AN APPROXIMATE LOGIC FOR MEASURESisaac/omq-final.pdfAN APPROXIMATE LOGIC FOR MEASURES 3 apply these...

AN APPROXIMATE LOGIC FOR MEASURES

ISAAC GOLDBRING AND HENRY TOWSNER

Abstract. We present a logical framework for formalizing connections be-tween finitary combinatorics and measure theory or ergodic theory that haveappeared in various places throughout the literature. We develop the basicsyntax and semantics of this logic and give applications, showing that themethod can express the classic Furstenberg correspondence as well as provideshort proofs of the Szemeredi Regularity Lemma and the hypergraph removallemma. We also derive some connections between the model-theoretic notionof stability and the Gowers uniformity norms from combinatorics.

1. Introduction

Since the 1970’s, diagonalization arguments (and their generalization to ul-trafilters) have been used to connect results in finitary combinatorics with resultsin measure theory or ergodic theory [14]. Although it has long been known thatthis connection can be described with first-order logic, a recent striking paper byHrushovski [22] demonstrated that modern developments in model theory can beused to give new substantive results in this area as well.

Papers on various aspects of this interaction have been published from a va-riety of perspectives, with almost as wide a variety of terminology [30, 10, 9, 1,32, 13, 4]. Our goal in this paper is to present an assortment of these techniquesin a common framework. We hope this will make the entire area more accessible.

We are typically concerned with the situation where we wish to prove a state-ment 1) about su�ciently large finite structures which 2) concerns the finiteanalogs of measure-theoretic notions such as density or integrals. The classic ex-ample of such a statement is Szemeredi’s Theorem (other examples can be foundlater in the paper and in the references):

Theorem 1.1 (Szemeredi’s Theorem). For each ✏ > 0 and each k 2 N>0, there

is an N such that whenever n > N and A ✓ [1, n] := {1, . . . , n} with

|A|n

� ✏,

there exist a, d such that a, a + d, a + 2d, . . . , a + (k � 1)d 2 A.

Suppose this statement were false, and therefore that for some particular ✏,some k, and infinitely many n, there were sets A

n

✓ [1, n] serving as counter-examples to the theorem. For each n, we could view the set [1, n], together withthe subset A

n

and the function + mod n, as a structure Mn

of first-order logic.Using the ultraproduct construction, these finite models can be assembled into asingle infinite model M satisfying the Los Theorem, which, phrased informally,states:

Theorem 1.2. A first-order sentence is true in M if and only the sentence is

true in Mn

for “almost every” n.

2010 Mathematics Subject Classification. Primary 03B48; Secondary 03C20, 60A10.Key words and phrases. first-order logic, ultraproducts, measure theory.Goldbring’s work was partially supported by NSF grant DMS-1007144 and Towsner’s work

was partially supported by NSF grant DMS-1001528.

1

2 ISAAC GOLDBRING AND HENRY TOWSNER

The existence of a and d satisfying the theorem can be expressed by a first-order sentence, so if we could show that this sentence was true in the infinitestructure M, then the same sentence would be true in many of the finite structuresM

n

, which would contradict our assumption.In each of the finite models, we have a natural measure on [1, n], namely the

(normalized) counting measure µ

n

, and the assumption that |An|n

� ✏ is simplythe statement that µ

n

(An

) � ✏. We would like to produce a measure on M whichsatisfies some analog of the Los theorem.

One way to accomplish this is to expand the language of our models withmany new predicates for measures, one for each definable set S and rational �,and specify that the formula m

S,�

holds in Mn

exactly if µ

n

(S) < �. Then, forinstance, the failure of m

A,✏

in the finite structures Mn

implies the failure ofthe same formula in M. With a bit of work, these new predicates are enough toensure that we can construct a genuine measure, the Loeb measure (see [15]),on the structure M whose properties are related to those of the finite structuresvia the Los theorem. The extension of a language of first-order logic by thesepredicates has been used in [22], but the details of the construction have notappeared in print.

The applications of these predicates for measures are su�ciently compellingto justify the study of this extension of first-order logic in its own right. It wouldbe quite interesting to see certain results about first-order logic proven for thisextension; for example, some of the results in [7] can be viewed as studying theaddition of a “random” predicate P to a language of first-order logic, a notionwhich has natural application in the presence of measures if the result can beextended to include AML.

In this paper we define approximate measure logic, or AML, to be an exten-sion of first-order logic in which these new predicates are always present. Thisserves two purposes. First, to give the full details of extending first-order logicby measure predicates in a fairly general way. Second, we hope that providinga precise formulation will spur the investigation of the distinctive properties ofAML.

In general, a logic which intends to talk about measure has to make somekind of compromise, since there is a tension between talking about measures in anatural way and talking about the usual first-order notions (which, for instance,impose the presence of projections of sets). AML is not the first attempt at such acombination (see [24, 2]), but di↵ers because it stays much closer to conventionalfirst-order logic. The main oddity, relative to what a naive attempt at a logicfor measures might look like, is that the description of measures in the logic isapproximate: there can be a slight mismatch between which sentences are true ina structure and what is true about the actual measure. For instance, it is possibleto have a structure M together with a measure µ so that M satisfies a sentencesaying the measure of a set B is strictly less than ✏ while in fact the measure of B

is precisely ✏. This is necessary to accommodate the ultraproduct construction.The next three sections lay out the basic definitions of AML and describe some

properties of structures of AML, culminating in the construction of ultraproductswhich have well-behaved measures. In Section 5, we present two examples of sim-ple applications, giving a proof of the Furstenberg correspondence between finitesets and dynamical systems, and a proof of the Szemeredi Regularity Lemma.

In Section 6 we prove a Downward Lowenheim-Skolem theorem for AML. InSection 7, we illustrate several useful model-theoretic techniques in AML and

AN APPROXIMATE LOGIC FOR MEASURES 3

apply these to show a connection between definability, the model-theoretic no-tion of stability, and combinatorial quantities known as the Gowers uniformitynorms. We intend this section to be a survey of methods that have proven useful.These methods are either more fully developed in the references, or await a fullerdevelopment.

In this article, m and n range over N := {0, 1, 2, . . .}. For a relation R ✓ X⇥Y ,x 2 X, and y 2 Y , we set R

x := {y 2 Y : (x, y) 2 R} and R

y

:= {x 2 X : (x, y) 2R}.

The authors thank Ehud Hrushovski, Lou van den Dries, and the membersof the UCLA reading seminar on approximate groups: Matthias Aschenbrenner,Greg Hjorth, Terence Tao, and Anush Tserunyan, for helpful conversations onthe topic of this paper.

2. Syntax and Semantics

A signature for approximate measure logic (AML) is nothing more than afirst-order signature. Suppose that L is a first-order signature. One forms AMLL-terms as usual, by applying function symbols to constant symbols and variables.We let T denote the set of L-terms.

Definition 2.1. We define the L-formulae of AML as follows:• Any first-order atomic L-formula is an AML L-formula.• The AML L-formulae are closed under the use of the usual connectives ¬

and ^.• If ' is an AML L-formula and x is a variable, then 9x' and 8x' are AMLL-formulae.

• If ~x = (x1, . . . , xn

) is a sequence of distinct variables, q is a non-negativerational, ./2 {<,}, and ' is an L-formula, then m

~x

./ q.' is an L-formula.

The last constructor, which we will refer to as the measure constructor, in theabove definition behaves like a quantifier in the sense that any of the variablesx1, . . . , xn

appearing free in ' become bound in the resulting formula.From now on, ./ will always denote < or .

Convention 2.2. It will become convenient to introduce the following notations:• m

~x

� q.' is an abbreviation for ¬m

~x

< q.'

• m

~x

> q.' is an abbreviation for ¬m

~x

q.'

Remark 2.3. It is not necessary that AML contain precisely one measure foreach tuple length. A more general definition could allow multiple measures ateach arity. In this case, one would also want to track which measures on pairs,for instance, correspond to the products of which measures on singletons, andso on. There are even situations where it is natural to have a measure on pairswhich does not correspond to any product of measures on singletons.1 Furthergeneralizations could involve such things as measures valued in ordered rings. Afully general definition of AML would include in the signature L, for each arity, alist of measures together with the ordered ring in which they are valued. In this

1A typical example is the proofs in extremal graph theory of sparse analogs of density state-ments, as in the regularity lemma for sparse graphs [26], where the counting measure on pairsis replaced by counting the proportion of pairs from some fixed (typically small) set of pairs. Ifthe fixed set of pairs satisfies certain randomness properties, this measure cannot be viewed asany product measure.


paper we eschew these complications and focus only on the simplest case, with asingle real-valued measure; the changes needed to handle more general cases arelargely routine.

Particular examples of AML structures are given by finite structures with thenormalized counting measure and by structures with definable Keisler measures.However we would like to define an a priori notion of semantics for AML: anal-ogously to the definition of a structure of first-order logic, we would like to firststate that a certain object is an AML structure and then interpret formulae andsentences within that object. The heart of such a definition should be that anAML structure consists of a first-order structure together with a measure suchthat all definable sets are measurable. Since identifying the definable sets requiresinterpreting the logic, this makes it di�cult to recognize a valid AML structurewithout simultaneously interpreting the logic.

We adopt a compromise: an AML quasistructure is an object which would bean AML structure except that definable sets may not be measurable. We definethe rank of a formula to be the level of nesting of measure quantifiers, and wethen, by simultaneous recursion, identify the definable sets of rank n and thequasistructures in which all sets defined by formulae of rank n are measurable; aslong as all sets defined by formulae of rank n are measurable, it will be possible tointerpret formulae of rank n+1. An AML structure will then be a quasistructurein which this recursion continues through all ranks.

We face one additional obstacle: a first-order structure together with a mea-sure does not provide enough information to completely specify the semantics. Inorder to satisfy compactness, we need to allow the possibility that

M ✏ m

~x

< r + ✏.'(~x)

for all ✏ > 0, butM 6✏ m

~x

r.'(~x).

In such a structure, the measure of the set defined by ' should be r, but thisstructure must have a di↵erent semantics from a structure N where the measureof the set defined by ' is r and

N ✏ m

~x

r.'(~x).

To accomodate this, we pair the measure portion of a structure with a function v

which assigns to each set X a value �, �, or . The values � and indicate thatthe structure can only identify the measure of the set X as it is approximated fromabove or from below, respectively, while the value � indicates that the structureactually identifies the measure of the set X exactly. This issue can only occurwhen the value of the measure is rational, so if the measure of X is irrational, wealways set v(X) = �.

Definition 2.4. Let L be a first-order signature. An AML L-quasistructure con-sists of:

• a first-order L-structure M;• for each n � 1, an algebra BM

n

of subsets of M

n and a finitely additivemeasure µ

Mn

on BMn

such that, for any m, n, BMm+n

extends BMm

⌦BMn

andµ

Mm+n

extends µ

Mm

⇥ µ

Mn

;• for each set X 2 BM

n

, an assignment of a value v

Mn

(X) 2 {�, ,�} suchthat v

Mn

(X) = � whenever µ

Mn

(X) /2 Q.


If the structure M and the arity n are clear from context, we will write µ andv instead of µ

Mn

and v

Mn

.Suppose that M is an AML L-quasistructure. Let V be the set of variables

and suppose also that s : V ! M is a function; we call such an s a valuation on

M. Given any sequence of distinct variables ~v = (v1, . . . , vn

) and any sequence~c = (c1, . . . , cn

) from M , we let s(~c/~v) be the valuation obtained from s byredefining (if necessary) s at v

i

to take the value c

i

.We would now like to define the satisfaction relation M |= '[s] by recursion

on complexity of formulae in such a way that this extends the satisfaction relationfor first-order logic. However, in order to specify the semantics of the measureconstructor, one needs to know that the set defined by the formula followingthe measure constructor is measurable. Thus, we need to define the semantics instages while ensuring that the sets defined at each stage are measurable.

Towards this end, we define the rank of an AML L-formula ', denoted rk('),by induction:

• If ' is a classical L-formula, then rk(') = 0;• rk(¬') = rk(8x') = rk(');• rk(' _ ) = max(rk('), rk( ));• rk(m

~x

./ r.'(~x, ~y)) = rk(') + 1.Suppose that the satisfaction relation M |= '(~x)[s] has been defined for for-

mulae of rank n. (Note that this is automatic for n = 0.) As is customary, weoften write M |= '(~a) if '(~x) is a formula with free variables among ~x, s is avaluation for which s(x

i

) = a

i

, and M |= '[s]. If '(~x, ~y) is a formula of rank n

and b 2M

|~y|, we then set '(M,

~

b) := {a 2M

|~x| : M |= '(~a,

~

b)}. We also use thenotation v(',

~

b) to denote v('(M,

~

b)).We say that the AML L-quasistructure M satisfies the rank n measurability

condition if, for all formulae '(~x, ~y) of rank n and all b 2 M

|~y|, we have'(M, b) 2 BM

|~x|.Assume that M satisfies the rank n measurability condition. We now define

the satisfaction relation |= for rank n + 1 formulae. We treat the connectiveand quantifier cases as in first-order logic and only specify how to deal with thesemantics of the measure constructor. Suppose that '(~x, ~y) is a formula of rankn and s is a valuation such that s(~y) = ~

b. We then declare:

• M |= m

~x

< r.'(~x,

~

b) if and only if:– µ('(M,

~

b)) < r, or– µ('(M,

~

b)) = r and v(',

~

b) = .• M |= m

~x

r.'(~x;~b) if and only if:– µ('(M,

~

b)) < r, or– µ('(M,

~

b)) = r and v(',

~

b) 6= �.

Definition 2.5. We say that the AML quasistructure M is a structure if itsatisfies the rank n measurability condition for all n.

If ' has no free variables, then ' is called a sentence. If T is a set of sentencesand M is a structure, we write M |= T if and only if M |= ' for all ' 2 T ; in thiscase, we say that M is a model of T .

Example 2.6. Let L be the signature consisting of a single binary functionsymbol · and a single constant symbol e. Let G be a group with |G| = m. Weview G as a first-order L-structure G by interpreting · as multiplication in G and


e as the identity of G. In order to view G as an AML L-quasistructure, we setBG

n

:= P(Gn) and µ

n

(X) := |X|m

n , that is, we equip each cartesian power G

n withthe (normalized) counting measure. It is clear that the AML quasistructure G isactually a structure.

Now fix g 2 G and let '(x, y) be the first-order L-formula x · y = y · x. ThenG |= m

x

< r.'(g) if and only if the cardinality of C

G

(g), the centralizer of g inG, is < r ·m. In particular, if r < 1, then G 6|= 8ym

x

< r.' as the centralizer ofthe identity has cardinality m.

As in classical logic, we say that D ✓ M

n is definable (in M) if there is aformula '(~x, ~y), with |~x| = n, and ~a 2 M

|~y| such that D = '(M,~a). If A ✓ M

and the tuple ~a as above lies in A, we also say that D is A-definable (in M) ordefinable (in M) over A. We let Def

n

(M) denote the Boolean algebra of definablesubsets of M

n. A function h : M

m !M

n is definable (in M) if the graph of h isa definable subsets of M

m+n.Convention 2.7. From now on, when we say that M is an L-(quasi)structure,we mean that M is an L-(quasi)structure of AML. We use fraktur letters M andN (sometimes decorated with subscripts) to denote quasistructures. The corre-sponding roman letter denotes the underlying universe of the quasistructure. (So,for example, the underlying universe of M will be M and the underlying universeof N

i

will be N

i

.) Likewise, L-formulae will always mean AML L-formulae. If wewish to consider a structure from classical first-order logic, then we will speak ofa classical structure and denote it using calligraphic letters such as M. Likewise,we will speak of classical formulae.

Suppose L0 is signature extending the signature L. Suppose that M is an L0-quasistructure. Then the L-reduct of M, denoted M|L, is the L-quasistructureobtained from M by “forgetting” to interpret symbols from L0 \ L. We also referto M as an expansion of M|L.Definition 2.8. Suppose that M and N are L-structures. We say that M is asubstructure of N if:

(1) the first-order part of M is a substructure of the first-order part of N,and

(2) for each B 2 BNn

, we have B \M

n 2 BMn

.We further say that M is an elementary substructure of N if, for all formulae'(~x) and a 2M

|~x|, we have M |= '(~a) if and only if N |= '(~a).It may seem a bit strange that the notion of substructure does not mention the

measure part of the structures in question. However, given that the constructorm

~x

< r acts as a quantifier and arbitrary substructures need not respect thetruth of quantified formulae, this lack of a requirement need not be so surprising.

We define the notions of embedding and elementary embedding in the obviousway so as to align with our notion of substructure and elementary substructure.

3. The Canonical Measures on an @1-saturated AML structure

An extremely important source of AML structures are those that are equippedwith an actual measure.Definition 3.1. A measured L-structure is an AML L-structure M such thatthere is a measure ⌫n on a �-algebra Bn of subsets of M

n extending Defn

(M)such that:


(1) For definable B ✓M

n, we have ⌫n(B) = µ

Mn

(B).(2) For each m, n � 1, the measure ⌫

m+n is an extension of the productmeasure ⌫m ⌦ ⌫n

(3) (Fubini property) For every m, n � 1 and B 2 Bm+n, we have(a) for almost all x 2M

m, B

x 2 Bn;(b) for almost all y 2M

n, B

y

2 Bm;(c) the functions x 7! ⌫

n(Bx) are y 7! ⌫

m(By

) are ⌫m- and ⌫n-measurable,respectively, and

⌫

m+n(B) =Z⌫

n(Bx)d⌫m(x) =Z⌫

m(By

)d⌫n(y).

For example, any finite L-structure equipped with its normalized countingmeasure is a measured structure; in fact, these structures (and their ultraprod-ucts) are our primary examples of measured structures.

Even if an AML structure does not come equipped with a measure, often thereis still a “natural” measure that can be placed on the structure, a process thatwe now describe. First, we say that an L-structure M is @1-compact if whenevern � 1 and (D

m

: m 2 N) is a family of definable subsets of M

n with thefinite intersection property, then

Tm2N D

m

6= ;. (If the signature L is countable,then this coincides with the notion of an @1-saturated structure more commonlyencountered in model theory; when the signature is countable, we will often usethe term saturated rather than compact.) For example, any finite L-structureis @1-compact; many ultraproducts are also @1-compact. In this section, we willshow that certain @1-compact AML structures can be equipped with a family ofcanonical measures, the so-called Loeb measures.

We write �(B) for the �-algebra generated by the algebra B.Suppose that M is an @1-compact AML L-structure and (A

m

: m 2 N)is a family of definable subsets of M

n for whichS

m2N A

m

is also definable.Then, by @1-compactness, we have that

Sm2N A

m

=S

k

m=0 A

m

for some k 2N. Consequently, µ

Mn

|Defn

(M) is a pre-measure. Thus, by the Caratheodoryextension theorem, there is a measure, which we again call µ

Mn

or simply µ

n

, on�(Def

n

(M)), extending the original µ

Mn

given by

µ

n(B) := inf{X

m

µ

Mn

(Dm

) : B ✓[

m

D

m

, D

m

2 Defn

(M) for all m}.

Moreover, if µ

Mn

|�(Defn

(M)) is �-finite, then µ

n is the unique measure on �(Defn

(M))extending the original µ

Mn

. Observe, however, that by @1-compactness again,µ

Mn

|Defn

(M) is �-finite if and only if µ

Mn

|Defn

(M) is finite (which occurs ifand only if µ

M1 (M) is finite by the product property). We refer to the family of

measures (µMn

) as the canonical measures on M. Although they are defined forall @1-compact L-structures, we mainly consider them in the case when µ

M1 (M)

is finite, for then they are truly “canonical.” (Indeed, when a pre-measure on analgebra A is not �-finite, there are many extensions of it to a measure on �(A);see Exercise 1.7.8 in [29].)

Until further notice, we assume that µ

M1 |Def1(M) is a finite pre-measure

on Def1(M) (and hence µ

Mn

|Defn

(M) is a finite pre-measure on Defn

(M) forall n � 1). Let �(Def

m

(M)) ⌦ �(Defn

(M)) denote the product �-algebra andµ

m

⌦ µ

n

denote the product measure. Observe that �(Defm

(M))⌦ �(Defn

(M))is generated by sets of the form A⇥B, with A 2 Def

m

(M)) and B 2 Defn

(M));since A⇥B 2 Def

m+n

(M) for such A, B, we have �(Defm

(M))⌦�(Defn

(M)) ✓


�(Defm+n

(M)). Now observe that, if A 2 Defm

(M) and B 2 Defn

(M), thenµ

m+n

(A⇥B) = µ

m

(A) · µn

(B) by the Product property. Thus

E := {D 2 �(Defm

(M))⌦ �(Defn

(M)) : µ

m+n

(D) = (µm

⌦ µ

n

)(D)}contains all sets of the form A ⇥ B, where A 2 Def

m

(M) and B 2 Defn

(M).One can easily check that E is a �-class. Since the sets of the form A⇥B, whereA 2 Def

m

(M) and B 2 Defn

(M), form a ⇡-class, we have, by the ⇡-� Theorem,that �(Def

m

(M))⌦ �(Defn

(M)) ✓ E. (For the definitions of �- and ⇡-classes aswell as the statement of the ⇡-�-theorem, see Section 17.A of [23].) Consequently,we see that µ

m

⌦ µ

n

= µ

m+n

|�(Defm

(M))⌦ �(Defn

(M)). This proves:

Lemma 3.2. If M is an @1-compact L-structure such that µ

M(M) is finite, then

M, equipped with its canonical measures, satisfies the first two axioms in the

definition of a measured structure.

The third axiom, requiring that Fubini’s Theorem hold, might still fail for setswhich do not belong to �(Def

m

(M)) ⌦ �(Defn

(M)). We call an @1-compact L-structure whose canonical measures satisfy the Fubini property a Fubini structure.(The measurability of the functions in (3) always holds for the canonical measures,so really a Fubini structure is one where the displayed equation in the statement ofthe Fubini property is required to hold.) Thus, by the preceding lemma, a Fubinistructure, equipped with its canonical measures, is a measured L-structure.

The following situation is also important:

Definition 3.3. Suppose that M is a classical L-structure which is equippedwith a family of measures ⌫n on a �-algebra Bn of subsets of Mn extending thealgebra of definable sets and such that ⌫m+n is an extension of ⌫m ⌦ ⌫

n for allm, n � 1. We call (M, (⌫n)) a classical measured structure. Then we can turn Minto an L-quasistructure M by declaring, for B ✓M

n, that v

Mn

(B) = �. We callM the AML quasistructure associated to (M, (⌫n)).

Remark 3.4. Usually, the AML quasistructure associated to a classical measuredstructure is not a structure: due to the added expressivity power, there are newdefinable sets which may not be measurable. (However this quasistructure willsatisfy the rank 0 measurability condition, and so satisfaction for formulae ofrank 1 can be defined.) For finite classical measured structures (which are thestructures most important for our combinatorial applications), the associatedAML quasistructure is in fact a structure because all sets are measurable.

Remark 3.5. Consider M, an @1-compact L-structure with µ

M1 (M) finite, equipped

with its family (µMn

) of canonical measures. Let M be the classical L-structureobtained from M by considering only the interpretations of the symbols in L.Then (M, (µM

n

)) is a classical measured structure. Let M0 be the AML qua-sistructure associated to (M, (µM

n

)). How do M and M0 compare? Suppose that'(x) is a classical L-formula and r 2 Q�0. It is straightforward to see thatM |= m

x

r.'(x) implies M0 |= m

x

r.'(x). However, the converse need nothold. For example, suppose that M0 |= m

x

0.'(x), whence µ('(M)) = 0. How-ever, if v

M(') = �, then M |= ¬m

x

0.'(x). (Such phenomena can, and will,happen in ultraproducts, as we will see in later sections.) Thus, in some sense,M0 is a “completion” of M.

We now briefly turn to the question of what our logic can say about these canon-ical measures. We present a couple of examples. Suppose that (X,S) is a measur-able space such that {x} 2 S for every x 2 X. (This happens, for example, when


(X,S) = (Mn

,�(Defn

(M))) for M an L-structure.) Then a measure µ on X issaid to be continuous if µ({x}) = 0 for each x 2 X. The following is immediate.

Proposition 3.6. Suppose that M is an @1-compact L-structure. Then we have

M |= {8xm

y

q(x = y) : q 2 Q>0} if and only if µ

1is a continuous measure.

More generally, there is a set �(x) consisting of L-formulae containing a singlefree variable x such that, for every L-structure M and every a 2M ,

µ

1({a}) = 0,M |= '(a) for all ' 2 �.

Indeed, take �(x) := {my

< q.(x = y) : q 2 Q>0}. In model-theoretic terms,the set of elements of µ

1-measure 0 is uniformly type-definable in all @1-compactL-structures. Below, as a consequence of the ultraproduct construction, we willsee that this set is not uniformly definable in all L-structures.

Here is another proposition along the same lines.

Proposition 3.7. There is a set T of AML sentences such that, for any @1-

compact L-structure M with µ

M1 (M) finite, M |= T if and only if µ

1is a proba-

bility measure.

Proof. Take T := {mx

1(x = x)} [ {¬m

x

q(x = x) : q 2 (0, 1) \Q}. ⇤Proposition 3.8. There is a set T of AML sentences such that, for any @1-

compact L-structure M with µ

M1 (M) finite, M |= T if and only if M is a Fubini

structure.

Proof. We take T to consist of the following two schemes:(1) For every '(~x, ~y, ~x), (~x,~z) and q, r, t with t < qr,

8~z⇥�8~x( ! m

~y

� r.') ^m

~x

� q.

�! m

~x,~y

> t.(' ^ )⇤,

(2) For every '(~x, ~y, ~x), (~x,~z) and q, r, t with qr < t,

8~z⇥�8~x( ! m

~y

r.') ^m

~x

q.

�! m

~x,~y

< t.(' ^ )⇤.

Note that may not contain variables from ~y.Suppose M ✏ T . It su�ces to show that for any definable set B ✓ M

m+n,we have µ

m+n(B) =R

µ

n(B~x)dµ

m(~x). Fix a formula '(~x, ~y,~z) and ~c 2 M

|~z

so that B = '(M,~c). Given rationals r, r

0, we set B

r,r

0 = {~x : M ✏ m

~y

�r.'(~x, ~y,~c)^m

~y

< r

0.'(~x, ~y,~c)}; we will also write B

r,r

0 as an abbreviation for itsdefining formula.

Observe that we can approximate the function ~x 7! µ

n(B~x) by step functionsof the form

Pi

r

i

�

Bri,ri+1for sequences of rationals r1 < · · · < r

k

. Let r1 < · · · <r

k

be such a sequence of rationals. For any rational q

i

< µ

m(Bri,ri+1), we must

have M ✏ m

~x

� q

i

.B

ri,ri+1 . Since M ✏ T , we must have

M ✏ m

~x,~y

> t.(' ^B

ri,ri+1)

for every t < q

i

r

i

, and so µ

m+n({(~x, ~y) 2 B : ~x 2 B

ri,ri+1}) � q

i

r

i

. Since theB

ri,ri+1 are pairwise disjoint, it follows that µ

m+n(B) �P

i

q

i

r

i

. Since we maychoose q

i

, r

i

as above so thatP

i

q

i

r

i

is arbitrarily close toR

µ

n(B~x)dµ

m(~x), wehave µ

m+n(B) �R

µ

n(B~x)dµ

m(~x).Similarly, for any q

i

> µ

m(Bri,ri+1), we must have M ✏ m

~x

q

i

.B

ri,ri+1 , andsince M ✏ T , we also have

M ✏ m

~x,~y

< t.(' ^B

ri,ri+1)


for any t > q

i

r

i+1. Therefore µ

m+n({(~x, ~y) 2 B : ~x 2 B

ri,ri+1}) q

i

r

i+1, andso µ

m+n(B) P

i

q

i

r

i+1. Again, sinceP

i

q

i

r

i+1 can be made arbitrarily close toRµ

n(B~x)dµ

m(~x), we have µ

m+n(B) R

µ

n(B~x)dµ

m(~x).Now suppose M is Fubini; we must show that every sentence in T holds.

For the first family of sentences, suppose M ✏ 8~x( ! m

~y

� r.') ^ m

~x

�q. (with parameters ~c 2 M

|~z|). Setting C = (M,~c), we see that µ(C) � q.Let B = '(M,~c). For each ~a 2 C, µ(B~a) � r. Since µ is Fubini, µ

m+n(B) =RC

µ

n(B~x)dµ

n(~x) � qr, and therefore for each t < qr we have M ✏ m

~x

> t.('^ ).For the second family of sentences, suppose M ✏ 8~x( ! m

~y

r.') ^m

~x

q. . Again setting C = (M,~c) and B = '(M,~c), for each ~a 2 C, µ(B~a) r.Since µ is Fubini, µ

m+n(B) =RC

µ

n(B~x)dµ

n(~x) qr, and therefore for eacht > qr we have M ✏ m

~x

< t.(' ^ )[s]. ⇤Remark 3.9. Fix T as in the proof of the previous proposition. The proof wehave given above shows that M |= T for any measured structure M.

4. Ultraproducts

We suppose that the reader is familiar with the standard ultraproduct con-struction from first-order logic. We sometimes use the phrase “P (i) holds a.e.”to mean that the set of i’s for which P (i) holds belongs to the ultrafilter.

Suppose that (Mi

: i 2 I) is a family of L-structures and suppose that Uis an ultrafilter on I. We let M denote the classical ultraproduct of the family(M

i

) with respect to U . We declare A 2 BMn

if and only if A =Q

U A

i

, whereeach A

i

2 BMin

; in this case, we declare µ

Mn

(A) = limU µ

Min

(Ai

) (the ultralimit ofµ

Min

(Ai

)). Observe that limU (µMin

(Ai

)) always exists and is unique since [0,1] isa compact hausdor↵ space.

If µ

Mn

(A) = r 2 Q, we set

v

Mn

(A) =

8>>>>>><

>>>>>>:

� if µ

Min

(Ai

) > r a.e. if µ

Min

(Ai

) < r a.e.� if µ

Min

(Ai

) = r and v

Min

(Ai

) = � a.e. if µ

Min

(Ai

) = r and v

Min

(Ai

) = a.e.� if µ

Min

(Ai

) = r and v

Min

(Ai

) = � a.e.

It is reasonably straightforward to check that each BMn

is an algebra and µ

Mn

isa finitely additive measure on BM

n

, whence the ultraproduct M is a quasistructure.Note that the assignment of ⌫M

n

(A) is consistent with the interpretations of� and as indicating approximation from above or below: if µ

Min

(Ai

) > µ

Mn

(A)a.e., we have obtained the measure µ

Mn

(A) by approximations from above, thus(irrespective of whether the values of µ

Min

(Ai

) were obtained exactly or obtainedby approximation themselves) v

Mn

(A) = �; if µ

Min

(Ai

) = r and v

Min

(Ai

) = �a.e. then almost every µ

Min

(Ai

) was found by approximations from above, so themeasure of µ

Mn

(A) is found the same way.The following theorem is of fundamental importance for our applications of

AML.

Theorem 4.1 ( Los Theorem for AML). Suppose that (Mi

: i 2 I) is a family of

L-structures, U is a ultrafilter on I, and M =Q

U Mi

is the ultraproduct of the

family (Mi

). Let ' be an L-formula and s a valuation on M. For each i 2 I, let

s

i

be a valuation on Mi

such that, for each variable v, we have s(v) = [si

(v)]U .

Then M |= '[s],Mi

|= '[si

] for U-almost all i.


Proof. As in the proof of the classical Los theorem, we proceed by inductionon the complexity of formulae. We only need to explain how to deal with themeasure constructor cases, that is, we consider a formula '(~x, ~y) such that thetheorem holds true for ' and we show that it holds for m

~x

./ r.'(~x, ~y), where./2 {<,}. We fix valuations s

i

and s as in the statement of the theorem and set~

b

i

:= s

i

(~y) and ~b := s(~y). We will repeatedly use the inductive hypothesis that'(M,

~

b) =Q

U '(Mi

,

~

b

i

).If µ

Mi('(Mi

,

~

b

i

)) < r a.e. then Mi

✏ m

~x

< r.'(~x,

~

b

i

) and Mi

✏ m

~x

r.'(~x,

~

b

i

)a.e.; also either µ

M('(M,

~

b)) < r or both µ

M('(M,

~

b)) r and v

M(',

~

b) = ,and in either case M ✏ m

~x

< r.'(~x,

~

b) and M ✏ m

~x

r.'(~x,

~

b).If µ

Mi('(Mi

,

~

b

i

)) > r a.e. then, symmetrically, we have Mi

✏ m

~x

> r.'(~x,

~

b

i

)and M

i

✏ m

~x

� r.'(~x,

~

b

i

) a.e., and also M ✏ m

~x

> r.'(~x,

~

b) and M ✏ m

~x

�r.'(~x,

~

b).Finally, if µ

M('(Mi

,

~

b

i

)) = r a.e. then v

M('(M,

~

b)) = v

Mi('(Mi

,

~

b

i

)) a.e., soM ✏ m

~x

< r.'(~x,

~

b) i↵ Mi

✏ m

~x

< r.'(~x,

~

b

i

) a.e. and M ✏ m

~x

r.'(~x,

~

b) i↵M

i

✏ m

~x

r.'(~x,

~

b

i

). ⇤

Corollary 4.2. The ultraproduct of structures is a structure.

Proof. Definable subsets of the ultraproduct are ultraproducts of definable setsin the factor structures. ⇤

Corollary 4.3. With the same hypotheses as in the previous theorem, if � is an

L-sentence, then

M |= � , Mi

|= � for U-almost all i.

Corollary 4.4 (Compactness Theorem for AML). If ⌃ is a set of L-sentences

such that each finite subset of ⌃ has a model, then ⌃ has a model.

Proof. The usual ultraproduct proof of the Compactness theorem from the Lostheorem applies; see, for example, Corollary 4.1.11 of [6]. ⇤

Corollary 4.5. For any L-structure M, any set I, and any nonprincipal ultra-

filter U on I, the diagonal embedding j : M ! MUgiven by j(a) = [(a)]U is an

elementary embedding.

The following result is standard in the classical context; this proof carries overimmediately to the framework of AML.

Proposition 4.6. Suppose that U is a countably incomplete ultrafilter on a set

I and suppose that L is countable. Suppose that (Mi

: i 2 I) is a family of

L-structures. Then

QU M

i

is @1-saturated. In particular, if I = N and U is any

nonprincipal ultrafilter on N, then

QU M

i

is @1-saturated.

Consequently, an ultraproduct M for which µ

M1 is finite can be equipped with

its canonical family of measures. (Observe that, by the Los theorem, µ

M1 is finite

if and only if the sequence (µMi1 ) is “essentially bounded.”) These measures are

instrumental for our applications of AML to problems in finite combinatorics andergodic theory.

We can also use the Los theorem to show that properties of the canonicalmeasures on the factor models are inherited by the canonical measures on theultraproduct.


Corollary 4.7. Suppose that (Mi

: i 2 I) is a family of Fubini structures, U is

a nonprincipal ultrafilter on I, and M =Q

U Mi

. Suppose further that there is a

c 2 R>0such that µ

M1 (M

i

) c for U-almost all i. Then M is a Fubini structure.

Proof. This is immediate from Proposition 3.8 and the Los theorem. ⇤Corollary 4.8. Suppose that (M

i

: i 2 I) is a family of @1-compact L-structures

such that µ

Mi1 is a continuous measure for almost all i. Suppose that U is a

nonprincipal ultrafilter on I and M =Q

U Mi

. Then µ

M1 is also a continuous

measure.

Proof. This is immediate from Proposition 3.6 and the Los theorem. ⇤Remark 4.9. Under some set-theoretic assumptions, the converse of the abovecorollary holds. A nonprincipal ultrafilter U on a (necessarily uncountable) indexset I is said to be countably complete if whenever (D

m

: m 2 N) is a sequenceof sets from U such that D

m

◆ D

m+1 for all m, we haveT

m2N D

m

2 U . Withthe same hypotheses as in the previous corollary, if we further assume that U iscountably complete, then continuity of µ

M1 implies continuity of almost all µ

Mi1 .

The existence of a countably complete ultrafilter is tantamount to the existenceof a measurable cardinal, which is a fairly mild set-theoretic assumption that mostset theorists are willing to assume when necessary. However, it is straightforwardto check that if U is countably complete and each M

i

is finite, thenQ

U Mi

willalso be finite, making such ultraproducts unusable for many purposes.

Corollary 4.10. There does not exist an L-formula '(x) such that for every

@1-compact L-structure M, we have

µ

M1 ({a}) = 0 , M |= '(a).

Proof. Suppose, towards a contradiction, that such an L-formula '(x) exists. Fori � 1, let M

i

be the classical measured L-structure whose universe is {1, . . . , i},whose measures are given by the normalized counting measures, and which inter-prets the L-structure in an arbitrary fashion. Let M

i

be the AML L-structure as-sociated with M

i

. Let U be a nonprincipal ultrafilter on N>0 and let M =Q

U Mi

.It is easy to see that µ

M1 (M) 1 and µ

M1 ({a}) = 0 for any a 2M . It follows that

M |= 8x'(x). By Los, we have that Mi

|= 8x'(x) for some i 2 N>0. Since Mi

is @1-compact, it follows that each m 2 M

i

has measure 0, which is a contradic-tion. ⇤Corollary 4.11. There does not exist an L-sentence ' such that, whenever Mis an @1-compact structure with µ

M1 (M) finite, then M |= ' if and only if µ

M1 is

a probability measure.

Proof. Suppose, towards a contradiction, that such a sentence ' exists. For n � 1,let M

n

be a finite L-structure such that µ

M1 (M

n

) = 1 � 1n

. Let M :=Q

U Mn

,where U is a nonprincipal ultrafilter on N>0. Observe that µ

M1 (M) = 1, whence

M |= '. Thus, Mn

|= ' for some n, contradicting the fact that µ

Mn1(Mn

) <

1. ⇤Corollary 4.12. There does not exist a set T of L-sentences such that, for any

L-structure M, M |= T if and only if µ

M1 is �-finite.

Proof. Suppose, towards a contradiction, that there is a set T of L-sentencessuch that M |= T if and only if µ

M1 is �-finite. Let M be a countably in-

finite set equipped with the unnormalized counting measure ⌫(B) = |B| and


turn M into an AML L-structure M by interpreting the symbols in L in anarbitrary fashion. Note that M |= T because M is the union of its singletons,which are definable and have measure 1. Let U be a nonprincipal ultrafilter onN and let N := MU . Then, by the Los theorem, we have that N |= T , so µ

N1

is �-finite. Since N is @1-compact, we have that µ

N1 is a finite measure. Thus,

for some n 2 N, N |= ¬9x1 · · ·9xn

(V

n

i=1 ¬m

x

< 1.(x = x

i

)). Consequently,M |= ¬9x1 · · ·9xn

(V

n

i=1 ¬m

x

< 1.(x = x

i

)), contradicting the fact that M con-tains infinitely many singletons of measure 1. ⇤

We now turn to the connection between integration and ultraproducts. Sup-pose that (M

i

: i 2 I) is a family of L-structures and h

i

: M

m

i

! M

n

i

arefunctions. We will say that the family (h

i

: i 2 I) is uniformly definable if thereexists a formula '(~x, ~y,~z), where |~x| = m and |~x| = n, and tuples ~c

i

2 M

|~z|i

such that, for all i 2 I, ~a 2 M

m

i

and ~b 2 M

n

i

, we have h

i

(~a) = ~

b if and onlyif M

i

|= '(~a,

~

b,~c

i

). If M =Q

U Mi

and ~c := [~ci

]U , then by Los, we have that'(~x, ~y,~c) defines a function h : M

m ! M

n, which we call the ultraproduct of theuniformly definable family (h

i

).We will also need to consider families of functions. Suppose (M

i

: i 2 I) isa family of measured L-structures, and, for each i, g

i

: M

m

i

! R is a function.We then say the family (g

i

: i 2 I) is uniformly layerwise definable

2 if, for eachq 2 Q, there is a formula '

q

(~x,~z) and tuples ~ci

2 M

|~z|i

such that, for ~a 2 M

m

i

,g

i

(~a) < q if and only if Mi

|= '

q

(~a,~c

i

).In the following theorem, the bound of 1 can easily be replaced by any positive

real number.

Theorem 4.13. Suppose that (Mi

: i 2 I) is a family of measured L-structures

with µ

1i

(Mi

) 1 for each i 2 I. Suppose that U is a nonprincipal ultrafilter on I

and M :=Q

U Mi

is equipped with its canonical measures (µm). For each i 2 I,

let g

i

: M

n

i

! [�1, 1] be a function and suppose that the family (gi

: i 2 I) is

uniformly layerwise definable. Then there is an expansion M0of M to a structure

and a measurable (with respect to BM0,n

) function g : M

n ! [�1, 1] such that,

for every uniformly definable family (hi

: i 2 I) of functions of m variables with

ultraproduct h, we have

R(g � h) dµ

m = limUR

(gi

� h

i

) dµ

m

i

.

Proof. Consider the extension L0 of L obtained by adding, for each q 2 Q\[�1, 1],a new unary predicate G

q

. We define expansions M0i

of Mi

by interpreting G

M0i

q

={~x 2 M

n

i

: g

i

(~x) < q}; since this is a definitional expansion, M0i

is indeed an L0-structure. Let M0 :=

QU M0

i

, and define g(~x) = inf{q 2 Q>0 : M0 ✏ G

q

(~x)}. It isclear that g is BM0

,n-measurable.Now suppose that (h

i

: i 2 I) is a uniformly definable family of functions, saydefined by '(~x, ~y) (and some parameters, which we suppress for sake of exposi-tion), with ultraproduct h. Set m := |~x| and n := |~y|. Fix c 2 Q>0 and choosea partition �1 = q0 < q1 < · · · < q

k

= 1 of [�1, 1] by rational numbers suchthat that |q

j

� q

j+1| < 1/c for each j < k. Fix also some rational q

k+1 > 1. For0 j k, set

S

qj ,qj+1

i

:= {~x 2M

m

i

: q

j

(gi

� h

i

)(~x) < q

j+1}

and take S

qj ,qj+1 =Q

U S

qj ,qj+1

i

, whence µ

m(Sqj ,qj+1) = limU µ

m

i

(Sqj ,qj+1

i

).

2This notion of definability of a function into R di↵ers from the usual notion in the literature,which requires the inverse image of a closed set be type-definable.


We have by definition thatX

j

q

j+1µm

i

(Sqj ,qj+1

i

)� 1c

Z

(gi

� h

i

)dµ

m

i

X

j

q

j+1µm

i

(Sqj ,qj+1

i

).

On the other hand, the sets S

qj ,qj+1 partition M

m and if ~x 2 S

qj ,qj+1 then, sinceg � h(~x) = limU g

i

� h

i

(~xi

), q

j

g � h(~x) q

j+1, so alsoX

j

q

j+1µm(Sqj ,qj+1)� 1

c

Z

(g � h)dµ

m X

j

q

j+1µm(Sqj ,qj+1).

Therefore we have | limUR

(gi

�h

i

)dµ

m

i

�R

(g �h)dµ

m| 1c

. Letting c go to 1,we have the desired result. ⇤

We call the function g as in the conclusion of the previous theorem the ultra-

product of the family (gi

). Observe that g(~x) = limU g

i

(~xi

) for ~x = [~xi

]U . We willparticularly be interested in the case that I = N, in which case, using the samenotation as in the statement of the previous theorem, if lim

i!1R

(g�hi

) dµ

m

i

= r,then

R(g � h) dµ

m = r.

Remark 4.14. In the previous theorem, one could drop the requirement thatthe family (g

i

) be uniformly layerwise definable and instead only require thateach g

i

be measurable. However, the M0i

and, consequently, M0, would only bequasistructures rather than structures.

5. Correspondence Theorems

5.1. The Furstenberg Correspondence Principle. One application of AMLis to give a general framework for correspondence principles between finite struc-tures and dynamical systems. We will only present a proof of the original corre-spondence argument given by Furstenberg [14]; since then, many variations andgeneralizations have been produced (for instance, [5]). The method given extendsimmediately to these generalizations.

Definition 5.1. Let E ✓ Z. The upper Banach density of E, d(E), is

lim supm�n!1

|[n, m] \ E|m� n

.

Definition 5.2. A dynamical system is a tuple (Y,B, µ, T ), where (Y,B, µ) is aprobability space and T : Y ! Y is a bimeasurable bijection for which µ(A) =µ(TA) for all A 2 B.

Furstenberg’s original correspondence can be stated as follows:

Theorem 5.3 (Furstenberg). Let E ✓ Z with positive upper Banach density

be given. Then there is a dynamical system (Y,B, µ, T ) and a set A 2 B with

µ(A) = d(E) such that for any finite set of integers U ,

d(\

i2U

(E � i)) � µ(\

i2U

T

�i

A).

Proof. Let (✏N

: N 2 N) be an increasing sequence of positive rational numberssuch that sup

N

✏

N

= d(E). Then for each N , we may find n, m such that m�n >

N and |[n,m]\E|m�n

� ✏

N

. Let f

n,m

: [n, m] ! [n, m] be the function such thatf

n,m

(x) = x+1 when x < m and f

n,m

(m) = n. We consider the classical measuredstructure ([n, m], E\[n, m], f

n,m

) equipped with the normalized counting measureand consider the corresponding AML structure M

n,m

.


Let (Y, A, T ) be the ultraproduct of the Mn,m

’s with respect to some nonprin-cipal ultrafilter on N. Let B := �(Def1(Y )) and let µ be the canonical measureon B. It is clear from Los’ theorem that (Y,B, µ, T ) is a dynamical system. Nextobserve that, by our construction, µ(A) = d(E). Now notice that, for any finiteset of integers U , any rational � < µ(

Ti2U

T

�i

A), and any N , we may find m, n

such that m� n > N and|[n, m] \

Ti2U

f

�i

n,m

(E \ [n, m])|m� n

> �.

Fix � > 0. Then for su�ciently large N , we also have that|[n, m] \

Ti2U

(E � i)|m� n

> � � �.

Consequently, d(T

i2U

(E � i)) � µ(T

i2U

T

i

A). ⇤

5.2. The Regularity Lemma.

Definition 5.4. Let M be an AML structure, let A ✓ M be a set, let n be apositive integer, and let I ✓ [1, n] be given. We define B0

n,I

(A) to be the Booleanalgebra of subsets of M

n generated by sets of the form

{(x1, . . . , xn

) 2M

n : M |= '(x1, . . . , xn

)},where ' is a formula with parameters from A whose free variables belong to I.

When k n, we define B0n,k

(A) to be the Boolean algebra generated by thealgebras B0

n,I

(A) as I ranges over subsets of [1, n] of cardinality k.In all cases, we drop 0 to indicate the �-algebra generated by the algebra.

When A = ;, we omit it and write Bn,k

.

These algebras were introduced, in a somewhat di↵erent context, by Tao in[30, 31].

Definition 5.5. Suppose that (G, E) is a finite (undirected) graph, U, U

0 ✓ G

are nonempty, and ✏ is a positive real number.(1) We set d(U, U

0) := |E\(U⇥U

0)|U⇥U

0 .(2) We say that U and U

0 are ✏-regular if whenever V ✓ U, V

0 ✓ V

0 are suchthat |V | � ✏|U | and |V 0| � ✏|U 0|, then we have

|d(U, U

0)� d(V, V

0)| < ✏.

Theorem 5.6 (Szemeredi’s Regularity Lemma, [28]). For any k and any ✏ >

0, there is a K � k such that whenever (G, E) is a finite graph, there is an

n 2 [k, K] and a partition G = U1 [ · · · [ U

n

such that, setting B = {(i, j) :U

i

and U

j

are not ✏-regular}, we have

��

[

(i,j)2B

U

i

⇥ U

j

�� ✏|G|2.

Proof. For a contradiction, suppose the conclusion fails. Then we may find k, ✏

such that for every K � k there is a finite graph (GK

, E

K

) with no partition intoat least k but at most K components satisfying the theorem. The partition of agraph into singleton elements consists entirely of ✏-regular pairs, so in particular|G

K

| > K for all K.


Let L0 := {E}, where E is a binary predicate symbol. Inductively assumethat AML signatures L0 ✓ L1 ✓ L2 ✓ · · · have been constructed. Then forany two AML L

n

-formulae '(x) and (x), at least one of which does not belongtoS

n�1i=1 Li

, with only the displayed free variable, we add new unary predicatesR

',

(x) and S

',

(x) to Ln+1. We let L :=

S1n=1 Ln

. We define finite measuredL-structures G

K

as follows:• The universe of G

K

is G

K

;• The measures on G

K

are given by the normalized counting measure;• E is interpreted in G

K

by E

K

; and• R

',

, S

',

are interpreted by induction on formulae. Having interpreted', , we may set U = '(G

K

) and U

0 = (GK

). If U, U

0 are not ✏-regularthen set R

GK',

= S

GK',

= ;. Otherwise, choose V ✓ U, V

0 ✓ U

0 witnessingthe failure of ✏-regularity and set R

GK',

= V and S

GK',

= V

0.Let G be the ultraproduct of the G

K

. We work in the canonical measure spaceon G

2. From here on, we will identify formulae ' with their interpretation '(G).Let h := E(�

E

| B2,1), the conditional expectation of �E

with respect to the�-algebra B2,1. Since h can be approximated by simple functions, we may write

h =X

iK0

↵

0i

�

Ci⇥Di + h

0

where ||h0||L

2 < ✏

4/4 and C

i

, D

i

are definable. Let U1, U2, . . . , Un

be the atoms ofthe finite algebra generated by {C

i

} [ {Di

}. Then we have

h =X

i,jn

↵

i,j

�

Ui⇥Uj + h

00

with ||h00||L

2 ||h0||L

2 and {Ui

}in

is a finite partition of G into definable sets.Let B be the collection of (i, j) such that R

Ui,Uj (and therefore S

Ui,Uj ) are

non-empty, and for each (i, j) 2 B, define �i,j

=µ(E\(RUi,Uj

⇥SUi,Uj))

µ(RUi,Uj)µ(SUi,Uj

) . By Los’

theorem, for each (i, j) 2 B, we have |�i,j

� ↵i,j

| � ✏. Let B

+ ✓ B be those (i, j)such that ↵

i,j

� �

i,j

� ✏. Suppose µ(S

(i,j)2B

U

i

⇥ U

j

) � ✏/2. We may assumeµ(S

(i,j)2B

+ U

i

⇥ U

j

) � ✏/4 (for otherwise we carry out a similar argument withB \B

+).Again by Los’ theorem, for each (i, j) 2 B

+, µ(RUi,Uj ) � ✏µ(U

i

) and µ(SUi,Uj ) �

✏µ(Uj

). Set Z =S

(i,j)2B

+(RUi,Uj ⇥ S

Ui,Uj ). Note that Z 2 B2,1 and

0 =Z�

E

�

Z

� h�

Z

dµ =Z�

E

�

Z

�X

i,jn

↵

i,j

�

Ui⇥Uj�Z

dµ +Z

h

0�

Z

dµ

so we have��Z

h

0�

Z

dµ

�� =

��

Z X

i,jn

↵

i,j

�

Ui⇥Uj�Z

� �E

�

Z

dµ

��

=

��

X

(i,j)2B

+

Z↵

i,j

µ(RUi,Uj )µ(S

Ui,Uj )� �E

�

RUi,Uj⇥SUi,Uj

dµ

��

�X

(i,j)2B

+

↵

i,j

µ(RUi,Uj )µ(S

Ui,Uj )� (↵i,j

� ✏)µ(RUi,Uj )µ(S

Ui,Uj )


=X

(i,j)2B

+

✏µ(RUi,Uj )µ(S

Ui,Uj )

�X

(i,j)2B

+

✏

3µ(U

i

)µ(Uj

)

� ✏

4/4.

But this is a contradiction, sinceR

h

0�

Z

dµ ||h0||L

2 ||�Z

||L

2 < ✏

4/4 · 1. It follows

that µ(S

(i,j)2B

U

i

⇥ U

j

) � ✏/2.Note that what we have constructed is a partition of G which roughly satisfies

the regularity conditions, but in the infinite model. Our final step is to pull thispartition down to give a finite model, giving a contradiction. By choosing K � n

large enough, we may ensure that the following hold:• Whenever (i, j) 62 B, R

GKUi,Uj

= ;, and therefore U

i

(GK

) and U

j

(GK

) are✏-regular,

• µ(Ui

(GK

)) p

2µ(Ui

(G)) for each i.This implies that

µ

0

@[

(i,j)2B

U

i

(GK

)⇥ U

j

(GK

)

1

A 2µ

0

@[

(i,j)2B

U

i

(G)⇥ U

j

(G)

1

A ✏

as desired. ⇤Remark 5.7. The regularity lemma is usually stated with an additional require-ment that the cardinalities of the partition pieces be nearly equal. This couldbe obtained with the following changes: for each integer n, include new unarypredicates P

n,1, . . . , Pn,n

in the language, and in the finite models interpret thesepredicates as a partition into nearly equal pieces. Using these predicates, refinethe partition {U

i

} into one where the components have nearly the same mea-sure. When we pull this down to the finite model, the sets will di↵er in size bya very small fraction (choosing parameters correctly, this fraction can be arbi-trarily small, say ✏

2), so we may redistribute a small number of points to makethese pieces have exactly the same size without making much change to the edgedensity of large subsets.

Remark 5.8. The method in this subsection extends very naturally to hyper-graphs, using the algebras B

k,k�1 in place of B2,1. An example of a related prooffor hypergraphs is given below in Section 7.3.

6. The Downward Lowenheim-Skolem Theorem

In this section, we prove the Downward Lowenheim-Skolem Theorem for AMLusing an appropriate version of the Tarski-Vaught test.

Proposition 6.1 (Tarski-Vaught Test). Suppose that M is a substructure of N.

Then M is an elementary substructure of N if and only if:

(1) for all formulae '(x; ~y) and ~a from M , if there is b 2 N with N |= '(b,~a),then there is c 2M with N |= '(c;~a); and

(2) for all formulae '(x; ~y) and ~a from M , we have µ

M('(N,~a) \ M) =µ

N('(N,~a)) and v

M('(N,~a) \M) = v

N('(N,~a)).

Observe that, assuming (1), '(N, a) \M = '(M,a), whence it makes senseto speak of µ

M('(N, a) \M) in (2).


Proof. First suppose that M is an elementary substructure of N. Then (1) isimmediate. To prove (2), first notice that '(N,~a) \M = '(M) by elementarity.Suppose, towards a contradiction, that µ('(M,~a)) = r while µ('(N,~a)) = s withr 6= s. Without loss of generality, suppose that r < s. Fix q 2 Q with r < q < s.Then M |= m

x

< q.'(x, ~y)[~a] while N 6|= m

x

< q.'(x, ~y)[~a], a contradiction.We now must show that v('(N,~a) \ M) = v('(N,~a)). Set X = '(N,~a),

Y = '(M,~a), and r := µ('(N,~a)) = µ('(M,~a)). Without loss of generality,we may assume that r 2 Q. First suppose that v(X) = �. If v(Y ) = , thenM |= m

~x

< r.'(~x;~a) while N 6|= m

~x

< r.'(~x;~a); if v(Y ) = �, then M |= m

~x

r.'(~x;~a) while N 6|= m

~x

r.'(~x;~a). Now suppose that v(X) = . If v(Y ) 6= ,then M 6|= m

~x

< r.'(~x;~a) while N |= m

~x

< r.'(~x;~a). Finally suppose thatv(X) = �. If v(Y ) = , then M |= m

~x

< r.'(~x;~a) while N 6|= m

~x

< r.'(~x;~a); ifv(Y ) = �, then M 6|= m

~x

r.'(~x;~a) while N |= m

~x

r.'(~x;~a).Conversely, suppose that (1) and (2) hold for every formula; we prove that M

is an elementary substructure of N by induction on complexity of formulae. Thequantifier case is handled as usual using (1) while the measure constructor caseis handled using (2). ⇤Theorem 6.2 (Downward Lowenheim-Skolem). Suppose that N is a structure

and X is a subset of N . Then there is an elementary substructure M of N con-

taining X with |M | max(|X|, |L|,@0).

Proof. We define a sequence (Xi

)i<!

of subsets of N as follows. Set X0 := X.Assume now that X

i

has already been defined. For each formula '(x) with pa-rameters from X

i

that is satisfiable in N, we fix a witness b

'

. We then let X

i+1

be the closure of X

i

and all the b

'

’s under the function symbols in L. Observethat, since we only added measure constructors m

~x

./ r for rational r, we have|X

i

| max(|X|, |L|,@0) for each i < !.We now define an elementary substructure M as follows. We take M :=S

i<!

X

i

and BM := BN \ M . By the usual Tarski-Vaught test, the first-orderpart of M is an elementary substructure of the first-order part of N. This isthe beginning of an inductive argument that allows one to define, for a 2 M ,µ

M('(N, a)\M) := µ

N('(N, a)) and v

M('(N, a)\M) := v

N('(N, a)). Anotherinductive argument allows one to show that the quasistructure M is actually astructure. Now the AML Tarski-Vaught Test (Proposition 6.1) yields that M isan elementary substructure of N. ⇤

7. Model Theory and Applications

7.1. More Model Theory. We introduce some standard (and less standard)notions from model theory. We often discuss the case where the language iscountable, M is @1-saturated, and we consider a countable substructure, sincethis is the most interesting case for our purposes. Most of these results would stillhold as long as the saturation of M is greater than the cardinality of either thesignature or the submodels we consider.

From now on, we fix a countable signature L and a measured L-structure M(although Lemmas 7.11, 7.10, and 7.13 below make no mention of the measureand thus these Lemmas go through for an arbitrary L-structure). For example,M could be a Fubini structure equipped with its canonical measures. Also, forsake of readability, we will write µ for the measure, regardless of the cartesianpower of M

n under discussion. Finally, we will often identify a formula '(x) withparameters from M with the subset of M

n that it defines; in this way, we may


write µ(') to denote the measure of the definable subset of M

n defined by '.Likewise, we may write ¬Y to indicate the complement of the definable set Y .

Definition 7.1. Suppose A ✓M .(1) A partial type of n-tuples (or an n-type) over A is a collection p(~x) of

formulae with parameters from A, where ~x = (x1, . . . , xn

), such that forevery '1, . . . ,'n

2 p, there is ~a 2 M

n such that M |= '

i

(~a) for everyi n.

(2) A partial type p is said to be a type over A if for every formula ' withsuitable free variables and parameters from A, either ' 2 p or ¬' 2 p.

(3) A global type is a type over M .(4) If ~a is a sequence of elements, we write tp(~a/A) for the set of formulae

with parameters from A satisfied by ~a.(5) If M is measured, then the partial type p is wide if for every ' 2 p,

µ(') > 0.

Remark 7.2. An L-structure M is @1-saturated if and only if, whenever A ✓M

is countable and p(~x) is a partial type over A, there is a 2 M

|~x| such thatM |= '(~a) for all ' 2 p. In this case, we write ~a |= p.

Lemma 7.3. Suppose that A ✓M is countable. Then for almost every ~a, tp(~a/A)is wide.

Proof. Any ~a such that tp(~a/A) is not wide must belong to some set of measure 0definable with parameters from A. There are countably many such sets, so theirunion has measure 0. ⇤Lemma 7.4. Suppose that A ✓ M is countable. Then for almost every (~a,

~

b),tp(~a/A [ {~b}) and tp(~b/A [ {~a}) are both wide.

Proof. By the previous lemma, for each ~b the set

T

~

b

= {~a : tp(~a/A [ {~b}) is not wide}

has measure 0. Therefore µ({(~a,

~

b) : ~a 2 T

~

b

}) =R

µ(T~

b

)dµ

|~b|(~b) = 0, so the set of(~a,

~

b) such that tp(~a/A[{~b}) is not wide has measure 0. By a symmetric argument,also the set of (~a,

~

b) such that tp(~b/A[{~a}) is not wide has measure 0. Thereforealmost every (~a,

~

b) is in neither of these sets; for such (~a,

~

b), tp(~a/A [ {~b}) andtp(~b/A [ {~a}) are both wide. ⇤Definition 7.5. Suppose that (~b

i

: i 2 I) is a sequence of r-tuples of elementsof M , where I is an initial segment of N, and A ✓ M . Then (~b

i

) is a sequence

of indiscernibles over A if whenever '(~x0, . . . , ~xn

) is a formula with parametersfrom A and the displayed variables and m0 < · · · < m

n

, we have

M |= '(~b0, . . . ,~

b

n

), '(~bm0 , . . . ,

~

b

mn).

That is, for every n there is a type p

n

of r·(n+1)-tuples such that tp(~bm0 , . . . ,

~

b

mn/A) =p

n

whenever m0 < · · · < m

n

is an increasing sequence.

7.2. Amalgamation. The arguments in this section are essentially derived from[22].

Lemma 7.6. Suppose that µ(M) is finite, (~bi

) is a sequence of indiscernibles

over A ✓ M , X is a definable set over A, and µ(X~

b0) > 0 holds. Then for any

n, µ(T

in

X

~

bi) > 0.


Proof. Suppose the conclusion of the lemma is false and take k > 0 minimalso that µ(

Tik

X

~

bi) = 0. For n � k � 1, let C

n

=T

i<k�1 X

~

bi\ X

~

bn. Then

µ(Cn

) = µ(T

ik�1 X

~

bi) > 0 by indiscernibility. However, when k � 1 n < m,

by indiscernibility again, µ(Cn

\C

m

) = µ(T

ik

X

~

bi) = 0. This is a contradiction

since µ is a finite measure. ⇤

Lemma 7.7. Suppose that M is @1-saturated, µ(M) is finite, and A ✓ M is

countable. If (~bi

) is a sequence of indiscernibles over A and tp(~a/A [ {~b0}) is

wide, then for any n, there is an ~a

0with tp(~a0,~b

i

/A) = tp(~a,

~

b0/A) for all i n.

Proof. Let X be a set definable over A and suppose (~a,

~

b0) 2 X. Then ~a 2X

~

b0and, since tp(~a/A [ {~b0}) is wide, µ(X

~

b0) > 0. By the previous lemma,

µ(T

in

X

~

bi) > 0, and is therefore non-empty. By @1-saturation, there is an a

0

such that a

0 2T

in

X

~

bisimultaneously for all X 3 (~a,

~

b0) that are definable overA. Therefore tp(~a0,~b

i

/A) = tp(~a,

~

b0/A) for all i n. ⇤

Remark 7.8. The previous lemma suggests a connection between wideness andthe model-theoretic notion of nonforking, as we now explain. First, a formula'(x, b) k-divides over a set A ✓ M if there is a sequence (b

i

: i 2 N) whichis indiscernible over A for which b0 = b and such that

Ti2I

'(M, b

i

) = ; forany I ✓ N with |I| = k. We say that '(x, b) divides over A if it k-dividesover A for some k � 2. We say that '(x, b) forks over A if there are formulae 1(x, b1), . . . , n

(x, b

n

) such that '(M, b) ✓S

n

i=1 i

(M, b

i

) and such that each

i

(x, b

i

) divides over A. Finally, we say that a type p(x) forks over A if it containsa formula which forks over A. In model theory, when tp(a/A[ {b}) does not forkover A, this implies that a is in some sense independent from b over A. Forexample, when working in an algebraically closed field K and k is a subfield ofK, then tp(a/k [ {b}) does not fork over k if and only if, whenever a is a genericpoint for a Zariski closed set V defined over k, then a does not lie in any Zariskiclosed set V

0 defined over k(b) with dim(V 0) < dim(V ). Lemma 7.7 shows that ifA ✓M is countable and tp(a/A [ {b}) is wide, then tp(a/A [ {b}) does not forkover A.

Definition 7.9. Suppose that A ✓M and p is a global type.

(1) p is A-invariant if whenever tp(~a1, . . . ,~an

/A) = tp(~a01, . . . ,~a0n/A), we haveX

~a1,...,~an 2 p , X

~a

01,...,~a

0n2 p.

(2) p is A-finitely satisfiable if whenever X 2 p, X \A 6= ;.

Lemma 7.10. Suppose that p is a global type and A ✓ M . If p is A-finitely

satisfiable then p is A-invariant.

Proof. Suppose not; then there are definable X and ~a1, . . . ,~an

,~a

01, . . . ,~a

0n

so thatX

~a1,...,~an 2 p and X

~a

01,...,~a

0n62 p. Since p is a global type, this means ¬X

~a

01,...,~a

0n2 p,

and since p is closed under intersections, X

~a1,...,~an \X

~a

01,...,~a

0n2 p. But then there

is an element m 2 A such that m 2 X

~a1,...,~an \ X

~a

01,...,~a

0n, contradicting the fact

that tp(~a1, . . . ,~an

/A) = tp(~a01, . . . ,~a0n/A). ⇤

Lemma 7.11. Let N be an elementary substructure of M. If p is a type over

N , then there is an extension of p to a global N -finitely satisfiable (and therefore

N -invariant) type.


Proof. Note that, since N is an elementary submodel of M, each X 2 p has theproperty that X \N

n 6= ;. Let

p

0 = p [ {¬Y : Y 2 Defn

(M) and Y \N

n = ;}.We will show that every finite subset of p

0 is satisfied by an element of N . If not,there is an X 2 p and Y1, . . . , Yn

with Y

i

\N = ; such that X \T

i

¬Y

i

= ;. SoX \N \

Ti

(Yi

[ ¬Y

i

) 6= ;, and since each Y

i

\N = ;, X \M \T

i

Y

i

6= ;.Let q be an arbitrary extension of p

0 to a (global) type. Suppose q is not finitelysatisfiable in M ; then there is a Y 2 q with Y \N = ;, which is impossible sincethen ¬Y 2 p

0 ✓ q. ⇤Definition 7.12. If S is a set and p a global type, write p | S for the restrictionof p to sets definable over S.

Lemma 7.13. Suppose p is a global A-invariant type, and recursively choose

~

b

n

✏ p | A [ {~bi

}i<n

. Then (~bi

) (if it exists) is a sequence of indiscernibles over

A. (This sequence always exists if M is @1-saturated.)

Proof. By induction on n, we show tp(~bm0 , . . . ,

~

b

mn/A) is constant whenever m0 <

· · · < m

n

. When n = 1, this follows since each tp(~bi

/A) = p | A. Supposenow that (~b

n+1,~

b

n

, . . . ,

~

b0) 2 X. Then also (~bmn+1 ,

~

b

n

, . . . ,

~

b0) 2 X holds becausetp(~b

mn+1/A[ {~b0, . . . ,~

b

n

}) = tp(~bn+1/A[ {~b0, . . . ,

~

b

n

}) = p | A[ {~b0, . . . ,~

b

n

}. Bythe induction hypothesis, we have that tp(~b0, . . . ,

~

b

n

/A) = tp(~bm0 , . . . ,

~

b

mn/A).Since X

~

bn,...,

~

b02 p, also X

~

bmn ,...,

~

bm02 p, whence we have (~b

mn+1 ,~

b

mn , . . . ,

~

b

m0) 2X. ⇤Lemma 7.14. Suppose that M is @1-saturated and µ(M) is finite. Suppose that

A ✓M is countable, q is an A-invariant type,

~

b ✏ q | A[{~a}, tp(~b0/A) = tp(~b/A),tp(~a0/A) = tp(~a/A), tp(~a0/A [ {~b0}) is wide, X,Y are definable with parameters

from A, and µ(X~a

\ Y

~

b

) > 0. Then µ(X~a

0 \ Y

~

b

0) > 0.

Proof. Suppose the claim fails, so µ(X~a

0 \Y

~

b

0) = 0. By @1-saturation and the factthat tp(~b0/A) = tp(~b/A), there an ~a00 with tp(~a00,~b/A) = tp(~a0,~b0/A); in this way,we may assume ~b = ~

b

0.Since q is A-invariant, whenever ~a⇤ ✏ tp(~a/A) and ~b⇤ ✏ q | A [ {~a⇤} we have

tp(~a⇤,~b⇤/A) = tp(~a,

~

b/A), and so in particular µ(X~a

⇤ \ Y

~

b

⇤) = µ(X~a

\ Y

~

b

) > 0.Choose ~a0 ✏ p and ~b0 ✏ q | A [ {~a0}. Let {~a

i

,

~

b

i

}in

be given so that (~bi

) isindiscernible over A. For each � > 0 and each C 2 tp(~a/A), by the wideness oftp(~a0/A [ {~b}), we have

µ(C \ {~x : M ✏⇣m

~z

< �.X(~z, ~x) ^ Y (~z,

~

b)⌘}) > 0,

and therefore by Lemma 7.7 and @1-saturation we may find an ~an+1 with tp(~a

n+1/A) =tp(~a/A) such that µ(X

~an+1 \ Y

~

bi) = 0 for i n. Let ~b

n+1 ✏ q � (A [ {~ai

,

~

b

i

}in

[{~a

n+1}); by Lemma 7.13, we have ensured that (~bi

) remains indiscernible over A.For each i, let C

i

= X

~ai \ Y

~

bi. Since each ~b

n

✏ q � ~an

, µ(Ci

) � ✏; while for i 6= j,µ(C

i

\ C

j

) = 0. This is a contradiction to the finiteness of µ(M). ⇤Lemma 7.15. Suppose that M is @1-saturated and µ(M) is finite. Suppose that

A ✓M is countable, q is an A-invariant type,

~

b ✏ q | A[{~a}, tp(~b0/A) = tp(~b/A),tp(~a0/A) = tp(~a/A), tp(~a0/A [ {~b0}) is wide, X,Y are definable with parameters

from A, and µ(X~a

\ Y

~

b

) = 0. Then µ(X~a

0 \ Y

~

b

0) = 0.


Proof. Suppose the claim fails, so µ(X~a

0\Y

~

b

0) > 0. As in the proof of the previouslemma, we may assume that~b = ~

b

0. Suppose that {~ai

,

~

b

i

}i<n

has been constructed(including the empty case where n = 0) so that ~a

i

|= tp(~a/A) for each i and sothat (~b

i

) is indiscernible over A. Choose ~bn

✏ q | A [ {~ai

,

~

b

i

}i<n

. There is some✏ 2 Q>0 such that for each C 2 tp(~a/A),

µ(C \ {~x : M ✏⇣m

~z

� ✏.X(~z, ~x) ^ Y (~z,

~

b)⌘}) > 0,

and therefore by Lemma 7.7 and @1-saturation we may find an ~an

with tp(~an

/A) =tp(~a/A) such that µ(X

~an \Y

~

bi) � ✏ for i n. For each i, let C

i

= X

~ai \Y

~

bi. Then

µ(Ci

) � ✏ for each i and, when i < j, µ(Ci

\ C

j

) = 0 since ~bj

✏ q | A [ {~ai

}.Thus µ(X

~ai \ Y

~

bj) = 0 by A-invariance. This contradicts the fact that µ(M) is

finite. ⇤Theorem 7.16. Suppose M is @1-saturated and µ(M) is finite. Let N be a

countable elementary substructure of M. Let p, q be types over N and suppose

tp(~a/N) = tp(~a0/N) = p, tp(~b/N) = tp(~b0/N) = q with tp(~a/N [ {~b}), tp(~a0/N [{~b0}) wide. Then for any definable sets X and Y over N , we have µ(X

~a

\Y

~

b

) > 0i↵ µ(X

~a

0 \ Y

~

b

0) > 0.

Proof. Fix an extension q

0 of q to an N -invariant global type; this is possible byLemma 7.11. Let ~a⇤ ✏ p and~b⇤ ✏ q

0 | N[{~a⇤}. If µ(X~a

⇤\Y

~

b

⇤) > 0 then by Lemma7.14, both µ(X

~a

\ Y

~

b

) > 0 and µ(X~a

0 \ Y

~

b

0) > 0. Otherwise, µ(X~a

⇤ \ Y

~

b

⇤) = 0,and by Lemma 7.15, both µ(X

~a

\ Y

~

b

) = 0 and µ(Xa

0 \ Y

b

0) = 0. ⇤

Definition 7.17. Let N be an elementary substructure of M. A set R ✓ M

2m

is wide-stable over N if R is N -invariant and whenever tp(~a/N) = tp(~a0/N),tp(~b/N) = tp(~b0/N), both tp(~a/N[{~b}) and tp(~a0/N[{~b0}) are wide, and (~a,

~

b) 2R, then also (~a0,~b0) 2 R.

R is stable over N if whenever (~ai

,

~

b

i

) is a sequence of indiscernibles over N

and R(~ai

,

~

b

j

) holds for every i < j, also R(~aj

,

~

b

i

) for some i < j.We say R 2 B2(N) is approximated by definable stable sets (ADS) (respec-

tively, approximated by definable wide-stable sets (ADWS)) over N if for every✏ > 0, there is a stable (respectively, wide-stable) set S definable over N suchthat µ(R4S) < ✏.

Observe that the wide-stable subsets of M

2m form a �-algebra. It can beshown that stable subsets of M

2m form a Boolean algebra. Note that if we replacewideness in the definition of wide-stability with non-forking, we obtain a standardconsequence of stability (see Lemma 3.3 of [25]). In particular, since wide typesare non-forking, stable implies wide-stable.

Theorem 7.18. Let M be @1-saturated, let N be a countable elementary substruc-

ture of M, and suppose µ is finite. If R ✓M

2is in B2(N), then the following are

equivalent:

(1) There is U 2 B2,1(N) such that µ(R4U) = 0;(2) R is ADS;

(3) R is ADWS.

Proof. (1) ) (2): First observe that every set of the form A ⇥ B with A, B

definable over N , is stable: if (ai

, b

i

: i 2 I) is a sequence of indiscernibles over N

and (ai

, b

j

) 2 A ⇥ B for i < j then (aj

, b

i

) 2 A ⇥ B for all i < j. Consequently,


the algebra generated by the sets of the form A⇥B, with A, B definable over N ,which is dense in B2,1(N), is contained in the algebra of stable subsets of M

2.Thus, if there is U 2 B2,1(N) such that µ(R4U) = 0, then for every ✏ > 0, thereis S 2 B0

2,1(N) such that µ(R4S) < ✏; such an S is stable.(2)) (3): Immediate since stable definable sets are wide-stable and definable.(3) ) (1): Let R ✓ M

2 be definable over N and wide-stable over N . SinceB2,1(N) is complete with respect to the pseudometric µ(·4·), it su�ces to prove(1) for such an R. For any N -definable A and B and any ✏ > 0, define A

B

✏

= {a 2A : M ✏ m

y

< ✏.B \R

a

}. Set

U = {A⇥B : µ((A⇥B)\R) = 0, A, B 2 B1(N)}[{\

n

(AB

1/n

⇥B) : A, B 2 B1(N)}.

Clearly U =SU belongs to B2,1(N) and satisfies µ(U \R) = 0.

Suppose, towards a contradiction, that µ(R \ U) > 0. Then we may find an(a, b) 2 R\U such that tp(a/N) and tp(b/N[{a}) are wide. For each A 2 tp(a/N)and B 2 tp(b/N), we have A⇥B 62 U , and therefore µ((A⇥B)\R) > 0. Thereforep = (tp(a/N)⇥ tp(b/N)) [ {¬R(x, y)} is a wide partial type. Set

p

0 = p [ {¬S(x, y) : (8✏ 2 Q>0)(my

< ✏.S(x, y) 2 p)}.We claim this is also a wide partial type. Towards this end, fix A and B definableover N such that a 2 A and b 2 B. Also fix S(x, y) definable over N so that, foreach ✏ 2 Q>0, m

y

< ✏.S(x, y) 2 p. Since (a, b) /2 U , we have some ✏ 2 Q>0 suchthat A

B

✏

62 tp(a/N). Set

A

0 = {a0 2 A : M |= (my

� ✏.(B \R

a

0)) ^ (my

< ✏.S(a0, y))}.Observe that A

0 2 tp(a/N), whence µ(A0) > 0. It is enough to show that µ((A0⇥B) \ (¬R \ ¬S)) > 0. Suppose this is not the case. Then

µ((A0 ⇥B) \R) = µ((A0 ⇥B) \ (S \R)).

However,

µ((A0 ⇥B) \R) =Z

a

02A

0µ(B \R

a

0)dµ � ✏µ(A0)

while

µ((A0 ⇥B) \ (S \R)) =Z

a

02A

0µ(B \ (S

a

0 \R

a

0))dµ < ✏µ(A0),

yielding a contradiction.Now take any extension of p

0 to a wide type q and suppose that (a0, b0) real-izes q. It follows immediately that tp(b0/N [ {a0}) is wide, tp(a0/N) = tp(a/N),and tp(b0/N) = tp(b/N). Since tp(b/N [ {a}) is wide and tp(a0/N) = tp(a/N),whenever B

a

2 tp(b/N [ {a}), µ(Ba

0) > 0. By @1-saturation, there is a b

00 withtp(b00/N [ {a0}) = tp(b/N [ {a}). But now we have (a0, b0) 62 R and (a0, b00) 2 R,contradicting wide-stability. ⇤

We could generalize these notions to stable relations on pairs of n-tuples, byusing the two algebras B2n,[1,n] and B2n,[n+1,2n] (the algebras of sets of 2n-tuplesdepending only on the first n or only on the second n coordinates, respectively);the arguments are exactly analogous, replacing the singletons with n-tuples.

Combining Theorem 7.16 and the proof of the previous theorem, we see:

Corollary 7.19. Suppose M is @1-saturated and µ(M) is finite. Suppose that Nis a countable elementary substructure of N and that X and Y are definable over

N . Then {(a, b) : µ(Xa

\ Y

b

) > 0} di↵ers from an element of B2,1 by a null set.


7.3. Hypergraph Removal. We now give a proof of the hypergraph removallemma [11, 17, 27]; this proof is closely related to the infinitary proof given byTao [30] in a di↵erent framework.

Suppose B is the �-algebra generated by some collection of formulae withparameters from an elementary substructure M of N. Two natural ways to extendB would be by extending the collection of formulae and by extending the set M .The following lemma states that these two methods are orthogonal in a certainsense. This lemma will be applied in the particular case of the algebras B

n,<I

(M),B

n,<I

(M [ {a}) (to be defined below), and Bn,I

(M) (as defined in Definition5.4), and the reader will not be mislead assuming these are the algebras used.(Recall our notation that B0 is the Boolean algebra of definable sets generatingthe algebra B.)

Lemma 7.20. Let � be a set of formulae, let M be an elementary substructure

of N, and let n be an integer. Suppose A0is the collection of sets of the form

'(Nn

,~a) where '(~x, ~y) 2 � and ~a 2M

|~y|, B0

is the set of M -definable sets of n-

tuples in N, and C0is the collection of sets of the form '(Nn

,~a) where '(~x, ~y) 2 �and ~a 2 N

|~y|.

Then for any f 2 L

2(B), ||E(f | A)� E(f | C)||L

2 = 0.

Proof.

3 Suppose not. Then setting ✏ := ||f � E(f | A)||L

2 and � := ||f � E(f |C)||

L

2 , we must have � < ✏. Since f is B-measurable, for some �1, . . . ,�m

, some 1, . . . , m

, and some ~b1, . . . ,~

b

m

in M , we have ||f �P

im

�

i

�

i(~x,

~

b

i

)||L

2 <

(✏� �)/4.Since ||f � E(f | C)||

L

2 = �, there are ↵1, . . . ,↵k

, formulae '1, . . . ,'k

2 �,and ~

d1, . . . ,~

d

k

in N such that ||f �P

ik

↵

i

�

'i(~x,

~

d

i

)||L

2 < ✏ � 3(✏ � �)/4, andtherefore

||X

im

�

i

�

i(~x,

~

b

i

)�X

ik

↵

i

�

'i(~x,

~

d

i

)||L

2 < ✏� (✏� �)/2.

This is not itself a formula, but we can construct a formula which corresponds toit.

Squaring both sides and expanding the product, we have

(✏� (✏� �)/2)2 >

R Pi,jm

�

i

�

j

�

i(~x,

~

b

i

)� j (~x,

~

b

j

)

�2R P

im,jk

�

i

↵

j

�

i(~x,

~

b

i

)�'j (~x,

~

d

i

)

+R P

i,jk

↵

i

↵

j

�

'i(~x,

~

d

i

)�'j (~x,

~

d

j

).

For each i m, j k, we may choose an r

ij

such thatR�

i(~x,

~

b

i

)�'j (~x,

~

d

i

)dµ <

r

ij

and for each i, j k we may choose an s

ij

such thatR�

'i(~x,

~

d

i

)�'j (~x,

~

d

i

)dµ <

s

ij

, and we may choose these so that

(✏� (✏� �)/2)2 >

Z X

i,jm

�

i

�

j

�

i(~x,

~

b

i

)� j (~x,

~

b

j

)�2X

im,jk

r

ij

�

i

↵

j

+X

i,jk

s

ij

↵

i

↵

j

.

3Hrushovski has pointed out that this proof is reminiscent of early proofs in the stabilitytheory of algebraically closed fields. The full connection between this argument and the amal-gamation methods from the previous section is not completely understood.


Now let ⇢(~b1, . . . ,~

b

m

,

~

d1, . . . ,~

d

k

) be the formula^

im,jk

m

~x

< r

ij

.'

i

(~x,

~

b

i

) ^ j

(~x,

~

d

j

) ^^

i,jk

m

~x

< r

ij

.

i

(~x,

~

d

i

) ^ j

(~x,

~

d

j

).

Since N |= ⇢(~b1, . . . ,~

b

m

,

~

d1, . . . ,~

d

k

), we have N |= 9~y1, . . . , ~yk

⇢(~b1, . . . ,~

b

m

, ~y1, . . . , ~yk

).By the elementarity of M, there exist ~d01, . . . , ~d0

k

in M such that M |= ⇢(~b1, . . . ,~

b

m

,

~

d

01, . . . ,

~

d

0k

).Consequently, for each i m, j k, we have

R�

i(~x,

~

b

i

)�'j (~x,

~

d

0i

)dµ r

ij

andfor each i, j k, we have

R�

'i(~x,

~

d

0i

)�'j (~x,

~

d

0i

)dµ s

ij

. Therefore

||X

im

�

i

�

i(~x,

~

b

i

)�X

ik

↵

i

�

'i(~x,

~

d

0i

)||L

2 < ✏� (✏� �)/2

and||f �

X

ik

↵

i

�

'i(~x,

~

d

0i

)||L

2 < ✏� (✏� �)/4.

SinceP

ik

↵

i

�

'i(~x,

~

d

0i

) is measurable with respect to A, this contradicts theassumption that ||f � E(f | A)||

L

2 = ✏. ⇤

The following theorem is essentially the infinitary version of hypergraph re-moval:

Theorem 7.21. Let M be an elementary substructure of N. Let n � k, I ✓�[1,n]k

�, and suppose that for each each I 2 I with |I| = k, we have a set A

I

2B

n,I

(M), and suppose there is a � > 0 such that whenever B

I

2 B0n,I

(M) and

µ

n(AI

\B

I

) < � for all I 2 I,

TI2I B

I

is non-empty. Then µ

n(T

I2I A

I

) > 0.

Proof. We proceed by main induction on k. When k = 1, the claim is trivial:we must have µ(A

I

) > 0 for all I, since otherwise we could take B

I

= ;; thenµ

n(T

A

I

) =Q

µ(AI

) > 0. So we assume that k > 1 and that whenever B

I

2B

n,I

(M) and µ

n(AI

\ B

I

) < � for all I,T

I2I B

I

is non-empty. Throughout thisproof, the variables I and I0 range over elements of I.

We write Bn,<I

(M) forS

J(I

Bn,J

(M).

Claim 1. For any I0,Z

(�AI0

� E(�AI0

| Bn,<I0(M)))

Y

I 6=I0

�

AI dµ

n = 0.

Proof. When k = n, this is trivial sinceQ

I 6=I0�

AI is an empty product, andtherefore equal to 1.

If k < n, we haveZ

(�AI0

� E(�AI0

| B�n,<I0

(M)))Y

I 6=I0

�

AI dµ

n

=Z

(�AI0

� E(�AI0

| B�n,<I0

(M)))Y

I 6=I0

�

AI dµ

k({xi

}i2I0)dµ

n�k({xi

}i62I0).

Observe that for any choice of {ai

}i62I0 , Bn,<I0(M),B

n,<I0(M[{ai

}i62I0),Bn,I0(M)

satisfy the preceding lemma, so

||E(�AI0

| B�n,<I0

(M))� E(�AI0

| B�n,<I0

(M [ {ai

}i62I0))||L2 = 0.


The functionQ

I 6=I0�

Ai({xi

}i2I0 , {ai

}i62I0) is measurable with respect to B�

n,<I0(M[

{~ai

}). Combining these two facts, we haveZ

(�AI0

� E(�AI0

| B�n,<I0

(M)))Y

I 6=I0

�

AI dµ

k({xi

}i2I0)

=Z

(�AI0

� E(�AI0

| B�n,<I0

(M [ {ai

}i62I0)))

Y

I 6=I0

�

Aidµ

k({xi

})

=0

Since this holds for any {ai

}i62I0 , the claim follows by integrating over all choices

of {ai

}. a

We next show that, without loss of generality, we may assume each A

I0 belongsto B

n,<I0(M) by showing that for each I0 2 I, there is some set A

0I02 B

n,<I

(M)with the property that, if we replace A

I0 by A

0I0

, the assumptions of the theoremall hold, and such that if we show the conclusion for the modified family of sets,the conclusion also holds for the original family.

Claim 2. For any I0, there is an A

0I02 B

n,<I0(M) such that:• Whenever B

I

2 B0n,I

(M) for each I, µ

n(AI

\B

I

) < � for each I 6= I0, andµ

n(A0I0\B

I0) < �,T

I2I B

I

is non-empty, and• If µ

n(A0I0\T

I 6=I0A

I

) > 0, µ

n(T

I2I A

I

) > 0.

Proof. Define A

0I0

:= {xI0 | E(�

AI0| B

n,<I0(M))(xI0) > 0}. If µ

n(A0I0\T

I 6=I0A

I

) >

0 then we have ZE(�

AI0| B

n,<I0(M))Y

I 6=I0

�

AI dµ

n

> 0,

and by the previous claim, this implies that that µ

n(T

A

I

) > 0.Suppose that for each I, B

I

2 B0n,I

(M) with µ

n(AI

\ B

I

) < � for I 6= I0 andµ

n(A0I0\B

I0) < �. Since

µ

n(AI0\A0I0) =

Z�

AI0(1��

A

0I0

) dµ

n =Z

E(�AI0

| Bn,<I0(M))(1��

A

0I0

) dµ

n = 0,

we have µ

n(AI0 \B

I0) < � as well, and thereforeT

I2I B

I

is non-empty. a

By applying the previous claim to each I 2 I, we may assume for the rest ofthe proof that for each I, A

I

2 Bn,<I

(M).Fix some finite algebra B ✓ B0

n,k�1(M) so that for every I, k�AI � E(�

AI |B)k

L

2(µn) <

p�p

2(|I|+1)(such a B exists because there are finitely many I and each

A

I

is Bn,k�1(M)-measurable). For each I, set A

⇤I

= {aI

| E(�AI | B)(a

I

) >

|I||I|+1}.

Claim 3. For each I, µ

n(AI

\A

⇤I

) �/2

Proof. A

I

\ A

⇤I

is the set of points such that (�AI � E(�

AI | B)) (~a) � 1|I|+1 . By

Chebyshev’s inequality, the measure of this set is at most

(|I|+ 1)2Z

(�AI � E(�

AI | B))2 dµ

n = (|I|+ 1)2k�AI � E(�

AI | B)k2L

2(µn) �

2.

a

Claim 4. µ

n(T

I

A

I

) � µ

n(T

I

A

⇤I

)/ (|I|+ 1).


Proof. For each I0,

µ

n((A⇤I0\A

I0) \\

I 6=I0

A

⇤I

) =Z�

A

⇤I0

(1� �AI0

)Y

I 6=I0

�

A

⇤Idµ

n

=Z�

A

⇤I0

(1� E(�AI0

| B))Y

I 6=I0

�

A

⇤Idµ

n

1|I|+ 1

Z Y

I2I�

A

⇤Idµ

n

=1

|I|+ 1µ

n(\

I2IA

⇤I

)

But then

µ

n(\

I2IA

⇤I

\\

I2IA

I

) X

I0

µ

n((A⇤I0\A

I0) \\

I 6=I0

A

⇤I

) |I||I|+ 1

µ

n(\

I2IA

⇤I

).

a

Each A

⇤I

may be written in the formS

irIA

⇤I,i

where A

⇤I,i

=T

J2( Ik�1)

A

⇤I,i,J

and A

⇤I,i,J

is an element of B0n,J

(M). We may assume that if i 6= i

0 then A

⇤I,i

\A

⇤I,i

0 = ;.We have

µ

n(\

I

A

⇤I

) = µ

n([

~

i2Q

I [1,rI ]

\

I

\

J2( Ik�1)

A

⇤I,iI ,J

).

For each ~i 2Q

I

[1, r

I

], let D

~

i

=T

I

TJ2( I

k�1)A

⇤iI ,J,I

. Each A

⇤I,iI ,J

is an element of

B0n,J

(M), so we may group the components and write D

~

i

=T

J2([1,n]k�1)

D

~

i,J

where

D

~

i,J

=T

I�J

A

⇤I,iI ,J

.Suppose, for a contradiction, that µ

n(T

I

A

⇤I

) = 0. Then for every~i 2Q

I

[1, r

I

],µ

n(D~

i

) = µ

n(T

J

D

~

i,J

) = 0. By the inductive hypothesis, for each � > 0, there isa collection B

~

i,J

2 B0n,J

(M) such that µ

n(D~

i,J

\ B

~

i,J

) < � andT

J

B

~

i,J

= ;. Inparticular, this holds with � = �

2( kk�1)(

QI rI)(maxI rI)

.

For each I, i r

I

, J ⇢ I, define

B

⇤I,i,J

= A

⇤I,i,J

\\

~

i,iI=i

2

4B

~

i,J

[[

I

0◆J,I

0 6=I

A

⇤I

0,iI0 ,J

3

5.

Claim 5. µ

n(A⇤I,i,J

\B

⇤I,i,J

) �

2( kk�1)(maxI rI)

.

Proof. Observe that if x 2 A

⇤I,i,J

\ B

⇤I,i,J

then for some ~i with i

I

= i, x 62 B

~

i,J

[S

I

0◆J,I

0 6=I

A

⇤I

0,iI0 ,J

. This means x 62 B

~

i,J

and x 2T

I

0◆J

A

⇤I

0,iI0 ,J

= D

~

i,J

. So

µ

n(A⇤I,i,J

\B

⇤I,i,J

) X

~

i2Q

I [1,rI ]

µ

n(D~

i,J

\B

~

i,J

) �

2�

k

k�1

�(max

I

r

I

).

a

Define B

⇤I

=S

irI

TJ

B

⇤I,i,J

.

Claim 6. µ

n(AI

\B

⇤I

) �.


Proof. Since µ

n(AI

\A

⇤I

) �/2, it su�ces to show that µ

n(A⇤I

\B

⇤I

) �/2.

µ

n(A⇤I

\[

i

\

J

B

⇤I,i,J

) = µ

n

[

i

\

J

A

⇤I,i,J

\[

i

\

J

B

⇤I,i,J

!

µ

n

[

i

\

J

A

⇤I,i,J

\\

J

B

⇤I,i,J

!!

X

irI

µ

n

\

J

A

⇤I,i,J

\\

J

B

⇤I,i,J

!

X

irI

X

J

µ

n(A⇤I,i,J

\B

⇤I,i,J

)

r

I

·✓

k

k � 1

◆· �

2�

k

k�1

�(max

I

r

I

) �/2.

a

Note that the sets B

⇤I

satisfy the assumption, and thereforeT

I

B

⇤I

6= ;.

Claim 7. \

I

B

⇤I

✓[

~

i

\

J

B

~

i,J

.

Proof. Suppose x 2T

I

B

⇤I

=T

I

SirI

TJ

B

⇤I,i,J

. Then for each I, there is ani

I

r

I

such that x 2T

J

B

⇤I,iI ,J

. Since B

⇤I,iI ,J

✓ A

⇤I,iI ,J

, for each I and J ⇢ I,x 2 A

⇤I,iI ,J

.For any J , let I � J . Then

x 2 B

⇤I,iI ,J

= A

⇤I,iI ,J

\\

~

i

0,i

0I=iI

2

4B

~

i,J

[[

I

0◆J,I

0 6=I

A

⇤I

0,iI0 ,J

3

5.

In particular, x 2hB

~

i,J

[S

I

0◆J,I

0 6=I

(A⇤I

0,iI0 ,J

)i

for the particular~i we have chosen.Since x 2 A

⇤I,iI0 ,J

for each I

0 � J , it must be that x 2 B

~

i,J

. This holds for any J ,so x 2

TJ

B

~

i,J

. a

SinceT

I

B

⇤I

is non-empty, there is some ~i such thatT

J

B

~

i,J

6= ;. But thisleads to a contradiction, so it must be that µ

n(T

I

A

⇤I

) > 0, and therefore, as wehave shown, µ

n(T

I2I A

I

) � 1|I|+1µ

n(T

I2I A

⇤I

) > 0. ⇤

Corollary 7.22. For each ✏ > 0, k and each k-regular hypergraph (W,F ) there

is a � > 0 such that whenever (V,E) is a k-regular hypergraph such that there are

at most �|V ||W |copies of (W,F ), it is possible to remove at most ✏|V |k edges to

obtain a hypergraph with no copies (W,F ).

Proof. Suppose not. Fix ✏ > 0, k, (W,F ) so that for each K, there is an k-regularhypergraph (V

K

, E

K

) with at most �|V ||W | copies of (W,F ) and such that everysub-hypergraph with at least |E

K

|� ✏|VK

|n edges contains a copy.We consider a signature of AML consisting of an k-ary predicate E. We con-

sider finite measured structures NK

defined as follows:


• The universe of NK

is V

K

,• The measures on N

K

are given by the normalized counting measure,• E is interpreted by E

K

.Let N be a (nonprincipal) ultraproduct of these structures and fix a countable

elementary submodel M. We may assume W = [1, n]. For each I 2�[1,n]

k

�, define

E

I 2 Bn,I

(M) to be the copy of E on the coordinates I. Observe that if ~x 2 N

n,~x is a copy of W exactly if W 2

TI2F

E

I . In particular, µ

n(T

I2F

E

I) = 0.By the preceding theorem, we can find definable sets B

I

such that µ

n(EI \B

I

) < ✏/|I| for each I 2�[1,n]

k

�so that

TI

B

I

= ;. Note that saying thisintersection of definable sets is empty is expressed by a single formula, andis therefore true for almost every structure (V

K

, E

K

). But then for some K,(V

K

, E

K

\T

B

I

(GK

)) is a sub-hypergraph obtained by removing fewer than✏|V

K

|k edges and containing no copies of (W,F ), yielding a contradiction. ⇤

The second author has extended this method [33] to the setting where (V,E)is a dense sub-hypergraph of a sparse random hypergraph. In this setting one re-places the normalized counting measure with a new counting measure normalizedby the ambient random hypergraph, causing substantial additional complications.

Remark 7.23. Szemeredi’s Theorem follows from hypergraph removal using thefollowing, now standard, encoding. Given A ✓ [1, n], to find an arithmetic pro-gression of length k + 1, we define a k + 1-partite k-regular hypergraph. Foreach i 2 [1, k], we take the part X

i

to be a copy of [1, n], and we take X

k+1 tobe a copy of [1, k

2n]. Given x

i

2 X

i

for all i, we say (x1, . . . , xk

) is an edge i↵Pik

i · xi

2 A, and when 1 i k, (x1, . . . , xi�1, xi+1, . . . , xk+1) is an edge i↵P

jk,j 6=i

j · xj

+ i(xk+1 �

Pjk,j 6=i

x

j

) 2 A.Suppose (x1, . . . , x

k+1) is a copy of the complete k-regular hypergraph onk + 1 vertices (that is, every k-tuple is an edge) with x

k+1 6=P

ik

x

i

. Then seta =

Pik

i · xi

and d = x

k+1 �P

ik

x

i

6= 0. Then we have a 2 A, and for eachi k, we have

a + id =X

jk

j · xj

+ i(xk+1 �

X

jk

x

j

) =X

jk,j 6=i

j · xj

+ i(xk+1 �

X

jk,j 6=i

x

j

) 2 A.

On the other hand, for any a 2 A and any sequence (x1, . . . , xk

) with a =Pik

i · xi

, the sequence (x1, . . . , xk

,

Pik

x

i

) is also a copy of the complete k-regular hypergraph on k+1 vertices. It is not possible to remove all such sequencesby removing a small number of edges, so hypergraph removal implies that theremust be many copies of the complete k-regular hypergraph on k +1 vertices; butthere are only a small number of sequences (x1, . . . , x

k

,

Pik

x

i

), so the remainingcopies must correspond to genuine arithmetic progressions.

7.4. Gowers Norms.

Definition 7.24. Let G be a finite abelian group and consider g : G ! R. Wedefine the k-th Gowers uniformity norm, || · ||

U

k , by

||g||2k

U

k =1

|G|k+1

X

x2G

X

~

h2G

k

Y

!2{0,1}k

g(x + ! · ~h).


It is not immediate that the right-hand side of the above display is nonnegative;however, this can be derived using an argument similar to that in the proof ofLemma 7.27 below.

The Gowers norms were introduced by Gowers in his proof of Szemeredi’sTheorem [16]. Since then, the norms have proven to be powerful tools in com-binatorics; for instance, these norms have been used to give a proof of the hy-pergraph regularity lemma [17] and in the proof of the Green-Tao theorem onarithmetic progressions in the primes [19]. Related norms for dynamical systems,the Gowers-Host-Kra norms, we introduced by Host and Kra [21], and have sim-ilarly been used to prove recurrence theorems in dynamical systems (for a fewexamples, see [3, 8]; [12] describes the general method and cites more than twentyexamples).

Roughly speaking, the U

2 norm is large when a function has a large corre-lation with some group character. The U

k norms for larger k are large when afunction correlates with a canonical function of a more general type. The exactcharacterization of these canonical functions was a substantial project [18, 20].Here we show that, in the setting of AML, these canonical functions can be takento be the simple functions generated by algebras B

k,k�1.

In the infinite setting, we give a slightly more general definition:

Definition 7.25. Let M be a measured AML structure and let f : M

k ! R bebounded and B

k

-measurable. Define || · ||U

k1

by:

||f ||2k

U

k1

=Z Y

!2{0,1}k

f(h!(1)1 , . . . , h

!(k)k

)dµ

2k(~h0,

~

h

1).

Suppose further that + is a definable group operation on M and g : M ! Ris bounded. Then define f : M

k ! R by f(h1, . . . , hk

) = g(P

k

i=1 h

i

) and define||g||

U

k1

:= ||f ||U

k1

.

Note that in the infinite setting, these are seminorms (and not even a seminormfor k = 1).

Lemma 7.26. Let (Gi

: i 2 N) be a sequence of finite abelian groups and, for

each i, let g

i

: G

i

! [�1, 1] be a function such that limi!1 ||g

i

||U

k exists. Let

L be the signature obtained by adding to the signature for abelian groups unary

predicates P

q

for q 2 Q \ [�1, 1]. Let Gi

be the AML L-structure associated to

G

i

, equipped with its normalized counting measure, by interpreting P

Giq

:= {a 2G

i

: g

i

(a) < q}. Let U be a nonprincipal ultrafilter on N, let G :=Q

U Gi

, and

let g : G ! [�1, 1] be the ultraproduct of (gi

). Then ||g||U

k1

= limi!1 ||g

i

||U

k .

Proof. First observe that

||gi

||2k

U

k =1

|Gi

|2k

X

~

h

0,

~

h

12G

k

Y

!2{0,1}k

g

i

(X

i

h

!(i)i

),

which is easily seen by making the substitutions x =P

i

h

0i

and h

i

= h

1i

�h

0i

. Define j

i

: G

k

i

! G

2k

i

by j

i

(~h) = (P

h

!(p)p

: ! 2 {0, 1}k); observe that(j

i

) is uniformly definable in Gi

. Now define g

0i

: G

2k

i

! [�1, 1] by g

0i

((x!

)) =Q!2{0,1}k g

i

(x!

). Then the lemma follows from Theorem 4.13 (applied to (g0i

)and (j

i

)). ⇤


For the rest of this subsection, we fix f : M ! R which is bounded andB

k

-measurable. We further assume that each µ

k is a probability measure on Bk

.

Lemma 7.27. ��Z

fdµ

k

�� ||f ||U

k1

.

Proof. Using Fubini’s theorem, we have��Z

f(~h)dµ

k(~h)��2k

=��Z ✓Z

f(~h)dµ(hk

)◆

dµ

k�1(h1, . . . , hk�1)

��2k

.

By Cauchy-Schwarz and Fubini again, we have��Z

f(~h)dµ

k(~h)��2k

✓Z

f(h1, . . . , hk�1, h

0k

)f(h1, . . . , hk�1, h

1k

)dµ

k+1

◆2k�1

.

Repeating this process, we arrive at��Z

f(~h)dµ

k(~h)��2k

0

@Z Y

!2{0,1}k

f(h!(1)1 , . . . , h

!(k)k

)dµ

2k(~h0,

~

h

1)

1

A

= ||f ||2k

U

k1

.

⇤Lemma 7.28. For each I ✓ [1, k] with |I| = k � 1, let B

I

be a definable set in

Bk,I

. Then 0 ||fQ

I

�

BI ||Uk1 ||f ||

U

k1

.

Proof. It su�ces to show that 0 ||f�BI ||Uk

1 ||f ||

U

k1

for a single I. Withoutloss of generality, we assume I = [1, k � 1]. Since f = f�

BI + f�

BI, we have

||f ||2k

U

k1

= ||f�BI + f�

BI||2k

U

k1

which in turn expands into a sum of 22k terms of the formZ Y

!2{0,1}k

(f�S!)(h!(1)

1 , . . . , h

!(k)k

)dµ

2k(~h0,

~

h

1),

where each S

!

is either �BI or �

BI. Observe that kf�

BIk2k

U

k1

corresponds to the

term when each S

!

= �

BI . Thus, it su�ces to show that all of the 22k terms arenon-negative.

Observe that if !(i) = !

0(i) for each i 2 I but S

!

6= S

!

0 then for any�

S!�S!0 = 0, whence the corresponding integral is 0. We thus restrict ourselvesto the case where, whenever ! � I = !

0 � I, S

!

= S

!

0 . In this case, we have

Z Y

!2{0,1}k

(f�S!)(h!(1)

1 , . . . , h

!(k)k

)dµ

2k(~h0,

~

h

1)

=Z Z Y

!2{0,1}k

(f�S!)(h!(1)

1 , . . . , h

!(k)k

)dµ

2(h0k

, h

1k

)dµ

2(k�1)

=Z 2

4Z Y

!2{0,1}k

(f�S!)(h!(1)

1 , . . . , h

!(k)k

)dµ(hk

)

3

52

dµ

2(k�1).


Since the inside of the integral is always non-negative, this term is non-negative.⇤

We set D(f)(~h0) :=R Q

!2{0,1}k,! 6=~0 f(h!(1)

1 , . . . , h

!(k)k

)dµ

k(~h1). Observe that

kfk2k

U

k1

=R

f ·D(f)dµ

k(~h0).

Lemma 7.29. D(f) is measurable with respect to Bk,k�1.

Proof. It su�ces to show that if ||E(f | Bk,k�1)|| = 0 then

Zf(~h0)D(f)(~h0)dµ

k(~h0) = 0.

We haveZf(~h0)D(f)(~h0)dµ

k(~h0) =Z

f(~h0)Y

!2{0,1}k,! 6=~0

f(h!(1)1 , . . . , h

!(k)k

)dµ

2k(~h0,

~

h

1)

=ZZ

f(~h0)Y

!2{0,1}k,! 6=~0

f(h!(1)1 , . . . , h

!(k)k

)dµ

k(~h0)dµ

k(~h1).

Observe that, for a given ~h1, we haveZ

f(~h0)Y

!2{0,1}k,! 6=~0

f(h!(1)1 , . . . , h

!(k)k

)dµ

k(~h0) = 0

sinceQ!2{0,1}k

,! 6=~0 f(h!(1)1 , . . . , h

!(k)k

) is Bk,k�1-measurable. The lemma now fol-

lows. ⇤Theorem 7.30. ||f ||

U

k1

> 0 if and only if ||E(f | Bk,k�1)|| > 0.

Proof. If ||f ||U

k1

> 0, thenR

fD(f)dµ

k(~h0) > 0; since D(f) is Bk,k�1-measurable,

||E(f | Bk,k�1)|| > 0.

For the other direction, if ||E(f | Bk�1)|| > 0, we may find B

I

-measurable B

I

so thatR

f

Q�

BI dµ 6= 0, whence

0 < |Z

f

Y

I

�

BI dµ| ||fY

I

�

BI ||Uk1 ||f ||

U

k1

.

⇤

References

[1] Tim Austin. Deducing the multidimensional Szemeredi theorem from an infinitary removallemma. J. Anal. Math., 111:131–150, 2010.

[2] Seyed-Mohammad Bagheri and Massoud Pourmahdian. The logic of integration. Arch.Math. Logic, 48(5):465–492, 2009.

[3] V. Bergelson, B. Host, B. Kra, and I. Ruzsa. Multiple recurrence and nilsequences. Inven-tiones Mathematicae, 160(2):261–303, May 2005.

[4] Vitaly Bergelson. Ergodic Ramsey theory. In Logic and combinatorics (Arcata, Calif., 1985),volume 65 of Contemp. Math., pages 63–87. Amer. Math. Soc., Providence, RI, 1987.

[5] Vitaly Bergelson. Ergodic theory and Diophantine problems. In Topics in symbolic dynamicsand applications (Temuco, 1997), volume 279 of London Math. Soc. Lecture Note Ser., pages167–205. Cambridge Univ. Press, Cambridge, 2000.

[6] C.C. Chang and J. Keisler. Model Theory, volume 73 of Studies in Logic and the Foundationsof Mathematics. Elsevier, 1990.

[7] Z. Chatzidakis and A. Pillay. Generic structures and simple theories. Ann. Pure Appl. Logic,95(1-3):71–92, 1998.


[8] Qing Chu. Convergence of weighted polynomial multiple ergodic averages. Proc. Amer.Math. Soc., 137(4):1363–1369, 2009.

[9] G. Elek. Weak convergence of finite graphs, integrated density of states and a cheegertype inequality. Journal of Combinatorial Theory, Series B, 98(1):62 – 68, 2008.10.1016/j.jctb.2007.03.004.

[10] G. Elek and B. Szegedy. Limits of hypergraphs, removal and regularity lemmas. a non-standard approach. unpublished, 2007.

[11] Peter Frankl and Vojtech Rodl. Extremal problems on set systems. Random StructuresAlgorithms, 20(2):131–164, 2002.

[12] N. Frantzikinakis. Some open problems on multiple ergodic averages. ArXiv e-prints, March2011.

[13] H. Furstenberg and Y. Katznelson. Idempotents in compact semigroups and Ramsey theory.Israel J. Math., 68(3):257–270, 1989.

[14] Harry Furstenberg. Ergodic behavior of diagonal measures and a theorem of sze-merdi on arithmetic progressions. Journal d’Analyse Mathmatique, 31:204–256, 1977.10.1007/BF02813304.

[15] R. Goldblatt. Lectures on the Hyperreals: An introduction to nonstandard analysis, volume188 of Graduate Texts in Mathematics. Springer-Verlag, 1998.

[16] W. T. Gowers. A new proof of Szemeredi’s theorem. Geometric And Functional Analysis,11(3):465–588, Aug 2001. 10.1007/s00039-001-0332-9.

[17] W. T. Gowers. Hypergraph regularity and the multidimensional Szemeredi theorem. Ann.of Math. (2), 166(3):897–946, 2007.

[18] Ben Green and Terence Tao. An inverse theorem for the Gowers U sp 3(G) norm. Proc.Edinb. Math. Soc. (2), 51(1):73–153, 2008.

[19] Ben Green and Terence Tao. The primes contain arbitrarily long arithmetic progressions.Ann. of Math. (2), 167(2):481–547, 2008.

[20] Ben Green, Terence Tao, and Tamar Ziegler. An inverse theorem for the Gowers U4-norm.Glasg. Math. J., 53(1):1–50, 2011.

[21] B. Host and B. Kra. Nonconventional ergodic averages and nilmanifolds. Ann. of Math.(2), 161(1):397–488, 2005. 10.4007/annals.2005.161.397.

[22] Ehud Hrushovski. Stable group theory and approximate subgroups.http://www.math.huji.ac.il/ ehud/NQF/nqf.pdf, December 2009.

[23] A. Kechris. Classical Descriptive Set Theory, volume 156 of Graduate Texts in Mathematics.Springer-Verlag, 1995.

[24] H. J. Keisler. Probability quantifiers. In Model-theoretic logics, Perspect. Math. Logic, pages509–556. Springer, New York, 1985.

[25] B. Kim and A. Pillay. Simple theories. Ann. Pure Appl. Logic, 88(2-3):149–164, 1997.[26] Y. Kohayakawa. Szemeredi’s regularity lemma for sparse graphs. In Foundations of compu-

tational mathematics (Rio de Janeiro, 1997), pages 216–230. Springer, Berlin, 1997.[27] Brendan Nagle, Vojtech Rodl, and Mathias Schacht. The counting lemma for regular k-

uniform hypergraphs. Random Structures Algorithms, 28(2):113–179, 2006.[28] Endre Szemeredi. Regular partitions of graphs. In Problemes combinatoires et theorie des

graphes (Colloq. Internat. CNRS, Univ. Orsay, Orsay, 1976), volume 260 of Colloq. Inter-nat. CNRS, pages 399–401. CNRS, Paris, 1978.

[29] T. Tao. An Introduction to Measure Theory, volume 126 of Graduate Studies in Mathemat-ics. American Mathematical Society, 2011.

[30] Terence Tao. A correspondence principle between (hyper)graph theory and probability the-ory, and the (hyper)graph removal lemma. Journal d’Analyse Mathmatique, 103:1–45, 2007.10.1007/s11854-008-0001-0.

[31] Terence Tao. Norm convergence of multiple ergodic averages for commuting transforma-tions. Ergodic Theory Dynam. Systems, 28(2):657–688, 2008.

[32] Henry Towsner. Convergence of diagonal ergodic averages. Ergodic Theory Dynam. Systems,29(4):1309–1326, 2009. 10.1017/S0143385708000722.

[33] Henry Towsner. An analytic approach to sparse hypergraphs: Hypergraph removal. sub-mitted, 2012.


University of Illinois at Chicago, Department of Mathematics, Statistics, andComputer Science, Science and Engineering Offices (M/C 249), 851 S. Morgan St.,Chicago, IL 60607-7045, USA

E-mail address: [email protected]: www.math.uic.edu/~isaac

University of Pennsylvania, Department of Mathematics, 209 S. 33rd Street,Philadelphia, PA 19104-6395

E-mail address: [email protected]: www.sas.upenn.edu/~htowsner

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