mathcal{M}}^{R}$
is the constant function which outputs $1/2$.

  2. (2)

For any measure $\nu$ such that $F_{\nu,\mathcal{M}}^{R}$ is a constant
function not equal to 1/2, $\nu$ is not pseudo-finite (with respect to this
indexing family and ultrafilter). In particular, any unfairly weighted coin-
flipping measures is not pseudo-finite in this context.

###### Proof.

We prove the statements.

  1. (1)

By (3) of Fact 4.12, $\mu(R(x,\mathbf{a}))=\frac{1}{2}$ for each
$\mathbf{a}\in\mathcal{M}$. Obviously this map can be extended to the constant
function $1/2$ over $S_{y}(\mathcal{M})$ which is continuous.

  2. (2)

Suppose $\nu$ is pseudo-finite. Assume that for every
$\mathbf{a}\in\mathcal{M}$, $\nu(R(x,\mathbf{a}))=r\neq 1/2$. Then $\nu$ is
$R^{*}$-definable and

$\nu\otimes\mu(R(x,y))=r\neq 1/2=\mu\otimes\nu(R(x,y)).$

This contradicts Remark 4.10. ∎

##  5\. Constructing idempotent measures on monster models of pseudo-finite
groups

We now wish to work in the “standard" context for Keisler measures, i.e.
measures over a monster model. In this section, we fix a monster model
$\mathcal{U}$ in the background and ultraproducts will always be small models
of $\mathcal{U}$. The purpose of this section is to construct idempotent
generically stable measures over pseudo-finite NIP groups. Idempotent Keisler
measures play an interesting role in the model theory of groups and have been
previously studied in [3, 4].

######  Definition 5.1.

Suppose that $\mathcal{U}$ is a monster model of a group and $G$ is a small
submodel. Suppose that $\mu\in\mathfrak{M}_{x}(\mathcal{U})$,
$\nu\in\mathfrak{M}_{y}(\mathcal{U})$. We assume that $|x|=1$.

  1. (1)

We say that $\mu$ is definable over $G$ if $\mu|_{G}$ is $G$-definable and for
any $\varphi(x,y)\in\mathcal{L}$ and $b\in\mathcal{U}$,
$\mu(\varphi(x,b))=F_{\mu|_{G},G}^{\varphi}(\operatorname{tp}(b/G))$.

  2. (2)

If $\mu$ is definable (over $G$), we let $\mu\tilde{\otimes}\nu$ denote the
standard Morley product (for global measures). In particular, for any formula
$\varphi(x,y,c)\in\mathcal{L}_{xy}(\mathcal{U})$, we have that

$\mu\tilde{\otimes}\nu(\varphi(x,y,b))=\int_{S_{y}(Gb)}F_{\mu,Gb}^{\varphi_{b}}d(\nu|_{Gb}).$

  3. (3)

If $\mu$ is definable (over $G$) then $\mu*\nu$ is the unique measure in
$\mathfrak{M}_{x}(\mathcal{U})$ such that for any formula
$\varphi(x)\in\mathcal{L}_{x}(\mathcal{U})$,

$\mu*\nu(\varphi(x))=\mu\tilde{\otimes}\nu(\varphi(x\cdot y)).$

  4. (4)

If $\mu$ is definable (over $G$) we say that $\mu$ is idempotent if
$\mu*\mu=\mu$.

A proof of the following fact can be found in [7, Theorem A.24].

######  Fact 5.2.

Suppose that $\mu\in\mathfrak{M}_{x}(M)$ and $\mu$ is $M$-definable. Then
there exists a unique measure $\hat{\mu}\in\mathfrak{M}_{x}(\mathcal{U})$ such
that $\hat{\mu}|_{M}=\mu$ and $\hat{\mu}$ is definable over $M$.

Finally, we recall the classification of idempotent measures over finite
groups. This follows immediately from the main results in [14] and/or [20]
since every finite group can be viewed as a is compact group under the
discrete topology.

######  Fact 5.3.

Let $G$ be any finite group and $\mu$ be a measure on $G$. Then the following
are equivalent:

  1. (1)

$\mu*\mu=\mu$

  2. (2)

There exists a subgroup $H$ of $G$ such that $\mu=\frac{1}{|H|}\sum_{h\in
H}\delta_{h}$.

######  Theorem 5.4.

Let $(G_{i})_{i\in I}$ be a sequence of groups and $H_{i}$ be any sequence of
finite subgroups where $H_{i}\leq G_{i}$. Let $\mathcal{G}=\prod_{D}G_{i}$,
$\mathcal{G}\subset\mathcal{U}$, $\mu_{i}:=\frac{1}{|H_{i}|}\sum_{h\in
H}\delta_{h}$, and $\mu=\lim_{D}\mu_{i}$. If $\mu$ is $\mathcal{G}$-definable,
then $\hat{\mu}$ is idempotent.

###### Proof.

By definition, $\hat{\mu}$ is the unique definable measure which extends $\mu$
from $\mathcal{L}_{x}(\mathcal{G})$ to $\mathcal{L}_{x}(\mathcal{U})$. By [3,
Proposition 3.15], it follows that $\hat{\mu}*\hat{\mu}$ is definable over
$\mathcal{G}$. Hence, if
$(\hat{\mu}*\hat{\mu})|_{\mathcal{G}}=\hat{\mu}|_{\mathcal{G}}$, then
$\hat{\mu}*\hat{\mu}=\hat{\mu}$. Let
$\varphi(x)\in\mathcal{L}_{x}(\mathcal{G})$,
$\varphi^{\prime}(x,y):=\varphi(x\cdot y)$, and consider the following
computation:

$\hat{\mu}*\hat{\mu}(\varphi(x))=\hat{\mu}_{x}\tilde{\otimes}\hat{\mu}_{y}(\varphi(x\cdot
y))=\int_{S_{y}(\mathcal{G})}F_{\hat{\mu},\mathcal{G}}^{\varphi^{\prime}}d(\hat{\mu}|_{\mathcal{G}})=\int_{S_{y}(\mathcal{G})}F_{\mu,\mathcal{G}}^{\varphi^{\prime}}d\mu$
$=\mu_{x}\otimes\mu_{y}(\varphi(x\cdot
y))=\mu_{x}\boxtimes\mu_{y}(\varphi(x\cdot
y))=\lim_{D}(\mu_{i}\otimes\mu_{i}(\varphi(x\cdot y)))$
$=\lim_{D}(\mu_{i}*\mu_{i}(\varphi(x)))=\lim_{D}(\mu_{i}(\varphi(x)))=\mu(\varphi(x))=\hat{\mu}(\varphi(x)).\qed$

######  Theorem 5.5.

Let $(G_{i})_{i\in I}$ be a sequence of groups and $H_{i}$ be any sequence of
finite subgroups where $H_{i}\leq G_{i}$. Let $\mathcal{G}=\prod_{D}G_{i}$,
$\mathcal{G}\subset\mathcal{U}$, $\mu_{i}:=\frac{1}{|H_{i}|}\sum_{h\in
H}\delta_{h}$, and $\mu=\lim_{D}\mu_{i}$. If $\mathcal{G}$ is NIP, then
$\hat{\mu}$ is a generically stable idempotent Keisler measure.

###### Proof.

Follows directly from Theorem 1.1 and Theorem 5.4. ∎

######  Question 5.6.

Does the converse of Theorem 5.5 hold? If $\mathcal{G}=\prod_{D}G_{i}$ is NIP
and $\mu\in\mathfrak{M}_{x}(\mathcal{G})$ is a generically stable idempotent,
does there exist a sequence of finite subgrups $H_{i}$ of $G_{i}$ such that
$\lim_{D}\mu_{i}=\mu$ where $\mu_{i}=\frac{1}{|H|}\sum_{h\in H}\delta_{h}$. We
remark that Theorem 3.5 implies that $\mu$ is pseudo-finite, but does not
directly imply that we can find a sequence of measures of the correct form.

Assume furthermore that the that the normal subgroups of $(G_{i})_{i\in I}$
are uniformly definable, i.e. there exists a single formula
$\gamma(x,\bar{y})$ such that for any $i\in I$ and normal subgroup $H$ of
$G_{i}$, there exists a parameter $b\in G_{i}^{y}$ such that
$G_{i}\models\gamma(a,b)$ if and only if $a\in H$. Then one can show that
$K:=\\{i\in I:\lim_{n\to\infty}\mu_{i}^{(*n)}$ converges$\\}$ is in $D$ (by
using the equivalence of $(i)$ and $(v)$ in [12, Corollary 2.1]). It is
routine to check that if $\lim_{n\to\infty}\mu_{i}^{(*n)}$ converges, then it
converges to an idempotent. Consider the sequence of measures

$\nu_{i}:=\begin{cases}\begin{array}[]{cc}\lim_{n\to\infty}\mu_{i}^{(*n)}&i\in
K,\\\ \delta_{e}&otherwise.\end{array}\end{cases}$

Does $\lim_{D}\nu_{i}=\mu$? If not, under which conditions does the equalty
hold?

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