H-structures and generalized measurable structures
aa r X i v : . [ m a t h . L O ] S e p H -STRUCTURES AND GENERALIZED MEASURABLESTRUCTURES ALEXANDER BERENSTEIN, DAR´IO GARC´IA, AND TINGXIANG ZOU
Abstract.
We study H -structures associated to SU -rank one measurable structures.We prove that the SU -rank of the expansion is continuous and that it is uniformlydefinable in terms of the parameters of the formulas. We also introduce notionsof dimension and measure for definable sets in the expansion and prove they areuniformly definable in terms of the parameters of the formulas. Introduction
We say that a theory T is geometric if for any model M | = T the algebraic closuresatisfies the exchange property and T eliminates the quantifier ∃ ∞ . When a theory isgeometric and M | = T , ( P ( M ) , acl ) is a pregeometry and elimination of ∃ ∞ guaranteesthat for every formula ϕ (¯ x, ¯ y ) and every k ≤ | ¯ x | , { ¯ a ∈ M | y | : dim( ϕ (¯ x, ¯ a )) = k } is adefinable set. There are many examples of geometric theories, among them SU-rank 1theories, dense o-minimal theories, the field of p-adic numbers in a single sort, etc.In this paper we will deal with dense codense expansions of geometric theories. Recallthat if we let H be a new unary predicate and M | = T , the expansion ( M, H ( M )) is dense-codense (see Definition 2.1) if it satisfies the density property (roughly, for anynon-algebraic L -formula ϕ ( x, ¯ b ) with parameters in M , ϕ ( H ( M ) , ¯ b ) = ∅ ) and M satisfiesthe extension property over H ( M ) (roughly, for any non-algebraic L -formula ϕ ( x, ¯ b ) withparameters in M , ϕ ( M, ¯ b ) \ acl ( H ( M )¯ b ) = ∅ ).Lovely pairs are dense-codense expansions where H ( M ) (cid:22) M and H -structures aredense-codense expansions where H ( M ) is algebraically independent.Many properties such as stability and NIP are preserved when going from a geometrictheory to the corresponding theories of dense/codense pairs. Similarly, when we startwith an SU-rank 1 theory, the algebraic dimension coincides with an abstract dimensioncoming from forking-independence, and the expansion corresponding to H -structureshas a supersimple theory of SU -rank less than or equal to ω (see [3]).A measurable structure (see [8]) is a structure equipped with a function that assigns adimension and a measure to every definable set, which is uniformly definable in terms oftheir parameters and satisfy certain conditions similar to those satisfied by ultraproductsof finite fields. In this setting not only do we have control on the dimension of definable Date : 17th September 2020.2010
Mathematics Subject Classification.
Key words and phrases. supersimple theories, H -structures, measurable structures, asymptoticclasses, pseudofinite dimension.The authors would like to thank Facultad de Ciencias Universidad de los Andes for its supportthrough the project Expansiones densas de teor´ıas geom´etricas: grupos y medidas . The second authorwas supported by the Programa Estancias Posdoctorales de Colciencias, at Universidad de los Andes.Contract FP44842-178-2018, Convocatoria 784-2017. The third author is supported by the ChinaScholarship Council and partially supported by ValCoMo (ANR-13-BS01-0006). She is also grateful tothe Lyon logic group, especially to Ita Ben-Yaacov for the support of her visit to Colombia. sets, but the measure provides a finer account of the size of the corresponding set. Somemeasurable structures can be built as ultraproducts of structures in a 1-dimensionalasymptotic class and are pseudofinite of SU -rank 1 (see [8, Lemma 4.1]). The work ofthe third author [11] shows that in this special case, the corresponding H -structure isalso pseudofinite. In constrast, the associated theory of a lovely pair (cf. [2], [9]) maynot be pseudofinite [11].It is a natural question to test whether these expansions, assuming the underly-ing theory is measurable of SU -rank one, become some sort of generalized measurablestructures . Namely, can we construct appropriate notions of dimension and measurewhich depend definably on the parameters of the defining formulas for H -structures ofmeasurable theories?The goal of this paper is to give a positive solution to this poblem. Our solutiondepends on a detailed study of SU -rank in H -structures, showing that it is continuousand Cantor additive. Drawing parallels with the notion of measurable structures, we useinformation associated to the SU -rank of a formula as an abstract notion of dimensionand prove that for each formula it is uniformly definable in terms of its parameters. Inaddition, and using the fact that the base theory comes from a measurable structures,we introduce a notion of measure for definable sets in H -structures, proving also uniformdefinability.We would like to point out that some expansions of measurable structures are alreadyknown to preserve measurability. For example, when T is measurable, eliminates ∃ ∞ and acl = dcl , then adding a generic predicate does not change forking (see [6, Theorem2.7]), and the result gives a new measurable structure (see [8, Theorem 5.11]). Thereis even some liberty on the choice of the size of the new predicate. By contrast, H -structures of non-trivial theories (and in some cases lovely pairs) increase the SU-rankfrom finite to infinite. As measurable structures are essentially finite-dimensional, wecould not expect that H-structures/lovely pairs to stay within this family. Instead, theresulting notions of dimension and measure for H -structures satisfy the properties of a generalized measurable structure , which is a generalization of the concept of measurablestructures. This idea has been studied by several authors and will appear in the paper[1] by S. Anscombe, D. Macpherson, C. Steinhorn and D. Wolf, which is currently inpreparation.This paper is divided as follows. In Section 2 we recall the definition of H -structuresand show some of its properties. In particular we characterize the SU -rank of a tupleand prove it is Cantor-additive. In Section 3 we show that the SU -rank of a formulais uniformly definable in terms of its parameters. In Section 4 we use these results toshow, under certain additional hypothesis, that if the SU-rank of a tuple is ω · n + k , thenthe large dimension n corresponds to the coarse pseudofinite dimension with respectto M , while the number k corresponds to the coarse dimension with respect to H ( M ).Finally in Section 5 we introduce measures for formulas and prove they are uniformlydefinable in terms of their parameters.2. Preliminaries: H -structures of geometric theories. In this section, we review the notions of H -structures of geometric theories from [2],[3]and their basic properties.Recall that a theory T in a language L is called geometric if (1) it eliminates thequantifier ∃ ∞ and (2) in all models of T , the algebraic closure satisfies the exchangeproperty. Whenever T is geometric and M | = T , ¯ a in a finite tuple in M and B, C ⊂ M -STRUCTURES AND GENERALIZED MEASURABLE STRUCTURES 3 we write dim(¯ a/B ) for the length of a maximal algebraically independent subtuple of¯ a and we write ¯ a | ⌣ B C when dim(¯ a/B ∪ C ) = dim(¯ a/B ). When A ⊂ M , we write A | ⌣ B C when ¯ a | ⌣ B C for all finite tuples ¯ a in A . We shall use the word independence to mean algebraic independence. Definition 2.1.
Given a geometric complete theory T in a language L and a model M | = T , let H be a new unary predicate symbol and let L H := L ∪ { H } be the extendedlanguage. Let ( M, H ( M )) denote an expansion of M to L H , where H ( M ) := { a ∈ M | H ( a ) } .(1) ( M, H ( M )) is called a dense expansion if, for any non-algebraic L -type p ( x ) ∈ S ( A ) where A ⊂ M has a finite dimension, p ( x ) has a realization in H ( M ).Whenever this is the case we say that ( M, H ( M )) has the density property .(2) ( M, H ( M )) has the extension property if, for any non-algebraic L -type p ( x ) ∈ S ( A ) where A ⊂ M has a finite dimension, p ( x ) has realizations in M \ acl L ( AH ( M )).(3) An expansion ( M, H ( M )) which satisfies the density and the extension propertyis called an H -structure if, in addition, H ( M ) is an L -algebraically independentsubset of M .It is easy to show by induction on the number of variables that the density propertyand the extension property also hold for types of independent tuples. Lemma 2.2.
Let T be a complete geometric theory and let ( M, H ( M )) be an expansionsatisfying the density property and the extension property. Then whenever A ⊂ M hasa finite dimension and p ( x , . . . , x n ) ∈ S n ( A ) is an L -type of dimension n , p ( x , . . . , x n ) has realizations both in H ( M ) n and in ( M \ acl ( AH ( M )) n . Furthermore, we can choose ¯ b ∈ ( M \ acl ( AH ( M ))) n realizing p ( x , . . . , x n ) with dim(¯ b/ acl ( AH ( M ))) = n . We will also use the expressions density property and extension property for thegeneralized versions of these properties. Although the class of H -structures is not afirst order class, it corresponds to the class of sufficiently saturated models of somecomplete theories: Theorem 2.3 ([2], [3]) . Given any geometric complete theory T , all the H -structuresassociated with T are elementarily equivalent to one another. Furthermore, any | T | + -saturated model of the common theory of H -structures is again an H -structure. Notation 2.4.
We will write T ind to denote the common complete theories of the H -structures associated with T . Whenever ( M, H ) | = T ind , and ¯ a is a tuple in M , we writetp L (¯ a ) for the type in the language L and tp L H (¯ a ) for the type in the language L H .Also, for A ⊂ M , we write H ( A ) for A ∩ H . We will also write H (¯ a ) for the subtupleof ¯ a consisting of those elements that also belong to H . Definition 2.5.
Let (
M, H ( M )) | = T ind . A subset A ⊂ M is called H -independent if A | ⌣ H ( A ) H ( M ). Lemma 2.6 ([2], [3]) . Let ( M, H ( M )) | = T ind . Then for any H -independent tuples ¯ a and ¯ b , tp L H (¯ a ) = tp L H (¯ b ) ↔ tp L (¯ aH (¯ a )) = tp L (¯ bH (¯ b )) Lemma 2.7 ([2], [3]) . Any formula in T ind is equivalent to a boolean combination offormulas of the form ∃ y ∈ H · · · ∃ y k ∈ Hϕ (¯ x, ¯ y ) , where ϕ (¯ x, ¯ y ) is a L -formula. ALEXANDER BERENSTEIN, DAR´IO GARC´IA, AND TINGXIANG ZOU
Furthermore, definable subsets of H n are just traces of L -formulas in H n : if ( M, H ( M )) | = T ind and X ⊂ H ( M ) n is L H -definable, there is an L -formula ϕ (¯ x, ¯ y ) and there is ¯ c ∈ M such that X = ϕ ( H n , ¯ c ) . Definition 2.8.
Let (
M, H ( M )) | = T ind be sufficiently saturated. For any subset A ⊂ M , scl ( A ) := acl L ( A ∪ H ( M ))is called the small closure of A . Any subset B ⊂ scl ( A ) is called A -small . A definableset X is small if there is a finite set A such that X is A -small. For any tuple b andany set A , we write ldim( b/A ) = dim( b/AH ( M )), it is called the large dimension of thetuple b over A (see [3]) and it corresponds to the dimension of the tuple b with respectto A in the pregeometry localized at H ( M ).The following properties and definitions come from [2], [3]; the properties follow easilyfrom the definitions. Proposition 2.9.
Let ( M, H ( M )) | = T ind . (1) For any tuple ¯ a , there exists some finite tuple ¯ h in H ( M ) such that ¯ a | ⌣ ¯ h H ( M ) .And, for such ¯ h , ¯ a ¯ h is H -independent. (2) If ¯ a is any H -independent tuple, then for any finite tuple ¯ h in H ( M ) , ¯ a ¯ h is also H -independent. (3) If ( M, H ) is an H -structure and ¯ a is a tuple and A ⊂ M is H -independent, thenthere is a minimal ¯ h in H ( M ) such that ¯ a ¯ h | ⌣ A H ( M ) . This tuple is unique upto permutation , we call it the H -basis of ¯ a over A and we denote it by HB(¯ a/A ) . (4) Assume ( M, H ( M )) | = T ind . Then for any tuple ¯ b and for any finite tuple ¯ h in H ( M ) , we have HB(¯ b ¯ h ) = HB(¯ b )¯ h . Proposition 2.10.
Let ( M, H ( M )) | = T ind . Then for any tuple ¯ a , acl L H (¯ a ) = acl L (¯ a HB(¯ a )) . Note that by the previous proposition, whenever (
M, H ( M )) | = T ind and A ⊂ M , acl L H ( A ) is always H -independent. So if we are given tuples ¯ a , ¯ b tuples, the setHB(¯ b/ acl L H (¯ a )) is well defined. This observation allows us to prove the following ad-ditivity property: Proposition 2.11 (Additivity of HB) . Let ( M, H ( M )) | = T ind and let ¯ a , ¯ b be tuples.Then HB(¯ a ¯ b ) = HB(¯ a ) ∪ HB(¯ b/ acl L H (¯ a )) . In particular, if in the proposition above the tuple ¯ a is H -independent, we haveHB(¯ a ¯ b ) = H (¯ a ) ∪ HB(¯ b/ ¯ a ). Sometimes we will abuse the notation and write HB(¯ b/ ¯ a )for HB(¯ b/ acl L H (¯ a )). Definition 2.12.
Let T be a SU -rank 1 theory and let M | = T be | T | + -saturated. Let A ⊂ M with | A | ≤ | T | and let p ( x ) ∈ S ( A ) be non-algebraic. We say p ( x ) is non-trivial if for some (all) realization a | = p ( x ) there are finite subsets ¯ b, ¯ c of M such that a | ⌣ A ¯ b , a | ⌣ A ¯ c and a ∈ acl ( A, ¯ b, ¯ c ). Note that we may take ¯ c = c to be a singleton and ¯ b to bean independent tuple over A .We say T is nowhere trivial if for all A ⊂ M with | A | ≤ | T | and p ( x ) ∈ S ( A ), if p ( x )is non-algebraic, then it is non-trivial. Example 2.13.
There are several examples of SU -rank one nowhere trivial theories,for example pseudofinite fields and the theory of torsion free divisible abelian groups; inboth cases there is a group structure and using group generics is easy to build algebraictriangles. -STRUCTURES AND GENERALIZED MEASURABLE STRUCTURES 5 Now consider the language L = { R + , R × , P } , where R + , R × are ternary relations, and P is a binary relation. Let M be a model where we interpret P ( M ) as a pseudofinitefield where R + , R × are the graphs of addition and the multiplication. On the otherhand, for mixed elements between P ( M ) and ¬ P ( M ) the graphs R + , R × are alwaysempty and for the elements in ¬ P ( M ), we interpret R + ( a, a, b ) as a random graph and R × is always empty. Then T h ( M ) is not nowhere trivial, inside ¬ P ( M ) all types aretrivial.Note that in Definition 2.12 the subsets ¯ b, ¯ c can be taken so that each of them is A -independent and ¯ b | ⌣ A ¯ c . It is proved in [3] that whenever T be a SU -rank 1 theory, T ind is supersimple of SU -rank less than or equal to ω and it is equal to ω whenever T has a non-trivial type. We will revisit the arguments from [3] and calculate explicitlythe SU-rank of types when T is nowhere trivial. We will need the following two resultsfrom [3]: Proposition 2.14.
Let T be a SU -rank 1 theory. Let ( M, H ( M )) | = T ind , let B ⊂ M be H -independent, let ¯ a be tuple and assume that ¯ a ∈ H ( M ) . Let k = dim(¯ a/B ) , then SU(tp L H (¯ a/B )) = k . Proposition 2.15. (Corollary 5.8 [3] ) Let T be a SU -rank 1 theory. Let ( M, H ( M )) | = T ind , let B ⊂ M be H -independent, let a be a singleton and assume that a acl L ( BH ( M )) and that tp( a/B ) is not trivial. Then SU(tp L H ( a )) = ω . Proposition 2.16.
Let T be a SU -rank 1 theory which is nowhere trivial. Let ( M, H ( M )) | = T ind , let B ⊂ M be H -independent and let ¯ a be tuple and write ¯ a = ¯ a ¯ a so that ¯ a is independent over BH ( M ) and ¯ a ∈ acl (¯ a H ( M ) B ) . Let n = dim(¯ a/BH ( M )) = | ¯ a | and let k = | HB (¯ a/B ) | . Then SU(tp L H (¯ a/B )) = ω · n + k .Proof. We may assume (
M, H ( M )) | = T ind is sufficiently saturated. We will prove theargument for B = ∅ , it is an easy exercise to generalize the proof for the general case.Case 1. First assume that ¯ a = a and that a acl ( H ( M )). Then by Proposition 2.15we have that SU(tp L H ( a )) = ω .Case 2. Now assume that ¯ a = ¯ a is an H -independent tuple and let n = | ¯ a | . Thenby Lascar’s inequalities and Case 1, we have SU(tp L H (¯ a )) = ω · n .Case 3. Now assume that ¯ a = ¯ a ¯ a where ¯ a is independent over H ( M ) and ¯ a ∈ acl (¯ a H ( M )). By Case 2 we have SU(tp L H ( ¯ a )) = ω · n . Since ¯ a and HB (¯ a ) are L H -interalgebraic over ¯ a , it follows from Proposition 2.14 that SU(tp L H ( ¯ a / ¯ a )) = | HB (¯ a ) | = k . Again using Lascar’s inequalities we get that SU(tp L P (¯ a )) = ω · n + k . (cid:3) The result above can be generalized to arbitrary SU -rank 1 theories, but the fulldescription is more elaborated. Here we provide a particular case. Observation 2.17.
Let T be a SU -rank 1 theory. Let ( M, H ( M )) | = T ind , let B ⊂ M be H -independent and let ¯ a be tuple and write ¯ a = ¯ a ¯ a ¯ a so that:(1) The tuple ¯ a is H -independent and for each element c ∈ ¯ a , tp( c/B ) is trivial.(2) The tuple ¯ a is H -independent and for each element c ∈ ¯ a , tp( c/B ) is nottrivial.(3) ¯ a ∈ acl (¯ a ¯ a BH ( M )).Let m = | ¯ a | , n = | ¯ a | , k = | HB (¯ a/B ) | . Then SU(tp L H (¯ a/B )) = ω · n + k + m .In this paper we will deal with SU -rank 1 theories which are nowhere trivial, whereProposition 2.16 holds. The SU-rank has nice properties under this assumptions. Forinstance, it has the following additivity property. ALEXANDER BERENSTEIN, DAR´IO GARC´IA, AND TINGXIANG ZOU
Lemma 2.18.
Let T be a SU -rank 1 theory which is nowhere trivial. Let ( M, H ( M )) | = T ind , let C ⊂ M be H -independent and let ¯ a and ¯ b be tuples. Then SU(tp L H (¯ a ¯ b/C )) =SU(tp L H (¯ a/C )) ⊕ SU(tp L H (¯ b/C ¯ a )) .Proof. Write ¯ a = ¯ a ¯ a so that ¯ a is independent over H ( M ) C and ¯ a ∈ acl (¯ a H ( M ) C ).Let n = | ¯ a | and let k = | HB (¯ a/C ) | , then by Proposition 2.16, SU(tp L H (¯ a/C )) = ω · n + k . Similarly write ¯ b = ¯ b ¯ b so that ¯ b is independent over C ¯ aH ( M ) and¯ b ∈ acl (¯ aH ( M ) C ). Let n = | ¯ b | and let k = | HB (¯ b/ ¯ aC ) | , so by Proposition 2.16,SU(tp L H (¯ b/C ¯ a )) = ω · n + k .Now consider the tuple ¯ a ¯ b . The subtuple ¯ a ¯ b is independent over CH ( M ) and¯ a ¯ b ∈ acl (¯ a ¯ b H ( M ) C ). By Proposition 2.11, we have that HB (¯ a ¯ b/C ) = HB (¯ a/C ) ∪ HB (¯ b/ ¯ aC ) and the later union is a disjoint union, so | HB (¯ a ¯ b/C ) | = k + k . Again usingProposition 2.16 we get SU(tp L H (¯ a ¯ b/C )) = ω · ( n + n ) + ( k + k ) as we wanted. (cid:3) Remark 2.19.
Cantor-additivity of SU -rank seems not to be a usual property of su-persimple theories of infinite SU -rank. It does not hold for ACFA (see 4.17 in [5]): if a is transformally trascendental and b = σ ( a ) − a , then SU( a, b ) = ω , SU( a/b ) = 1and SU( b ) = ω . A similar example can be build for T = DCF taking a to be genericand b = δ ( a ) − a . Are there structural consequences of supersimple theories of infinite SU -rank that have Cantor-additivity?3. Existential L H -formulas and definability of SU-rank In this section we deal with L H -formulas of the form ∃ ¯ z ∈ H | ¯ z | φ ( x, ¯ z, ¯ y ), where φ ( x, ¯ z, ¯ y ) be an L -formula. We will call such formulas existential H -formulas , and wewill see that these formulas are fundamental in the analysis of ranks for definable setsin H -structures: they witness the continuity of the SU -rank, and any other L H -formulacan be approximated uniformly by existential H -formulas, up to smaller SU -rank. Wewill use them to show that in T ind the SU-rank of a formula is definable in terms of itsparameters. Lemma 3.1.
Let ( M, H ) be an H -structure and ¯ c be a tuple. Suppose h ∈ acl L H (¯ c ) ∩ H .Then h ∈ HB(¯ c ) .Proof. By Proposition 2.10, acl L H (¯ c ) = acl L (¯ c, HB(¯ c )). Therefore h ∈ acl L (¯ c, HB(¯ c ))and h ∈ H . By definition of H -bases, ¯ c, HB(¯ c ) | ⌣ HB(¯ c ) H , so h ∈ acl L (HB(¯ c )). Since H is an acl L -independent set, we get h ∈ HB(¯ c ). (cid:3) Lemma 3.2.
Let φ ( x, z, y ) be an L -formula and consider the L H -formula defined by ψ ( x, y ) := ∃ z ∈ H | z | φ ( x, z, y ) . Then, the set E = { a ∈ M | y | : ψ ( x, a ) is non-algebraic } is L H -definable.Proof. We work in a sufficiently saturated H -structure ( M, H ), and let us first fix atuple a . By elimination of ∃ ∞ in T , there is an L -formula χ φ ( z, y ) such that for any c, a ∈ M , φ ( x, c, a ) is non-algebraic if and only if M | = χ φ ( c, a ). So, the L H -formula ∃ z ∈ H ( χ φ ( z, y )) defines precisely the set E = { a ∈ M | y | : φ ( x, h, a ) is non-algebraic for some h ∈ H | z | } , which is contained in { a ∈ M | y | : ψ ( x, a ) is non-algebraic } . Thus, we can restrict ourattention to E \ E , and so we can assume that φ ( x, h, a ) is algebraic for every tuple h ∈ H . Again by elimination of ∃ ∞ in the theory T , there is a formula η ( z, y ) such thatfor every tuple c, a from M , M | = η ( c, a ) if and only if φ ( x, c, a ). Thus, by working with -STRUCTURES AND GENERALIZED MEASURABLE STRUCTURES 7 φ ( x, z, y ) ∧ η ( z, y ) instead of φ ( x, z, y ), we can even assume that for every c, a from M ,the formula φ ( x, c, a ) is algebraic.We will now prove by induction on the length of the tuple z that the set E \ E isdefinable. To start, suppose z = z has length 1. Claim 1:
The L -formula ∃ z ( φ ( x, z, a )) is not algebraic if and only if the L H -formula ∃ z ∈ H ( φ ( x, z, a )) is not algebraic.Proof of Claim 1: The right to left direction is clear. Suppose now that ∃ z ( φ ( x, z, a ))is not algebraic. For every n ∈ N , let us consider the formula θ n ( z , . . . , z n , a ) := ∃ x . . . ∃ x n ^ i ≤ n φ ( x i , z i , a ) ∧ ^ ≤ i For n > , the L H -formula ∃ z ∈ H . . . ∃ z n ∈ H ( φ ( x, z , . . . , z n , a )) is notalgebraic if and only if there is i ≤ n and h i ∈ H such that the formula Ψ i ( x, h i , a ) := ∃ z ∈ H . . . ∃ z i − ∈ H ∃ z i +1 ∈ H . . . ∃ z n ∈ H ( φ ( x, z , . . . , z i − , h i , z i +1 , . . . , z n , a )) is not algebraic.Proof of Claim 2: As before, the right to left direction is immediate. Suppose now that ∃ z ∈ H . . . ∃ z n ∈ H ( φ ( x, z , . . . , z n , a )) is not algebraic. If Ψ ( x, h , a ) is not algebraicfor some h ∈ H we are done. Similarly, if Ψ ( x, h , a ) is not algebraic for some h ∈ H ,we are done.Thus, assume that for every h , h ∈ H the formulas ψ ( x, h , a ) and ψ ( x, h , a ) areboth algebraic. By saturation, there is a realization b of ∃ z ∈ H . . . ∃ z n ∈ H ( φ ( x, z , . . . , z n , a )) ALEXANDER BERENSTEIN, DAR´IO GARC´IA, AND TINGXIANG ZOU that is independent from a, HB( a ). For this element, there are h , h ∈ H such that( M, H ) | = ∃ z ∈ H . . . ∃ z n ∈ H ( φ ( b, h , h , z , . . . , z n , a )) . In particular, b realizes the formulas ψ ( x, h , a ) and ψ ( x, h , a ), and we have b ∈ acl L H ( h , a ) = acl L ( h , a, HB( a, h )) = acl L ( h , a, HB( a )) . Similarly, b ∈ acl L ( h , a, HB( a )), and since b is independent from a, HB( a ), b acl L ( a, HB( a ).Thus, by the exchange property, we have h ∈ acl L ( b, a, HB( a )) ⊆ acl ( h , a, HB( a )) . So by Lemma 3.1 and item 4 in Proposition 2.9, h ∈ HB(HB( a ) h ) = HB( a ) h ,and since h = h , h ∈ HB( a ). This would imply b ∈ acl ( a, HB( a )), a contradic-tion. (cid:3) Claim 2 By induction hypothesis, for every formula ψ i ( x ; z i , y ) there is an L H -formula χ i ( z i , y )such that for every tuple c i , a ∈ M , ψ i ( x, c i , a ) is non-algebraic if and only if χ i ( c i , a )holds. Therefore, by Claim 2, the set E = { a ∈ M y : ∃ z , . . . , z n ∈ H ( φ ( x, z , . . . , z n , a )) is non-algebraic } is defined by the L H -formula ∃ z ∈ H ( χ φ ( z, y )) ∨ n _ i =0 ( ∃ z i ∈ H χ i ( z i , a )) . This concludesthe proof. (cid:3) Lemma 3.3. Let ( M, H ) be an H -structure, suppose ¯ c is a tuple and ¯ h is an enumera-tion of HB(¯ c ) with | ¯ h | = n . Then there is an L -formula ϕ ¯ c (¯ z, ¯ y ) such that the followinghold: (1) ( M, H ) | = ϕ ¯ c (¯ h, ¯ c ) . (2) Whenever ¯ h ′ is an n -tuple in H such that ( M, H ) | = ϕ ¯ c (¯ h ′ , ¯ c ) we have that ¯ h ′ isanother enumeration of HB(¯ c ) . (3) For all ¯ d ∈ M | ¯ c | , the set ϕ ¯ c ( H n , ¯ d ) is finite.Proof. Fix ¯ h an enumeration of HB(¯ c ) and write ¯ c = ¯ c ¯ c with ¯ c independent over H and ¯ c ∈ acl (¯ c , ¯ h ). Let ϕ (¯ y ; ¯ c , ¯ h ) be an algebraic L -formula satisfied by ¯ c . Since H -bases as sets are unique (see Proposition 2.9), it is easy to see that if there is ¯ h ′ ∈ H n with ϕ (¯ c ; ¯ c , ¯ h ′ ) then ¯ h ′ is another enumeration of HB(¯ c ).For 0 ≤ j < n , we can reorganize the object variables and define the L -formulas ϕ j ( z ; t , . . . , t j − , t j +1 , . . . , t n − , ¯ y ) := ϕ (¯ y , ¯ y , t , . . . , t j − , z, t j +1 , . . . , t n − ) , where ¯ y = ¯ y ¯ y and let ψ j ( t , . . . , t j − , t j +1 , . . . , t n − , ¯ y ) be the L -formula stating that ϕ j is an algebraic formula in the variable z . Claim: For every j = 0 , . . . , n , the formula ψ j ( h , . . . , h j − , h j +1 , . . . , h n − , ¯ c ) holds. Otherwise, ϕ ( h , . . . , h j − , M, h j +1 , . . . , h n − , ¯ c ) is infinite, and by the density prop-erty we have that ϕ ( h , . . . , h j − , H, h j +1 , . . . , h n − , ¯ c ) is also infinite, contradicting that ϕ ( H n , ¯ c ) is a set of some enumerations of HB(¯ c ).Let ϕ ¯ c (¯ z, ¯ y ) := ϕ (¯ y , ¯ y , ¯ z ) ∧ ^ i = j ( z i = z j ) ∧ ^ ≤ j Combining Lemma 3.1 and the previous lemma, we know that for any ϕ ¯ c defined as in Lemma 3.3, we have that for any tuple d , ϕ ¯ c ( H n , ¯ d ) is a collection ofdifferent enumerations of HB( ¯ d ).Recall that the SU-rank of a formula ϕ ( x ), which is defined over a set of parameters c , is given by: SU( ϕ ) := sup { SU( p ) : ϕ ∈ p, p ∈ S T ind ( c ) } . We will now prove that this supremum can be attained as a maximum. Lemma 3.5. Let M := ( M, H ) be a sufficiently saturated H -structure. Let X ⊂ M n be L H -definable over c where c is H -independent. Then there is a type p ∈ S T ind ( c ) such that SU( X/c ) = SU( p ) . That is, SU( X/c ) = max { SU( p ) : X ∈ p, p ∈ S T ind ( c ) } . Proof. Let ϕ (¯ x, ¯ c ) be an L H -formula defining X and let p ( x ) be a complete L H -typeover c extending ϕ ( x, c ). By saturation, there is a ∈ X with a | = p ( x ). Suppose a = ( a , . . . , a | x | ). Let a = ( a i , . . . , a i r ) be a maximal independent subtuple of a over Hc and let a = ( a j , . . . , a j t ) be the rest of tuple a . Then a ∈ acl L ( a H ¯ c ). LetHB( a/c ) = { h , . . . , h k p } and fix an enumeration h = ( h , . . . , h k p ) of HB( a/c ). Since c is H -independent, by additivity of the H -bases we have HB(¯ a ¯ c ) = HB(¯ a/ ¯ c ) H (¯ c ) ⊂ ¯ h ¯ c so we also have a ∈ acl L ( a h ¯ c ).Let Ψ p ( x p ; x p , z, c ) be an L -algebraic formula with the separation of variables x p =( x i , . . . , x i r ) and x p = ( x j , . . . , x j t ) (which depends on p ) and let ℓ > M | = ∀ x p ∀ z ∃ ≤ ℓ x p Ψ p ( x p ; x p , z, c )and M | = Ψ p ( a ; a , h, c ). Then Θ p ( x ) = ∃ z ∈ H k p Ψ p ( x p ; x p , z, c ) ∈ p ( x ). Note thatfor any b satisfying Θ p ( x ), we have SU( b/c ) ≤ ω · r + k p (recall that r = | x p | ). On theother hand, SU( p ) = ω · r + k p . Therefore, SU(Θ p ) = SU( p ).Notice that Θ p ( x ) is an existential H -formula and { Θ p ( x ) : p ∈ S T ind ( c ) , X ∈ p } isan open covering of ϕ ( x, c ). By compactness, there are finitely many types p , . . . , p N and formulas Θ p ( x ) , . . . , Θ p N ( x ) such that ϕ ( x, c ) ⊢ W i ≤ N Θ p i ( x ). Hence,SU( X ) = SU( ϕ ( x, c )) ≤ SU _ i ≤ N Θ p i ( x ) = max { SU( p i ) : i ≤ N } , and since by definition we have ϕ ∈ p i for each i ≤ N , we also haveSU( X ) = SU( ϕ ) := sup { SU( p ) : ϕ ∈ p, p ∈ S T ind ( c ) } ≥ max { SU( p i ) : i ≤ N } . Therefore, SU( ϕ ( x, c )) = max { SU( p ) : ϕ ∈ p, p ∈ S T ind ( c ) } = SU _ i ≤ N Θ p i ( x ) . (cid:3) We will use repeatedly the previous construction, so it is important to fix somenotation for the rest of the paper. Notation 3.6. Let M := ( M, H ) be a sufficiently saturated model of T ind . Let c ∈ M be H -independent and let p ( x ) be a complete L H -type over c . Let a | = p ( x ) andchoose a = ( a i , . . . , a i r ) which is maximal independent subtuple of a over Hc and let a = ( a j , . . . , a j t ). Also let h = ( h , . . . , h k p ) be an enumeration of HB( a/c ).Then a ∈ acl ( a h ), so we can fix an L -formula Ψ p ( x p ; x p , z, c ) with x p = ( x i , . . . , x i r )and x p = ( x j , . . . , x j t ) and ℓ > M | = Ψ p ( a ; a , h, c ) ∧ ∃ ≤ ℓ x p Ψ p ( x p ; a p , h, c )We can do this process for each enumeration h of the H -basis of a , so we may assumethat for any enumeration h of the H -basis of a we have that M | = Ψ p ( a ; a , h, c ) ∧ ∃ ≤ ℓ x p Ψ p ( x p ; a p , h, c )By elimination of ∃ ∞ , we may choose Ψ p so that( M, H ) | = ∀ x p ∀ z ∃ ≤ ℓ x p Ψ p ( x p ; x p , z, c ) , and we will also write Θ p ( x ) = ∃ z ∈ H k p Ψ p ( x p ; x p , z, c ) . Corollary 3.7. Let M = ( M, H ) be a sufficiently saturated H -structure and let c bean H -independent tuple. Let p ( x ) be a complete type over c of SU-rank ω · n + k . Let Ψ p as described in the Notation 3.6. Then for any a = a , a with SU( a/c ) = ω · n + k and any ¯ h ∈ H k , Ψ p ( a , a , h, c ) holds if and only if h is an enumeration of HB( a/c ) .Proof. If ( M, H ) | = Ψ p ( a ; a , h, c ), since Ψ p ( x , a , h, c ) is algebraic in the variables x ,we have a ∈ acl L ( a hc ). As | x | = n and SU( a/c ) ≥ ω · n , we must have dim( a /H, c ) = n . Therefore, a | ⌣ h,c H . Since | HB( a/c ) | = k and { h } has size at most k , we must havethat h is minimal, hence it is an enumeration of HB( a/c ). For the other direction, notethat Ψ p was chosen so that Ψ p ( a ; a , h, c ) holds for every enumeration h of HB( a/c ). (cid:3) Notation 3.8. Since HB (¯ a ) is a set and not a tuple, we may need to consider itspermutation group. Whenever k ≥ 1, we write S k for the permutation group on k elements. Also, whenever z = z z . . . , z k is a tuple of variables and σ ∈ S k , we let σ ( z )denote the tuple z σ (1) z σ (2) . . . z σ ( k ) . This notation will be used for the rest of this paper. Lemma 3.9. Let M := ( M, H ) be a sufficiently saturated H -structure and let c be an H -independent tuple. Let X ⊂ M m be L H -definable over c and SU( X/ ¯ c ) = ω · n + k .Then there is Y ⊂ M m defined by an L H -formula η (¯ x, ¯ c ) , such that (1) η (¯ x, ¯ c ) is of the form ∃ ¯ z ∈ H k η ′ (¯ x ; ¯ z, ¯ c ) , where η ′ (¯ x ; ¯ z, ¯ c ) is an L -formula; (2) There are finitely many complete types p , . . . , p t of SU -rank ω · n + k such that M | = ∀ x ∀ z η ′ ( x, z, ¯ c ) → _ i ≤ t Ψ p i ( x, z, c ) , where Ψ p i is defined as in Notation 3.6. (3) Whenever ¯ a ∈ X satisfies SU(¯ a/ ¯ c ) = ω · n + k , and ¯ h is an enumeration of HB(¯ a/ ¯ c ) , we have M | = η ′ (¯ a, ¯ h, ¯ c ) ; (4) SU( Y △ X ) < ω · n + k . -STRUCTURES AND GENERALIZED MEASURABLE STRUCTURES 11 Furthermore, we may choose η ′ (¯ x ; ¯ z, ¯ c ) such that ¯ z is invariant under permutations.That is, for any σ ∈ S k we have M | = ∀ ¯ x ( η ′ (¯ x ; ¯ z, ¯ c ) ↔ η ′ (¯ x ; σ (¯ z ) , ¯ c ) .Proof. Let us consider the set E := { p ∈ S T ind ( c ) : X ∈ p, SU( p ) = ω · n + k } , whichis non-empty by Lemma 3.5. For every p ∈ E , we can consider the open Θ p ( x thatcontains p . Notice also that E := { p ∈ S T ind ( c ) : X ∈ p, SU( p ) ≥ ω · n + k } , so it isclosed, and by compactness there are finitely many types p , . . . , p N in E such that E is covered by Θ p ( x ) ∨ · · · ∨ Θ p N ( x ). Note that SU( p i ) = ω · n + k for all i ≤ N . Let Y be the set defined by the L H -existential formula _ i ≤ N Θ p i ( x ) = _ i ≤ N [ ∃ z ∈ H k (Ψ p i ( x, z, c ))] = ∃ z ∈ H k _ i ≤ N Ψ p i ( x, z, c ) . Then SU( Y ) = ω · n + k . If SU( Y △ X ) < SU( X ) we are finished. So, let us assumethat SU( Y △ X ) = ω · n + k . Since all types p with X ∈ p and SU( p ) = ω · n + k belongto Y , we get that SU( X \ Y ) < ω · n + k , and so SU( Y \ X ) = ω · n + k . Claim: For any q ∈ Y \ X with SU( q ) = ω · n + k there is a set Z q defined by anexistential H -formula such that E ⊆ Z q and q Z q . First, let us fix a type p ∈ X with SU( p ) = ω · n + k and choose realizations a | = p, b | = q . Let h a , h b be enumerations of HB( a/c ) , HB( b/c ) respectively. Notethat both tuples h a , h b have length k . Since p = tp L H ( a/c ) = tp L H ( b/c ) = q , byLemma 2.6 we get that tp L ( ah a /c ) = tp L ( bh b /c ) and there is an L -formula η pq ( x, z, c ) ∈ tp L ( a, h a , c ) \ tp L ( b, h b , c ). This process can be done uniformly in all enumerations h a , h b of HB( a/c ) , HB( b/c ) so we can choose η pq ( x, z, c ) to be always the same formula.Consider now the set Z pq defined by the existential H -formula ∃ z ∈ H k ( η pq ( x, z, c ) ∧ Ψ p ( x, z, c )) . Clearly Z pq ∈ p = tp L H ( a/c ), since M | = η pq ( a, h a , c ) ∧ Ψ p ( a, h a , c ). On the other hand, if Z pq ∈ q = tp L H ( b/c ), then there is a tuple h ∈ H k such that M | = η pq ( b, h, c ) ∧ Ψ p ( b, h, c ).By Lemma 3.7, this would imply that h is an enumeration of HB( b/c ), and since η pq ( x, z, c ) was chosen to be independent from the enumerations of HB( a/c ) , HB( b/c ),we necessarily obtain M = η pq ( b, h, c ). Hence, Z pq is an existential L H -formula thatseparates p from q .Note that the set { Z pq : p ∈ E } is an open cover of E and does not contain q . Since E is closed, we obtain again by compactness that there are types p , . . . , p m ∈ E suchthat Z q = m _ i =1 Z p i q = m _ i =1 [ ∃ z ∈ H k ( η p i q ( x, z, c ) ∧ Ψ p i ( x, z, c ))]= ∃ z ∈ H k m _ i =1 η p i q ( x, z, c ) ∧ Ψ p i ( x, z, c ) ! is a single existential H -formula Z q that separates E from q , as desired. (cid:3) Claim Given a type q ∈ Y \ X with SU( q ) = ω · n + k , let us write χ q ( x, z, c ) for an L -formula such that ∃ z ∈ H k χ q ( x, z, c ) extends E and does not contain q , which exists by the previous claim. Note that χ q ( x, z, c ) → W N q i =1 Ψ p i,q ( x, z, c ) for some N q , where thetypes p i,q all satisfy SU( p i,q ) = ω · n + k .Consider now the L H -type-definable (closed) set E = { q : q ∈ Y \ X and SU( q/c ) ≥ ω · n + k } . The collection { Z cq : q ∈ E } forms an open cover of E , and by compactness thereare finitely many types q , . . . , q s such that E ⊆ Z cq ∪ · · · ∪ Z cq s . Moreover, the sets Z q , . . . , Z q s extend E and the formulas defining them have the form ∃ z ∈ H k χ q i ( x, z, c ),for L -formulas χ q i ( x, z, c ), i = 1 , . . . , s . To simplify the notation, for 1 ≤ i ≤ s , write χ i ( x, z, c ) = χ q i ( x, z, c ).The set Y ′ = Y ∩ Z q ∩ · · · ∩ Z q s has the property that SU( Y ′ △ X ) < ω · n + k and if wesuppose that Y is defined by ∃ z ∈ H k χ ( x, z, c ), then Y ′ is defined by the conjunction Y ′ : ^ ≤ i ≤ s ∃ z ∈ H k χ i ( x, z, c ) . To finish the proof, we will show that by moving the existential quantifier ∃ z ∈ H k out of the conjunction we obtain a definable set with the required properties.Let Y be the definable set given by the existential H -formula Y : ∃ z ∈ H k ^ ≤ i ≤ s χ i ( x, z, c ) . Observation 1. For each 0 ≤ i ≤ s there is N i such that M | = ∀ x ∀ z χ i (¯ x, z, c ) → _ j ≤ N i Ψ p qi,j ( x, z, c ) . Therefore, by Corollary 3.7, for any a ∈ M | x | and h ∈ H k , if SU( a/c ) = ω · n + k and M | = χ i ( a, h, c ) then h is an enumeration of HB( a/c ). Claim The definable set Y satisfies all the conditions of the lemma.Note that condition (1) is satisfied with η ′ ( x, z, c ) = V ≤ i ≤ s χ i ( x, z, c ), and condition(2) follows from Observation 1 by taking { p , . . . , p t } = { p q i ,j : 1 ≤ i ≤ s, j ≤ N i } . Condition (4): We show first that SU( Y ′ △ Y ) < ω · n + k . We know that Y ⊂ Y ′ , soit remains to show that SU( Y ′ \ Y ) < ω · n + k . That is, it suffices to show that every a ∈ Y ′ with SU( a/c ) = ω · n + k belongs to Y .Let a ∈ Y ′ with SU( a/c ) = ω · n + k . There are h i ∈ H k such that M | = χ q i ( a, h i , c )for 0 ≤ i ≤ s . By Observation 1, we have that h i is an enumeration of HB( a/c ). Let h be a fixed enumeration of HB( a/c ). Then χ q i ( a, h, c ) holds, so a ∈ Y .Hence, since Y △ X = ( Y △ Y ′ ) △ ( Y ′ △ X ) ⊆ ( Y △ Y ′ ) ∪ ( Y ′ △ X ) , we get SU( Y △ X ) = max { SU( Y △ Y ′ ) , SU( Y ′ △ X ) } < ω · n + k . Condition (3): Let a ∈ X with SU( a/c ) = ω · n + k . Also let h be an enumerationof HB( a/c ). By condition (4) we have that a ∈ Y , so for all i ≤ s there is h ′ such that -STRUCTURES AND GENERALIZED MEASURABLE STRUCTURES 13 χ i ( a, h ′ , c ). By item 2, h ′ is permutation of h and since η ′ is invariant under permutationof HB( a/c ) we get M | = η ′ ( x, h, c ), as desired.For the furthermore part, note that each Ψ p i (¯ x, ¯ z, ¯ y ) for i ≤ s can be chosen to beinvariant under the permutation of ¯ z . It is easy to see now that we can replace η ′ (¯ x ; ¯ z, ¯ c )by W σ ∈ S k η ′ (¯ x ; σ (¯ z ) , ¯ c ) and the conclusion will hold. (cid:3) Notation: For a tuple ¯ z = ( z , . . . , z | ¯ z |− ) and i < | ¯ z | , we will write ˆ z i to denote thetuple ( z , . . . , z i − , z i +1 , . . . , z | ¯ z |− ), where the coordinate z i is ommitted.In all the results we have obtained so far we are using that the definable sets aredefined by formulas with parameters that are H -independent. However, given an arbit-rary tuple c and a c -definable set X defined by ϕ ( x, c ), note that the tuple c ′ = c HB( c )is H -independent and the formula ϕ ′ ( x, c ′ ) := ϕ ( x, c ) defines the same set X . Usingthis, and Lemma 3.3, we can obtain more general results that we present below. Lemma 3.10. Let M := ( M, H ) be a sufficiently saturated H -structure. Suppose η (¯ x, ¯ z ; ¯ y ) is an L -formula with | ¯ z | = k , and c ∈ M | y | is a tuple satisfying SU( ∃ ¯ z ∈ H k η (¯ x, ¯ z ; ¯ c )) = ω · n + k (over ¯ c HB(¯ c ) ).Assume that M | = ∀ ¯ x ∀ ¯ z η (¯ x, ¯ z ; ¯ c ) → _ i ≤ t Ψ p i (¯ x, ¯ z ; ¯ c ) , where p i is a complete typeof SU -rank ω · n + k and the formulas Ψ p i are defined as in Notation 3.6.Given ¯ z = z . . . z k − , and ≤ j ≤ k − , let us consider the formula η j ( z j ; ¯ x, ˆ z j , ¯ y ) := η (¯ x, ¯ z, ¯ y ) (obtained by choosing z i as object variable) and let ζ j (¯ x, ˆ z j , ¯ y ) be a formulasaying that η j is an algebraic formula over ¯ x, ˆ z j , ¯ y .Finally, consider the formula ˜ η (¯ x, ¯ z ; ¯ y ) := η (¯ x, ¯ z ; ¯ y ) ∧ ^ ≤ j Let ( M, H ) be a sufficiently saturated H -structure. Then for any L H -formula ϕ (¯ x ; ¯ y ) , there is a finite subset D ϕ = { ( n i , k i ) : i ≤ N } ⊆ N × N , existential L H -formulas { ϕ i (¯ x ; ¯ y ) : i ≤ N } and L H -formulas { ψ i (¯ y ) : i ≤ N } such that the followinghold: (1) The formulas { ψ i ( y ) : i ≤ N } are existential L H -formulas that form a cover of M y (2) For any i ≤ N and c ∈ M | y | , M | = ψ i (¯ c ) implies that SU( ϕ i (¯ x ; ¯ c )) = SU( ϕ (¯ x ; ¯ c )) = ω · n i + k i and SU( ϕ i (¯ x ; ¯ c ) △ ϕ (¯ x ; ¯ c )) < SU( ϕ (¯ x ; ¯ c )) .Proof. Let us first fix a tuple ¯ c ∈ M | y | and let ¯ h be a fixed enumeration of HB(¯ c ). ByLemma 3.3, there is a formula ξ ¯ c (¯ z, ¯ y ) such that ξ ¯ c ( H | ¯ z | , ¯ c ) consists only on permuta-tions of ¯ h and ξ ¯ c ( H | ¯ z | , ¯ c ′ ) is finite for any ¯ c ′ . (If HB(¯ c ) = ∅ , we set ξ ¯ c (¯ z ) := “¯ z = ¯ z ”.)By Lemma 3.9, there is an L H -existential formula η ¯ c (¯ x ; ¯ h, ¯ c ) such thatSU( η ¯ c (¯ x ; ¯ h, ¯ c ) △ ϕ (¯ x ; ¯ c )) < SU( ϕ (¯ x ; ¯ c )) . Clearly, we have SU( η ¯ c (¯ x ; ¯ h, ¯ c )) = SU( ϕ (¯ x ; ¯ c )) (over the parameter set ¯ h ¯ c ). SupposeSU( ϕ (¯ x ; ¯ c )) = ω · n + k . Note that by Lemma 3.9, η ¯ c (¯ x ; ¯ h, ¯ c ) is of the form ∃ ¯ z ∈ H k η ′ ¯ c (¯ x ; ¯ z, ¯ h, ¯ c ) where η ′ ¯ c (¯ x ; ¯ z, ¯ h, ¯ c ) is an L -formula withdim( η ′ ¯ c (¯ x, ¯ z ; ¯ h, ¯ c )) = n + k. By Lemma 3.10, we may replace η ′ ¯ c by a formula ˜ η ′ ¯ c such that all properties of η ′ ¯ c inLemma 3.9 also hold for ˜ η ′ ¯ c . In particular dim(˜ η ′ ¯ c (¯ x, ¯ z ; ¯ h, ¯ c )) = n + k. Also by Lemma 3.9and Lemma 3.10 there are { p i : i ≤ K } complete types over ¯ h, ¯ c with SU -rank ω · n + k ,such that M | = ∀ ¯ x ∀ ¯ z ˜ η ′ ¯ y (¯ x, ¯ z ; ¯ h, ¯ y ) → _ i ≤ K Ψ p i (¯ x, ¯ z ; ¯ h, ¯ y ) . Let ˜ η ¯ c (¯ x, ¯ h, ¯ c ) := ∃ ¯ z ∈ H k ˜ η ′ ¯ y (¯ x ; ¯ z, ¯ h, ¯ c ) . By the definability of dimensions in the geometric structure M , there is an L -formula χ cn,k (¯ h, ¯ y ) such that M | = χ cn,k (¯ h, ¯ y ) if and only if dim(˜ η ′ ¯ c (¯ x, ¯ z ; ¯ h, ¯ y )) = n + k. We split the argument in cases to define a formula ψ ′ ¯ c (¯ h, ¯ y ). Case 1: If ω · n + k = 0, we have n = k = 0. By assumption we have M | = ∀ ¯ x (˜ η ¯ c (¯ x ; ¯ h, ¯ c ) ↔ ϕ (¯ x ; ¯ c )). So, we can define ψ ′ ¯ y (¯ h, ¯ y ) := ξ ¯ y (¯ h, ¯ y ) ∧ χ , (¯ h, ¯ y ) ∧ ∀ ¯ x (˜ η ¯ y (¯ x ; ¯ h, ¯ y ) ↔ ϕ (¯ x ; ¯ y )) . Case 2: If ω · n + k ≥ 1, then by compactness, there are algebraic L -formulas { ξ n,k,j (¯ x j ; ¯ x j , ¯ d j , ¯ h, ¯ c ) : j ≤ N n,k } such that ¯ x divides into subtuples ¯ x j ; ¯ x j and,either | ¯ x j | < n , or we have | ¯ x j | = n and | ¯ d j | < k . Also, we know that ∀ ¯ x ˜ η ¯ c (¯ x ; ¯ h, ¯ c ) △ ϕ (¯ x ; ¯ c ) → _ j ≤ N n,k ∃ ¯ d j ∈ H k j ξ n,k,j (¯ x j , ¯ x j ; ¯ d j , ¯ h, ¯ c ) . -STRUCTURES AND GENERALIZED MEASURABLE STRUCTURES 15 Let us consider the formula ψ ′ ¯ c (¯ h, ¯ y ) := ξ ¯ c (¯ h, ¯ y ) ∧ χ n,k (¯ h, ¯ y ) ∧ ∀ ¯ x ∀ ¯ z ˜ η ′ ¯ c (¯ x, ¯ z ; ¯ h, ¯ y ) → _ i ≤ K ψ p i (¯ x, ¯ z ; ¯ h, ¯ y ) ∧ ∀ ¯ x ˜ η ¯ c (¯ x ; ¯ h, ¯ y ) △ ϕ (¯ x ; ¯ y ) → _ j ≤ N n,k ∃ ¯ d j ∈ H k j ξ n,k,j (¯ x j , ¯ x j ; ¯ d j , ¯ h, ¯ y ) . In both cases, let ψ ¯ c (¯ y ) := ∃ ¯ h ∈ H | ¯ h | ψ ′ ¯ c (¯ h, ¯ y ) and ϕ ¯ c (¯ x ; ¯ y ) := ∃ ¯ h ∈ H | ¯ h | ( ψ ′ ¯ c (¯ h, ¯ y ) ∧ ˜ η ¯ c (¯ x ; ¯ h, ¯ y )) . Claim 3.12. For any ¯ c ′ such that M | = ψ ¯ c (¯ c ′ ) we have SU( ϕ ¯ c (¯ x ; ¯ c ′ )) = SU( ϕ (¯ x ; ¯ c ′ )) = ω · n + k and SU( ϕ ¯ c (¯ x ; ¯ c ′ ) △ ϕ (¯ x ; ¯ c ′ )) < ω · n + k (over the parameter set ¯ c ′ HB(¯ c ′ ) ). We will prove this claim later. Let us conclude now the proof of the theorem assum-ing this result.We have shown that for every tuple c ∈ M | y | there is an existential H -formula ϕ c ( x, y )and an L H -formula ψ c ( y ) such that M | = ψ c ( c ) and the Claim 3.12 holds. Hence, { ψ c ( y ) : c ∈ M | y | } is a cover of M | y | , and by compactness, there are finitely manyformulas ψ ( y ) := ψ c ( y ) , . . . , ψ N ( y ) := ψ c N ( y ) that cover M | y | , so condition (1) issatisfied. If SU( ϕ ( x, c i )) = ω · n i + k i for i ≤ N , we can take the finite set D ϕ = { ( n i , k i ) : i ≤ N } ⊆ N × N , and condition (2) follows from Claim 3.12. (cid:3) Theorem 3.11 Remark 3.13. Notice that by taking boolean combinations, we can obtain an ∅ -definable partition { ψ ′ ( y ) , . . . , ψ ′ N ( y ) } from the L H -formulas ψ ( y ) , . . . , ψ N ( y ). How-ever, ψ ′ ( y ) , . . . , ψ ′ N ( y ) are not necessarily existential L H -formulas, and we keeping themas existential L H -formulas will be a crucial later in the paper.We show now a slightly improved version of Claim 3.12 Claim 3.14. For any ¯ c ′ such that M | = ψ ¯ c (¯ c ′ ) we have SU( ϕ ¯ c (¯ x ; ¯ c ′ )) = SU( ϕ (¯ x ; ¯ c ′ )) = ω · n + k and SU( ϕ ¯ c (¯ x ; ¯ c ′ ) △ ϕ (¯ x ; ¯ c ′ )) < ω · n + k (over the parameter set ¯ c ′ HB(¯ c ′ ) ).Furthermore, for any ¯ c ′ , ¯ h ′ such that ψ ′ ¯ c (¯ h ′ , ¯ c ′ ) holds and ¯ h ′ ∈ H | ¯ h | , we have (1) SU(˜ η ¯ c (¯ x ; ¯ h ′ , ¯ c ′ )) = ω · n + k ; (2) The definable set ˜ η ′ ¯ c (¯ a, H k ; ¯ h ′ , ¯ c ′ ) is finite over any tuple ¯ a, ¯ h ′ , ¯ c ′ and SU(˜ η ¯ c (¯ x ; ¯ h ′ , ¯ c ′ ) △ ϕ (¯ x ; ¯ c ′ )) < ω · n + k ;(3) For any ¯ e, ¯ b satisfying ˜ η ′ ¯ c (¯ e, ¯ b, ¯ h ′ , ¯ c ′ ) with dim (¯ e, ¯ b/ ¯ c ′ , HB(¯ c ′ )) = n + k , there are ¯ a, ¯ d such that tp L (¯ e, ¯ b/ ¯ c ′ , HB(¯ c ′ )) = tp L (¯ a, ¯ d/ ¯ c ′ , HB(¯ c ′ )) , and SU(¯ a, ¯ d/ ¯ c ′ , HB(¯ c ′ )) = ω · n + k with ¯ d an enumeration of HB(¯ a/ ¯ c ′ , HB(¯ c ′ )) .(Note that by Lemma 3.1, ¯ h ′ ⊆ HB(¯ c ′ ) .) (4) for any ¯ a satisfying ϕ (¯ a, ¯ c ′ ) such that SU(¯ a/ ¯ c ′ , HB(¯ c ′ )) = ω · n + k , and for ¯ d ∈ H k , we have ˜ η ′ ¯ c (¯ a, ¯ d, ¯ h ′ , ¯ c ′ ) holds if and only if ¯ d is an enumeration of HB(¯ a/ ¯ c ′ , HB(¯ c ′ )) .Moreover, we can make ψ ′ ¯ c (¯ h, ¯ y ) and ˜ η ′ ¯ c (¯ x, ¯ z, ¯ h, ¯ y ) invariant under permutation of ¯ h .That is, for any σ ∈ S | ¯ h | , for any ¯ c ′ , ¯ h ′ and ¯ a, ¯ d , if ψ ′ ¯ c (¯ h ′ , ¯ c ′ ) holds then so does ψ ′ ¯ c ( σ (¯ h ′ ) , ¯ c ′ ) , and if ˜ η ′ ¯ c (¯ a, ¯ d, ¯ h ′ , ¯ c ′ ) holds, then ˜ η ′ ¯ c (¯ a, ¯ d, σ (¯ h ′ ) , ¯ c ′ ) also holds.Proof. By assumption, we have that ξ ¯ c ( H | ¯ h | , ¯ c ′ ) is finite for any ¯ c ′ . Let ¯ h ′ ∈ H | ¯ h | be suchthat ψ ′ ¯ c (¯ h ′ , ¯ c ′ ) holds. We only need to show that the desired result holds for ˜ η ¯ c (¯ x ; ¯ h ′ , ¯ c ′ ).By the construction of ˜ η ′ in Lemma 3.10, we have that ˜ η ′ ¯ c (¯ x, H k ; ¯ h ′ , ¯ c ′ ) is finite overany ¯ x, ¯ h ′ , ¯ c ′ .We start by showing item 2: SU(˜ η ¯ c (¯ x ; ¯ h ′ , ¯ c ′ ) △ ϕ (¯ x ; ¯ c ′ )) < ω · n + k .Fix any ¯ h ′ such that ψ ′ ¯ c (¯ h ′ , ¯ c ′ ) holds. By definition, M | = ∀ ¯ x ˜ η ¯ c (¯ x ; ¯ h ′ , ¯ c ′ ) △ ϕ (¯ x ; ¯ c ′ ) → _ j ≤ N n,k ∃ ¯ d j ∈ H k j ξ n,k,j (¯ x j , ¯ x j ; ¯ d j , ¯ h ′ , ¯ c ′ ) . Given any j ≤ N n,k SU (cid:16) ∃ ¯ d j ∈ H k j ξ n,k,j (¯ x j , ¯ x j ; ¯ d j , ¯ h ′ , ¯ c ′ ) (cid:17) < ω · n + k so we have SU(˜ η ¯ c (¯ x ; ¯ h ′ , ¯ c ′ ) △ ϕ (¯ x ; ¯ c ′ )) < ω · n + k over ¯ c ′ , HB(¯ c ′ ).By assumption M | = ∀ ¯ x ∀ ¯ z (cid:16) ˜ η ′ ¯ c (¯ x, ¯ z ; ¯ h ′ , ¯ c ′ ) → W i ≤ K ψ p i (¯ x, ¯ z ; ¯ h ′ , ¯ c ′ ) (cid:17) . Let ¯ x be suchthat M | = ˜ η ¯ c (¯ x ; ¯ h ′ , ¯ c ′ ). Then there is ¯ h ∈ H k such that M | = ˜ η ′ ¯ c (¯ x , ¯ h ; ¯ h ′ , ¯ c ′ ).Therefore, M | = W i ≤ K ψ p i (¯ x , ¯ h ; ¯ h ′ , ¯ c ′ ). Then by the definition of each ψ p i , we haveSU(¯ x / ¯ h ′ , ¯ c ′ ) ≤ ω · n + k . Since ¯ h ′ ⊆ HB(¯ c ′ ), we conclude SU( ϕ ¯ c (¯ x, ¯ c ′ ) / ¯ c ′ , HB(¯ c ′ )) ≤ ω · n + k .Now we show item 3 of the claim, which will also imply that SU(˜ η ¯ c (¯ x ; ¯ h ′ , ¯ c ′ ) / ¯ c ′ , HB(¯ c ′ )) ≥ ω · n + k .Since χ n,k (¯ h ′ , ¯ c ′ ) holds, dim(˜ η ′ ¯ c (¯ x, ¯ z ; ¯ h ′ , ¯ c ′ )) = n + k . Let ¯ e, ¯ b be such that M | =˜ η ′ ¯ c (¯ e, ¯ b, ¯ h ′ , ¯ c ′ ) and dim(¯ e, ¯ b/ ¯ c ′ , HB(¯ c ′ )) = n + k . By assumption, there is p i such that M | = ψ p i (¯ e, ¯ b ; ¯ h ′ , ¯ c ′ ). By definition of ψ p i , we have dim(¯ e/ ¯ b, ¯ h ′ , ¯ c ′ ) ≤ n , hence dim(¯ e/ ¯ b, ¯ c ′ , HB(¯ c ′ )) ≤ n . Therefore, dim(¯ b/ ¯ c ′ , HB(¯ c ′ )) = k and dim(¯ e/ ¯ b, ¯ c ′ , HB(¯ c ′ )) = n . Consider tp L (¯ b/ ¯ c ′ , HB(¯ c ′ )),which has dimension k . By the density property in H -structures, we an choose ¯ d ∈ H k such that tp L ( ¯ d/ ¯ c ′ , HB(¯ c ′ )) = tp L (¯ b/ ¯ c ′ , HB(¯ c ′ )) . Suppose ¯ e = ¯ e ¯ e , where | ¯ e | = dim(¯ e / ¯ b, ¯ c ′ , HB(¯ c ′ )) = n and ¯ e ∈ acl (¯ e , ¯ b, ¯ c ′ , HB(¯ c ′ )).Consider q := tp L (¯ e / ¯ b, ¯ c ′ , HB(¯ c ′ )), it has dimension n . Let q ′ be the type over ¯ d, ¯ c ′ , HB(¯ c ′ )by replacing ¯ b by ¯ d . Since ¯ b and ¯ d has the same L -type over ¯ c ′ , HB(¯ c ′ ), we also have q ′ has dimension n . By the extension property in H -structures, we can choose ¯ a ∈ M n such that ¯ a | = q ′ and dim(¯ a /H, ¯ c ′ ) = n . Let ¯ a be such thattp L (¯ a , ¯ a , ¯ d/ ¯ c ′ , HB(¯ c ′ )) = tp L (¯ e , ¯ e , ¯ b/ ¯ c ′ , HB(¯ c ′ ))and let ¯ a = ¯ a ¯ a . Then by construction, we havetp L (¯ a, ¯ d/ ¯ c ′ , HB(¯ c ′ )) = tp L (¯ e, ¯ b/ ¯ c ′ , HB(¯ c ′ )) . We claim that SU(¯ a/ ¯ c ′ , HB(¯ c ′ )) = ω · n + k . We know that(1) ¯ a | ⌣ ¯ d, ¯ c ′ , HB(¯ c ′ ) H. -STRUCTURES AND GENERALIZED MEASURABLE STRUCTURES 17 We want to show that ¯ d = HB(¯ a/ ¯ c ′ , HB(¯ c ′ )). We only need to show that ¯ d is aminimal tuple in H for which (1) holds. Suppose not, then there is d i ∈ ¯ d = d . . . d k − such that ¯ a ∈ acl (¯ a , ¯ d, ¯ c ′ , HB(¯ c ′ )) \ { d i } . Thus,dim(¯ a, ˆ d i / ¯ c ′ , HB(¯ c ′ )) = dim(¯ a , ˆ d i / ¯ c ′ , HB(¯ c ′ )) = n + k − . Note that by the construction of ˜ η ′ ¯ c in Lemma 3.10, we must have that d i ∈ acl (¯ a, ˆ d i , ¯ h ′ , ¯ c ′ ) . Therefore,dim(¯ a, ¯ d/ ¯ c ′ , HB(¯ c ′ )) =dim( d i / ¯ a, ˆ d i , ¯ c ′ , HB(¯ c ′ )) + dim(¯ a, ˆ d i / ¯ c ′ , HB(¯ c ′ ))=0 + n + k − n + k − . This contradicts that dim(¯ a, ¯ d/ ¯ c ′ , HB(¯ c ′ )) = n + k .Therefore SU(¯ a/ ¯ c ′ , HB(¯ c ′ )) = ω · n + k and we conclude that SU(˜ η ¯ c (¯ x ; ¯ h ′ , ¯ c ′ )) = ω · n + k .Since SU(˜ η ¯ c (¯ x ; ¯ h ′ , ¯ c ′ ) △ ϕ (¯ x ; ¯ c ′ )) < ω · n + k , we must have SU( ϕ ( x, ¯ c ′ )) = ω · n + k .Now we prove item 4 of the claim.Suppose ( M, H ) | = ϕ (¯ a, ¯ c ′ ) and SU(¯ a/ ¯ c ′ , HB(¯ c ′ )) = ω · n + k . SinceSU(˜ η ¯ c (¯ x ; ¯ h ′ , ¯ c ′ ) △ ϕ (¯ x ; ¯ c ′ )) < ω · n + k, we must have ( M, H ) | = ˜ η ¯ c (¯ a ; ¯ h ′ , ¯ c ′ ). Therefore, there is ¯ d ′ ∈ H k such that M | =˜ η ′ ¯ c (¯ a, ¯ d ′ ; ¯ h ′ , ¯ c ′ ). By assumption, there is some p i complete type of SU -rank ω · n + k suchthat M | = ψ p i (¯ a, ¯ d ′ ; ¯ h ′ , ¯ c ′ ). By Lemma 3.7, ¯ d ′ is an enumeration of HB(¯ a/ ¯ c ′ , HB(¯ c ′ )).On the other hand, if ¯ d be an enumeration of HB(¯ a/ ¯ c ′ , HB(¯ c ′ )), then it is a permutationof ¯ d ′ . By construction, the formula ˜ η ′ ¯ c (¯ x, ¯ z ; ¯ h ′ , ¯ c ′ ) is invariant under permutations of thevariables ¯ z , therefore, M | = ˜ η ′ ¯ c (¯ a, ¯ d ; ¯ h ′ , ¯ c ′ ).We prove the moreover part at the end of the statement of the Claim. Note thatthe formulas ψ ′ ¯ c (¯ h, ¯ y ) and ˜ η ′ ¯ c (¯ x, ¯ z, ¯ h, ¯ y ) depend on the enumeration ¯ h of HB(¯ y ). Let ¯ h ′ be another enumeration of HB(¯ y ). Then there is σ ∈ S k such that σ (¯ h ) = ¯ h ′ . Withthis enumeration, we can find ψ ′ ¯ y,σ ( σ (¯ h ) , ¯ y ) and ˜ η ′ ¯ y,σ (¯ x, ¯ z, σ (¯ h ) , ¯ y ) such that the sameproperties hold. Redefine ψ ′ ¯ c (¯ h, ¯ y ) := _ σ ∈ S k ψ ′ ¯ y,σ ( σ (¯ h ) , ¯ y ) , and ˜ η ′ ¯ c (¯ x, ¯ z, ¯ h, ¯ y ) := _ σ ∈ S k (cid:0) ψ ′ ¯ y,σ ( σ (¯ h ) , ¯ y ) ∧ ˜ η ′ ¯ y,σ (¯ x, ¯ z, σ (¯ h ) , ¯ y ) (cid:1) . Now it is easy to see that the new formulas have the desired properties and are invariantunder the permutation of ¯ h . (cid:3) Remark: Theorem 3.11 establishes that the SU -rank in T ind is definable, in the sensethat given a uniformly definable family of sets, then the property “having fixed SU -rankequal to ω · n + k is a definable condition. In particular, T ind eliminates ∃ ∞ .4. Comparing two notions of dimension Let T be the limit theory of a 1-dimensional asymptotic class in a language L . Inparticular T is a SU -rank one theory and thus geometric. Assume as before that T is nowhere trivial and suppose that T is the theory of M = Q U M n here each M n is a finite L -structure in the asymptotic class and the sizes of the structures M n goto infinity. Consider the pseudofinite H -structure ( M, H ( M )) = Q U ( M n , H n ) builtin [11]. For each X ⊂ M definable in the extended language L H , we will study twocounting dimensions associated to the ultraproduct: the pseudofinite coarse dimensions with respect to M and H ( M ). Formally, let δ H ( X ) = lim U log | X ( M n ) | / log | H ( M n ) | and let δ M ( X ) = lim U log | X ( M n ) | / log | M n | . Note that we may build ( M, H ( M )) suchthat δ M ( H ) = 0, that is δ H ( M ) = ∞ , see Remark 2.9 of [11].We will show that under mild assumptions δ H ( X ) = n corresponds to SU( X ) = n and δ M ( X ) = 1 corresponds to SU( X ) = ω . This provides some intuition for the restof the paper: instead of studying these ranks, we will study SU -rank and show that the SU -rank of a formula can described definably in terms of the parameters of the formula. Proposition 4.1. Let X ⊂ M m be type-definable defined by a collection of formulaswith parameters over A ⊂ M . Assume that SU( X ) = n . Then δ H ( X ) ≤ n .Proof. Let q ( x ) be a complete type over A extending X and let b | = q ( x ). Let HB ( b/A ) = r , then r ≤ n and there are h = h , . . . , h r ∈ H all distinct such that b ∈ acl ( Ah , . . . , h r ).Let ϕ q ( x ; y ) be a formula and a k q > M | = ∀ y ∃ ≤ k q xϕ q ( x ; y ) and M | = ϕ q ( b ; h ). By adding extra dummy variables if necessary, we may choose y of length n . Bycompactness, there are finitely many formulas ϕ ( x ; y ) , . . . , ϕ ℓ ( x ; y ) and a single k suchthat X ⊂ ∃ h ∈ H n ϕ ( M m ; h ) ∨ · · · ∨ ∃ h ∈ H n ϕ ℓ ( M m ; h ) and M | = ∀ y i ∃ ≤ k x ( ϕ i ( x, y i ))for i = 1 , . . . , ℓ .Then, δ H ( X ) ≤ δ H ( ∃ h ∈ H n ϕ ( M m ; h ) ∪ · · · ∪ ∃ h ∈ H n ϕ ℓ ( M m ; h )) ≤ max ≤ i ≤ ℓ { δ H ( ∃ h ∈ H n ϕ i ( M m ; h )) } ≤ δ H ( H n ) ≤ n. (cid:3) To proof the converse we will need an extra assumption. The definable subsets of H ( M ) are just the traces of L -formulas with parameters in M , thus we expect when Y ⊂ H n has dimension n and Y = Y ∩ H ( M ) n , where Y is L -definable, the size of | Y | / | H n | should be roughly that of | Y | / | M n | .Assumption ( ∗ ): whenever Y ⊂ H n is L H -definable of dimension n , there is µ > | Y ( M ℓ ) | ≥ µ | H ( M ℓ ) | n for l large enough. Proposition 4.2. Let X ⊂ M m be definable over A and that A is H -independent.Assume that SU( X ) = n and that the condition ( ∗ ) holds. Then δ H ( X ) ≥ n .Proof. By Theorem 3.10 and Claim 3.11 we can find X defined by ∃ h ∈ H n η A ( x, h )such that SU( X △ X /A ) < n , where η A is an L -formula where η A ( c, H n ) is finite forany c ∈ M | x | and η A ( M m , d ) is finite for any d ∈ M | d | . By the previous Proposition δ H ( X △ X ) < n . If we show that δ H ( X ) ≥ n , then δ H ( X ) = δ H ( X △ ( X △ X )) ≥ δ H ( X \ ( X △ X )) ≥ n . Thus it suffices to prove δ H ( X ) ≥ n . Let Y ⊂ H n be theset of realizations of ∃ xη ( x, H n ). Since SU( X /A ) = n , we know that dim( Y ) = n .By condition ( ∗ ), we get | Y ( M ℓ ) | ≥ µ | H ( M ℓ ) | n for sufficiently large l . On the otherhand, by compactness, there is k such that | η A ( c, H n ) | ≤ k for any c ∈ M | x | . Hence, k | X ( M ℓ ) | ≥ µ | H ( M ℓ ) | n . This shows that δ H ( X ) ≥ n as we wanted. (cid:3) We will need the following result from [2, Prop 3.12], it provides a description ofdefinable sets of H -structures in terms of definable sets in the original language andsmall sets. Lemma 4.3. If ϕ ( x ; y ) is an L H -formula, and a is an H -independent tuple, there areformulas θ ( x ; b ) ∈ L and ψ ( x ; b ) ∈ L H such that: (1) ψ ( x ; b ) ⊆ acl ( b, H ) (the set ψ ( M ; b ) is small). (2) ϕ ( x ; b ) ≡ θ ( x ; b ) △ ψ ( x ; b ) . -STRUCTURES AND GENERALIZED MEASURABLE STRUCTURES 19 This will allow us to prove that we can detect large and small sets using the | M | -dimension coming from the asymptotic class. Proposition 4.4. Let ϕ ( x, y ) be an L H -formula with x a single variable. For any b ∈ M | y | we have δ M ( ϕ ( M ; b )) ∈ { , } . Moreover, { y ∈ M y : ϕ ( M ; y ) ⊆ acl L ( y, H ) } = { y ∈ M y : δ M ( ϕ ( M ; y )) = 0 } and this set is L H -definable.Proof. Consider the formula given by b ϕ ( x ; b, HB ( b/A )) := ϕ ( x ; b ). Since b, HB ( b/A ) isan H -independent tuple, by Lemma 4.3, there are formulas θ ( x ; b, HB ( b/A )) ∈ L and ψ ( x ; b, HB ( b/A )) ∈ L H such that(1) ψ ( x ; b, HB ( b/A )) ⊆ acl ( b, H ) . (2) ϕ ( M ; b ) = b ϕ ( M ; b, HB ( b/A ))) ≡ θ ( x ; b, HB ( b/A )) △ ψ ( x ; b, HB ( b/A )) . Since X = ψ ( M ; b, HB ( b/A )) is a small set, SU( X ) = n (and n depends on tp L H ( b )), sofor every a ∈ ψ ( M ; b, HB ( b/A )) there is an n -tuple h a = h a , . . . , h an (possibly with re-petitions) such that a ∈ acl ( b, h a , . . . , h an ). Let η a ( x, b, HB ( b, y ) be an algebraic formulain x such that a | = η a ( x, b, HB ( b ) , h a ).By compactness, there are finitely many algebraic formulas η , . . . , η r such that ψ ( M ; b, HB ( b/A )) ⊆ r [ i =1 ∃ h ∈ H n ( η i ( M ; b, HB ( b/A ) , h )) . Let k > i ≤ r we have ∀ y ∃ ≤ k xη i ( x ; b, HB ( b/A ) , y ). Then δ M ( ψ ( M ; b, HB ( b/A ))) ≤ max (cid:8) δ M (cid:0) ∃ h ∈ H, . . . , ∃ h n ∈ H η i ( M ; b, HB ( b/A ) , h , . . . , h n ) (cid:1) : i = 1 , . . . , r (cid:9) ≤ kδ M ( H n ) = 0 . This proves that δ M ( ϕ ( M ; b )) = δ M ( θ ( M ; b, HB ( b/A ))) ∈ { , } , as desired.Now we show the definability condition. We have proved so far that for an arbitrarytuple b ∈ M y , with | HB ( b/A ) | = n we get an L -formula θ b ( x ; b, h , . . . , h n ) and an L H -formula ψ b ( x ; b, h , . . . , h n ) with the properties (1) and (2) as above. So, the L H -formula ρ k,θ b ,ψ b ( y ) := ∃ h ∈ H, . . . , h n ∈ H (cid:2) ϕ ( x ; y ) ≡ θ b ( x ; y, h , . . . , h n ) △ ψ b ( x ; b, h , . . . , h n ) (cid:3) belongs to tp L H ( b ). By compactness we can find finitely many formulas ρ ( y ) , . . . , ρ ℓ ( y )covering M | y | , providing also finitely many formulas θ j ( x, y, z , . . . , z n ) ∈ L and ψ j ( x, y, z , . . . , z n ) ∈L H for j = 1 , . . . , ℓ (all of which define small sets) with the following properties: forevery tuple b ′ ∈ M y , we must have M | = ρ j ( b ′ ) for some 1 ≤ j ≤ ℓ , and whenever M | = ρ j ( b ′ ) there are h ′ , . . . , h ′ n ∈ H with ϕ ( M ; b ′ ) ≡ θ j ( x ; b ′ , h ′ , . . . , h ′ n ) △ ψ j ( x ; b ′ , h ′ , . . . , h ′ n ) . We can refine the formulas ρ ( y ) , . . . , ρ ℓ ( y ) to ensure the definability condition by con-sidering ρ tj ( y ) := ∃ h ∈ H, . . . , ∃ h k ∈ H (cid:2) ( ϕ ( x ; y ) ≡ θ j ( x ; y, h , . . . , h k ) △ ψ ( x ; b, h , . . . , h k )) ∧ dim( θ ( x ; y, h , . . . , h k ) = t ) (cid:3) for j = 1 , . . . , ℓ and t = 0 , 1. Note that the condition dim( θ ( x ; y, z ) = t is defined byan L -formula in variables y, z , . . . , z t because M is an ultraproduct of structures in a1-dimensional asymptotic class and θ ( x ; y, z ) is an L -formula. Hence, the definabilitycondition is given by δ M ( ϕ ( M ; b )) = 1 if and only if M | = ρ ( b ) ∨ · · · ∨ ρ ℓ ( b ). (cid:3) By induction on fibers, it is not hard to get the following result, also see Corollary 9[12]. Corollary 4.5. Let ϕ (¯ x, ¯ y ) be an L H -formula without parameters. For any ¯ b ∈ M | ¯ y | ,we have δ M ( ϕ ( M | ¯ x | , ¯ b )) ∈ { , · · · , | ¯ x |} . Moreover, { ¯ y ∈ M | ¯ y | : δ M ( ϕ ( M, ¯ y )) = i } is L H -definable without parameters for each i ∈ { , , · · · , | ¯ x |} .In particular, the coarse dimension δ M is definable and additive. Lemma 4.6. Let a be an element in ( M, H ) and C be a countable subset of M . Then SU T ind ( a/C ) < ω if and only if δ M ( a/C ) = 0 Proof. Suppose SU T ind ( a/C ) < ω . Let X be a definable over set of finite SU-rank over C with a ∈ X . By Proposition 4.1, δ H ( X ) < ∞ . Thus, δ M ( X ) = 0. Therefore, δ M ( a/C ) ≤ δ M ( X ) = 0 . Suppose SU T ind ( a/C ) = ω , then a acl L ( C ∪ H ). Take any formula ψ ( x, ¯ c ) ∈ tp H ( a/C ), by Theorem 4.4, either δ M ( ψ ( M, ¯ c )) = 0 or δ M ( ψ ( M, ¯ c )) = 1 and the formerholds if only if ψ ( M, ¯ c ) ⊆ acl L (¯ c, H ). Therefore, δ M ( ψ ( M, ¯ c )) = 1. We conclude that δ M ( a/C ) = 1. (cid:3) Let ¯ a be a tuple in ( M, H ) and A be a countable subset of M . Suppose SU T ind (¯ a/A ) = ω · k + n for some 0 ≤ k ≤ | ¯ a | and n ∈ N . Recall from Definition 2.8 that ldim(¯ a/A ) = k is the large dimension of ¯ a over A . Theorem 4.7. For any tuple ¯ a in ( M, H ) and any countable subset A ⊆ M , we have ldim(¯ a/A ) = δ M (¯ a/A ) . Proof. it follows from additivity of both ldim and δ M . (cid:3) Measure and dimension In this section, we will define notions of dimension and measure for H -structurescoming from theories that are measurable of SU -rank one. We recall Definition 5.1 in[8], with a slight change of terminology. Definition 5.1. A structure M is measurable if there is a function h : Def( M ) → N × R > (where we denote h ( X ) = (dim( X ) , meas( X ))) satisfying the following conditions:(1) If X is finite and non-empty, then h ( X ) = (0 , | X | ).(2) (Finitely many values) For every formula ϕ ( x, y ) with | x | = n , | y | = m there isa finite set D ϕ ⊆ N × R > so that for all a ∈ M m , h ( ϕ ( M n ; a )) ∈ D ϕ .(3) (Definability Condition) For every formula ϕ ( x, y ) and each ( d, µ ) ∈ D ϕ , the set { a ′ ∈ M m : h ( ϕ ( M n , a ′ )) = ( d, µ ) } is 0-definable. In some parts of the literature these structures are called MS-measurable to distinguish this conceptfrom other notions of measure. -STRUCTURES AND GENERALIZED MEASURABLE STRUCTURES 21 (4) (Fubini property) Suppose f : X → Y is a definable surjection and considera partition Y = Y ∪ · · · ∪ Y r where Y i = { b ∈ Y : h ( f − ( b )) = ( d i , µ i ) } forfinitely many values ( d , µ ) , . . . , ( d r , µ r ) ∈ N × R > , which exists by conditions(2) and (3). In addition, suppose h ( Y i ) = ( e i , ν i ) for each i = 1 , . . . , r . Thendim( X ) = d = max ≤ i ≤ r { d i + e i } and meas( X ) = X j ∈ J µ j ν j , where J is the set of in-dices j ∈ { , . . . , r } where the maximum d = d j + e j is attained.Due to the Definability Condition, it is easily checked that if M ≡ N and M ismeasurable then N is also measurable. The main examples of measurable structurescorrespond to ultraproducts of . These are proved tobe supersimple of SU-rank 1 (cf. [8, Lemma 4.1]), so they are also geometric structures.In [11], the third author showed that for the ultraproducts of 1-dimensional asymp-totic classes, the corresponding H -structure is also pseudofinite. Hence, it is natural toask whether the construction of H -structures preserves measurability.A preliminary answer is that it is not possible to preserve measurability if the un-derlying theory is not trivial: a consequence of Definition 5.1 (cf. [7, Corollary 3.6])is that if M is a measurable structure, then for every definable set X the D -rank of X is bounded above by dim( X ). In particular, since dim( X ) ∈ N , this shows thatTh( M ) must be a supersimple theory of finite rank. Therefore, since the corresponding H -structure of a SU -rank one non-trivial theory will have SU-rank ω , it would not bemeasurable.However, there is a variation of measurability, intended to include some examplesof supersimple theories of infinite rank, the so-called generalized measurable structures .In these structures, the dimension and measure of definable sets are taken to be inan arbitrary ordered semi-ring, rather than in N × R > . This definition and the mainproperties of these structures, as well as several interesting examples and connectionswith asymptotic classes, will appear in the work [1] by S. Anscombe, D. Macpherson, C.Steinhorn and D. Wolf, currently in preparation. Even though we will not include theirdefinition here, the main results of this section will essentially show that H -structuresof measurable geometric structures are generalized measurable.In other words, we will restrict to the study of H -structures of SU -rank one measur-able structures, and show that there is an appropriate notion of dimension and measurefor definable sets (taking values in ( N × N ) × R ) that satisfies the appropriate analogsof the conditions given in Definition 5.1: they are uniformly definable in terms of theparameters of the formulas and satisfy the Fubini property, and have a strong connec-tion with the SU-rank for definable sets.The main idea is that dimension will be defined using data associated to the SU -rank (which in turn is related to large dimension and the size of an H -basis), while themeasure will be induced by the measure on L -definable sets in the underlying SU -rankone structure. Definition 5.2. Let R be the set N × N > × R > ∪ { (0 , , k ) : k ∈ N } . We will defineoperations ⊕ , ⊙ and a relation ≤ that will give R an ordered semiring structure. First,define ⊕ on R as:( x , y , r ) ⊕ ( x , y , r ) = ( x , y , r ) if ( x , y ) < lex ( x , y );( x , y , r ) if ( x , y ) < lex ( x , y );( x , y , r + r ) if ( x , y ) = ( x , y ) . Note that (0 , , 0) is the neutral element for ⊕ .Define the product ⊙ on R as: ( x , y , r ) ⊙ ( x , y , r ) = ( x + x , y + y , r r ),for r , r = 0. And (0 , , ⊙ ( x, y, r ) = ( x, y, r ) · (0 , , 0) = (0 , , ⊙ is (0 , , ≤ to be the lexicographic order on N × N > × R > ∪ { (0 , , k ) : k ∈ N } .For a triple ( x, y, r ) ∈ R , the pair ( x, y ) is called the dimension , and r is called the measure .Since we are dealing with structures that are both measurable and geometric, thereare two possible notions of dimension that we can use: one coming from the fact thatthe structure M is measurable that is defined in terms of the function h : Def( M ) → N × R > , and one coming from the fact that M is of SU -rank one, so it is geometricand the dimension is given in terms of algebraic independence. Definition 5.3. We say that an L -structure M is coherent measurable if:(1) Th( M ) is nowhere trivial,(2) Th( M ) is measurable and has SU -rank 1, and(3) For every L -formula ϕ ( x, y ) and every b ∈ M | y | , the dimension of ϕ ( M | x | , b ) (asa measurable structure) coincides with SU ( ϕ ( M | x | , b )).Note that a coherent measurable structure is geometric of SU -rank one. By theadditivity property of dimension and SU -rank, the SU -rank of a definable set coincideswith its dimension as a geometric structure. By (3) in Definition 5.3, these also coincidewith the dimension of the definable set as a measurable structure. Hence, the threenotions of dimension are equivalent for definable sets in this setting. We will need thetechnical assumption of being nowhere trivial in order to apply the results from Section3. Example 5.4. Every ultraproduct of a 1-dimensional asymptotic class has SU-rank1 and is a measurable structure. If the resulting structure is a group, then it isalso nowhere trivial. Hence, every ultraproduct of a 1-dimensional asymptotic classof groups or rings is a coherent measurable structure. These includes key examples suchas pseudofinite fields or infinite vector spaces over finite fields. Example 5.5. If T is the theory of a random graph, then T is the theory of an ul-traproduct of a 1-dimensional asymptotic class. It is measurable of SU -rank one, butthe theory is trivial. So the random graph is not coherent measurable. Example 5.6. In the language L = { P n : n < ω } where each P n is a unary predicate, wecan consider the finite L -structures M k with universe { , , . . . , k } , where we interpret P n ( M k ) to be { k · n + 1 , . . . , k · ( n + 1) } if n ≥ k , and P n ( M k ) = ∅ otherwise. Anyinfinite ultraproduct M = Q U M k models the theory of countably many infinite disjoint -STRUCTURES AND GENERALIZED MEASURABLE STRUCTURES 23 predicates. This theory has SU-rank 1 (although it is ω -stable of Morley rank 2) so thetheory is geometric with dimension 1.We can also see M as a measurable structure: for instance, for each n < ω wecan define dim( P n ( M )) = 1 and meas( P n ( M )) = 1. Finally, define dim( x = x ) = 2and meas( x = x ) = 1. Note that the dimension corresponds precisely with the coarsedimension of M with respect to the size of P ( M ) and agrees with Morley rank. Also,for a definable set, its measure agrees with its Morley degree. However, since each ofthe predicates has positive dimension and the predicates are disjoint, it is not possibleto see M as a measurable structure in such a way that dim( M ) = 1. Hence, M is notcoherent measurable.From now on we will work with coherent measurable structures. We will now assign,to each definable set in ( M, H ), a value in the semiring R that will correspond to adimension (a pair ( n, k ) ∈ N × N ) and a measure. For this, we will use the constructionin Claim 3.12. Definition 5.7. Let ( M, H ) be a sufficiently saturated H -structure. Let X ⊆ M m be a set definable over ¯ c of SU -rank ω · n + k . We call an L -formula ϕ (¯ x, ¯ z ) with | ¯ x | = m, | ¯ z | = k a measuring formula for X , if(i) ϕ is defined over parameters in ¯ c, HB(¯ c );(ii) ϕ (¯ x, H k ) is finite over any ¯ x and ∃ ¯ z ∈ H k ϕ (¯ x, ¯ z ) defines a set Y , such thatSU( X △ Y / ¯ c, HB(¯ c )) < ω · n + k ;(iii) For all ¯ a ∈ X with SU(¯ a/ ¯ c, HB(¯ c )) = ω · n + k and ¯ d ∈ H k , we have M | = ϕ (¯ a, ¯ d )if and only if ¯ d enumerates HB(¯ a/ ¯ c, HB(¯ c ));(iv) dim( ϕ ) = n + k and for all ¯ e, ¯ b with M | = ϕ (¯ e, ¯ b ) and dim(¯ e, ¯ b/ ¯ c, HB(¯ c )) = n + k ,there are ¯ a, ¯ d such thattp L (¯ a, ¯ d/ ¯ c, HB(¯ c )) = tp L (¯ e, ¯ b/ ¯ c, HB(¯ c ))SU(¯ a/ ¯ c, HB(¯ c )) = ω · n + k and ¯ d an enumeration of HB(¯ a/ ¯ c, HB(¯ c )). Remark: By Claim 3.14, for any definable set X there is is a measuring formula for X . Definition 5.8. Suppose M is a coherent measurable structure, and ( M, H ) is a suf-ficiently saturated H -structure. Let X be a definable set defined over ¯ c with SU-rank ω · n + k and let ϕ (¯ x, ¯ z ) be a measuring formula for X . Suppose ϕ has measure µ . Then,we call the pair ( n, k ) the dimension of X and we call µ/ | S k | the measure of X . Thetriple ( n, k, µ/ | S k | ) will be called the dimension and measure of X .We will now prove that the dimension and measure for a definable set X is well-defined, i.e., that the dimension and measure of X do not depend on the particularchoice of a measuring formula. Lemma 5.9. Suppose M is a coherent measurable structure, ( M, H ) is a sufficientlysaturated H -structure and X is a definable set. Suppose ϕ (¯ x, ¯ z ) and ϕ (¯ x, ¯ z ) are twomeasuring formulas of X . Then ϕ and ϕ have the same dimension and measure as L -formulas.Proof. Suppose X is defined over ¯ c and SU( X/ ¯ c, HB(¯ c )) = ω · n + k . By the definition ofmeasuring formula, dim( ϕ ) = dim( ϕ ) = n + k . It suffices to show that dim( ϕ △ ϕ ) Example 5.10. Let ( V , H ) = Q U ( F np , H n ) be a infinite ultraproduct of the finitevector spaces F np together with a predicate H n for n linearly independent vectors. Thereduct corresponding to the vector space is strongly minimal and pseudofinite, so it isa coherent geometric measurable structure. Note that the measure of a definable setsin this reduct coincides with its Morley degree. Now, since H is an infinite collectionof elements from V and the dimension of the quotient V / span( H ) is again infinite, thepair ( V , H ) is an H -structure.Consider the definable sets X = H + H = { h + h : h , h ∈ H } and Y = H + 2 H = { h + 2 h : h , h ∈ H } . Clearly SU( H + H ) = 2, SU( H + 2 H ) = 2. Notice that | H + 2 H | = | H × H | (as non-standard sizes), while 2 · | H + H | = | H × H | − | H | becausethe function that maps ( h , h ) to h + h is generically 2-to-1.Let ϕ ( x, z , z ) be the formula x = z + z , so X is defined by ∃ z ∃ z ∈ H ϕ ( x, z , z )and ϕ ( x, z , z ) is a measuring formula for X . Note that the L -measure of ϕ ( x, z , z )is 1 (it has Morley degree 1) and the measure of X is 1 / (2!) = 1 / ϕ ( x, z , z ) be the formula ( x = z + 2 z ∨ x = 2 z + z ). The set Y is definedby ∃ z ∃ z ∈ H ϕ ( x, z , z ) and ϕ ( x, z , z ) is a measuring formula for Y . Note thatwe had to use ϕ ( x, z , z ) instead of the formula x = z + 2 z to guarantee that clause(iii) holds in the Definition of a measuring formula. The L -measure of ϕ ( x, z , z ) is 2.Since the Morley degree of ϕ ( x, z , z ) is 2, the measure of Y is 2 / (2!) = 1. Lemma 5.11. Suppose M is a coherent measurable structure and that ( M, H ) is a suf-ficiently saturated H -structure. Then the dimension and measure defined in Definition L H -definable.Proof. By Theorem 3.11 and Claim 3.14, given an L H -formula ϕ (¯ x ; ¯ y ) there are formulas { ψ i (¯ y ) , ψ ′ i (¯ v i , ¯ y ) , ˜ η ′ i (¯ x, ¯ z i , ¯ v i , ¯ y ) : i ≤ N } and pairs { ( n i , k i ) ∈ N ≥ × N ≥ : i ≤ N } such that • the collection { ψ i ( y ) : i ≤ N } forms a partition of M | y | ; • ψ i (¯ y ) = ∃ ¯ v i ∈ H | ¯ v i | ψ ′ i (¯ v i , ¯ y ); • for every i ≤ N and ¯ a ∈ M | y | , if ( M, H ) | = ψ i (¯ a ) holds then the set X ¯ a definedby ϕ (¯ x ; ¯ a ) has SU -rank ω · n i + k i over ¯ a, HB(¯ a ); • for every i ≤ N and ¯ a ∈ M | y | , if ¯ h i ∈ H | ¯ v i | satisfies ( M, H ) | = ψ ′ i (¯ h i , ¯ a ), then˜ η ′ i (¯ x, ¯ z i ; ¯ h i , ¯ a ) is a measuring formula for X ¯ a .Since M is a measurable structure, there are L -formulas { ζ ij (¯ v i , ¯ y ) : j ≤ K i , i ≤ N } and { µ ij ∈ R ≥ : j ≤ K i , i ≤ N } such that ζ ij (¯ v i , ¯ a ) implies that the L -definable setdefined by ˜ η ′ i (¯ x, ¯ z i ; ¯ v i , ¯ a ) has measure µ ij and µ ij = 0 if and only if n i = k i = 0. Thus,we know that X ¯ a has dimension and measure ( n i , k i , µ ij / | S k i | ) if and only if( M, H ) | = ∃ ¯ v i ∈ H | ¯ v i | (cid:0) ψ ′ i (¯ v i , ¯ a ) ∧ ζ ij (¯ v i , ¯ a ) (cid:1) . (cid:3) -STRUCTURES AND GENERALIZED MEASURABLE STRUCTURES 25 Lemma 5.12. Suppose M is a coherent measurable structure and that ( M, H ) is asufficiently saturated H -structure. Then the dimension and measure are finitely additive,that is, if X , X ⊆ M m are disjoint definable sets whose dimensions and measures are ( n , k , µ ) , ( n , k , µ ) respectively, then the dimension and measure of X ∪ X is ( n , k , µ ) ⊕ ( n , k , µ ) .Proof. Let ϕ (¯ x, ¯ z ) and ϕ (¯ x, ¯ z ) be the measuring formulas for X and X respectively.Suppose ( n , k ) > ( n , k ). Then it is easy to see that ϕ (¯ x, ¯ z ) is also a measuringformula for the set X ∪ X . Therefore, the dimension and measure for X ∪ X is( n , k , µ ), which equals ( n , k , µ ) ⊕ ( n , k , µ ). The case ( n , k ) < ( n , k ) is ana-loguous.It remains to consider the case ( n , k ) = ( n , k ). Then ¯ z and ¯ z have both length k = k and we can rename them as ¯ z and write n instead of n or n and write k instead of k or k . It is easy to check that ϕ (¯ x, ¯ z ) ∨ ϕ (¯ x, ¯ z ) is a measuring formulafor X ∪ X . Claim dim( ϕ (¯ x, ¯ z ) ∧ ϕ (¯ x, ¯ z )) < n + k .We may assume that X and X are defined over ¯ c . Suppose, towards a contradiction,that there are ¯ e, ¯ b such that ϕ (¯ e, ¯ b ) ∧ ϕ (¯ e, ¯ b ) holds and that dim(¯ e, ¯ b/ ¯ c, HB(¯ c )) = n + k . By the definition of measuring formulas, there are ¯ a, ¯ d such that ¯ d ∈ H k ,SU(¯ a/ ¯ c, HB(¯ c )) = ω · n + k and tp L (¯ e, ¯ b/ ¯ c, HB(¯ c ) = tp L (¯ a, ¯ d/ ¯ c, HB(¯ c ). In particular ϕ (¯ a, ¯ d ) ∧ ϕ (¯ a, ¯ d ) holds. Therefore, ¯ a ∈ X and ¯ a ∈ X , contradicting that X ∩ X = ∅ .We know that both ϕ and ϕ have dimension n + k and the measure of ϕ is µ · | S k | and that of ϕ is µ ·| S k | . Since dim( ϕ (¯ x, ¯ z ) ∧ ϕ (¯ x, ¯ z )) < n + k , we get that the measureof ϕ ∨ ϕ is ( µ + µ ) · | S k | . Therefore, the measure of X ∪ Y is µ + µ as desired. (cid:3) Lemma 5.13. Let f : X → Y be an L H -definable surjective function and suppose ϕ (¯ x, ¯ y, ¯ z ) is a measuring formula for G ( f ) , the graph of f . Then: (a) The formula ϕ ′ (¯ x, ¯ y, ¯ z ) := ϕ (¯ x, ¯ y, ¯ z ) ∧ ∀ ¯ y ′ ( ϕ (¯ x, ¯ y ′ , ¯ z ) → ¯ y = ¯ y ′ ) is also a meas-uring formula for G ( f ) . (b) Assume that ρ ( x, z ) is a measuring formula for X . Then the formula ϕ ′′ (¯ x, ¯ y, ¯ z ) := ϕ ′ (¯ x, ¯ y, ¯ z ) ∧ ρ ( x, z ) is also a measuring formula for G ( f ) and ∃ y ϕ ′′ ( x, y, z ) is ameasuring formula for X .Proof. To simplify the notation along this proof, we will assume that f is defined over ∅ . In the generalize case one can rewrite the proof by relativizing all the notions to c, HB ( c ).(a) Suppose SU( G ( f )) = ω · n + k . Since ϕ ′ (¯ x, ¯ y, ¯ z ) implies ϕ (¯ x, ¯ y, ¯ z ), it is enough toshow that if ¯ a, ¯ b, ¯ d are such that ϕ (¯ a, ¯ b, ¯ d ) holds and dim(¯ a, ¯ b, ¯ d ) = n + k , then for all ¯ b ′ such that ϕ (¯ a, ¯ b ′ , ¯ d ) also holds, we have ¯ b = ¯ b ′ .Since dim(¯ a, ¯ b, ¯ d ) = n + k , by the definition of measuring formula, there are ¯ a , ¯ b , ¯ d with tp L (¯ a , ¯ b , ¯ d ) = tp L (¯ a, ¯ b, ¯ d ) such that SU(¯ a , ¯ b ) = ω · n + k and ¯ d is an enu-meration of HB(¯ a , ¯ b ). Also, (¯ a , ¯ b ) ∈ G ( f ). Note that ¯ b ∈ dcl L H (¯ a ), hence, byProposition 2.10, we have ¯ b ∈ acl L (¯ a , HB(¯ a )). Since ¯ a | ⌣ HB(¯ a ) H , this implies that¯ a ¯ b | ⌣ HB(¯ a ) H , we have HB(¯ a ¯ b ) = HB(¯ a ) and ¯ d is also an enumeration of HB(¯ a ). Claim: ¯ b ∈ dcl L (¯ a , ¯ d ). Let ¯ b ′ be such that tp L (¯ a , ¯ b , ¯ d ) = tp L (¯ a , ¯ b ′ , ¯ d ). As ¯ b ′ ∈ acl L (¯ a , ¯ d ) and (¯ a , ¯ d )is H -independent, so is (¯ a , ¯ b ′ , ¯ d ). Therefore, by Lemma 2.6, we havetp L H (¯ a , ¯ b , ¯ d ) = tp L H (¯ a , ¯ b ′ , ¯ d ) . Since ¯ b ∈ dcl L H (¯ a ), we get ¯ b = ¯ b ′ and the claim holds.Let g be an L -definable function over ∅ such that b = g ( a , d ). Since tp L (¯ a , ¯ b , ¯ d ) =tp L (¯ a, ¯ b, ¯ d ), we also have that ¯ b = g (¯ a, ¯ d ). By compactness, we may assume thatwhenever dim( a , b , d ) = n + k and ϕ ( a , b , d ) holds, then b = g ( a , d ). In partic-ular, if ϕ ( a, b ′ , d ) holds, we have n + k ≥ dim(¯ a, ¯ b ′ , ¯ d ) ≥ dim(¯ a, ¯ d ) = dim(¯ a, ¯ b, ¯ d ) = n + k, and thus b ′ = g ( a, d ) = b , as desired.(b) The proof that ϕ ′′ ( x, y, z ) is a measuring formula for G ( f ) is similar to the proofof (a), and we leave it to the reader. We will now prove that ∃ y ϕ ′′ ( x, y, z ) is a measuringformula for X . First choose a ∈ X with maximal SU -rank and let b := f ( a ). Just as inthe proof of part (a), we know that ¯ a | ⌣ HB(¯ a H , and this implies that ¯ a ¯ b | ⌣ HB(¯ a ) H andthus HB( ab ) = HB( a ), so k = | z | = | HB( a ) | . So we can take a defining formula for X with the same number of extra variables as the one we used for G ( f ).We will show that ∃ y ϕ ′′ ( x, y, z ) is a measuring formula for X , so we check properties(i) to (iv) in Definition 5.7.(i) First, notice that G ( f ) is definable over parameters, say c , then X is also definableover c . Since ϕ ( x, y, z ) is a measuring formula for G ( f ), it is defined over c, HB( c ), andso the formula ∃ y ϕ ′′ ( x, y, z ) is also defined over c, HB( c ).(ii) Consider a and the collection of tuples d ∈ H k such that ∃ y ϕ ′ ( a, y, d ) ∧ ρ ( x, d )holds. This collection is finite since it is a subset of ρ ( a, H k ) which has finitely manysolutions because ρ ( x, z ) is a measuring formula for X .Let X be the set defined by ∃ z ∈ H k ∃ y ϕ ′′ ( x, y, z ). Assume that a ∈ X is suchthat SU ( a ) = SU ( X ). Let b = f ( a ), then SU ( ab ) = SU ( X ) = SU ( G ( f )). As in theproof of (a), we have that HB ( ab ) = HB ( a ). Let d ∈ H k be an enumeration of thiscommon H -basis. Since ϕ ′ ( x, y, z ) is a measuring formula, ϕ ′ ( a, b, d ) holds. Similarly,since ρ ( a, H k ) is a measuring formula for X , ρ ( a, d ) also holds, thus a ∈ X .Now assume that a ∈ X is such that SU (tp( a )) = SU ( X ) and choose d ∈ H k suchthat ∃ y ϕ ′′ ( a, y, d ) holds. Let b be such that ϕ ′ ( a, b, d ) ∧ ρ ( a, d ) holds. Since ϕ ′ ( x, y, z )is a measuring formula for G ( f ) and the rank of the tuple ab coincides with SU ( G ( f )),then b = f ( a ) and a ∈ X .(iii) Assume that SU ( X ) = ω · n + k and choose a ∈ X such that SU (tp( a )) = ω · n + k .Let ¯ d ∈ H k . We have to show that ∃ y ϕ ′′ ( a, y, d ) holds if and only if ¯ d is an enumerationof HB( a ). So assume that ∃ y ϕ ′ ( a, y, d ) holds. Then ρ ( a, d ) holds we have that d is anenumeration of HB( a ). Conversely, let d be an enumeration of HB( a ). Let b = f ( a ).Then d be an enumeration of HB( ab ) = HB( a ). Since ϕ ′ is measuring formula for G ( f ),we have that ϕ ′ ( a, b, d ) holds. Also, since ρ is a measuring formula for X then ρ ( a, d )holds, so does ∃ y ϕ ′′ ( a, y, d ).(iv) Finally, choose e, d such that dim( e, d ) = n + k and ∃ y ϕ ′′ ( e, y, d ) holds. Since ρ ( x, z ) is a measuring formula, we can find a, d such thattp L (¯ a, ¯ d ) = tp L (¯ e, ¯ d ) -STRUCTURES AND GENERALIZED MEASURABLE STRUCTURES 27 with SU(¯ a ) = ω · n + k and ¯ d an enumeration of HB(¯ a ) as we wanted. (cid:3) Theorem 5.14. Let M be a coherent measurable structure and let ( M, H ) be a suf-ficiently saturated H -structure. Then, the dimension and measure of definable sets in ( M, H ) satisfy the Fubini condition: If f : X → Y is an L H -definable surjective func-tion such that for any ¯ b ∈ Y the definable set f − ( { ¯ b } ) has dimension and measure ( n , k , µ ) , and Y has dimension and measure ( n , k , µ ) . Then X has dimension andmeasure ( n , k , µ ) ⊙ ( n , k , µ ) . Proof. We may assume f is defined over ∅ . For every ¯ b ∈ Y , let X ¯ b := f − ( { ¯ b } ).Consider the definable family { X ¯ b : ¯ b ∈ Y } . We apply compactness, Theorem 3.11 andClaim 3.14 to the parameter space Y ∩ { ¯ b : SU(¯ b ) = ω · n + k } (which is a closed setin the space of L H -types by continuity of SU-rank) to get formulas { ψ i (¯ y ) , ψ ′ i (¯ y, ¯ v ) , ˜ η ′ i (¯ x, ¯ z, ¯ y, ¯ v ) : i ≤ N } where { ˜ η ′ i ( x, z, y, v ) : i ≤ N } are L -formulas, and the following conditions hold:(1) | ¯ v | = k and ψ ′ i (¯ b, H k ) is finite for any ¯ b ∈ Y . Also, if ¯ b ∈ Y and SU(¯ b ) = ω · n + k then for any ¯ h ∈ H k satisfying ψ ′ i (¯ b, ¯ h ), we have ¯ h is an enumerationof HB(¯ b ).(2) ψ i (¯ y ) := ∃ ¯ v ∈ H k ψ ′ i (¯ y, ¯ v );(3) Y ∩ { ¯ b : SU(¯ b ) = ω · n + k } is contained in the union S i ≤ N ψ i ( M | ¯ y | );(4) Whenever ¯ b ∈ Y and ¯ h ∈ H | ¯ v | is such that ψ ′ i (¯ b, ¯ h ) holds, then ˜ η ′ i (¯ x, ¯ z ; ¯ b, ¯ h ) is ameasuring formula of X ¯ b with | ¯ z | = k . In particular, ˜ η ′ i (¯ x, ¯ z ; ¯ b, ¯ h ) has dimension n + k over ¯ b, ¯ h and measure | S k | · µ . Furthermore, as in Lemma 3.9, we have ∀ ¯ x, ¯ z ˜ η ′ i (¯ x, ¯ z ; ¯ b, ¯ h ) → _ j i ≤ N i Ψ p ij (¯ x, ¯ z ; ¯ b, ¯ h ) where p ij are complete L H -types of SU -rank ω · n + k over parameter ¯ h, ¯ y ;(5) Both formulas ψ ′ i (¯ y, ¯ h ) and ˜ η ′ i (¯ x, ¯ z ; ¯ y, ¯ h ) are invariant under permutations of ¯ h .That is, for any σ ∈ S k , M | = ψ ′ i (¯ y, ¯ h ) ↔ ψ ′ i (¯ y, σ (¯ h )) and M | = ˜ η ′ i (¯ x, ¯ z ; ¯ y, ¯ h ) ↔ ˜ η ′ i (¯ x, ¯ z ; ¯ y, σ (¯ h )).By finite additivity, we only need to show the Fubini Property for every set Y i := Y ∩ ψ i ( M | ¯ y | ) \ [ j
The formula τ ′ (¯ x, ¯ y, ¯ z, ¯ v ) is a measuring formula for the definable set G ( f ) := { (¯ a, f (¯ a )) : ¯ a ∈ X } . We will first finish the proof of this theorem assuming the claim.By the additivity of SU-rank, SU( G ( f )) = ω · ( n + n ) + ( k + k ), and by the claimabove the formula τ ′ (¯ x, ¯ y, ¯ z, ¯ v ) has dimension ( n + n ) + ( k + k ). Suppose it hasmeasure µ , then the measure of G ( f ) is µ/ | S k + k | . By Lemma 5.13 part (a),˜ τ (¯ x, ¯ y, ¯ z, ¯ v ) := τ ′ (¯ x, ¯ y, ¯ z, ¯ v ) ∧ ∀ ¯ y ′ (cid:0) τ ′ (¯ x, ¯ y ′ , ¯ z, ¯ v ) → ¯ y = ¯ y ′ (cid:1) is also a measuring formula of G ( f ). Hence the measure of ˜ τ is also µ .Assume that ρ ( x, z ) is a measuring formula for X . Then by part (b) of Lemma 5.13the formula τ ′′ (¯ x, ¯ y, ¯ z ) := ˜ τ (¯ x, ¯ y, ¯ z ) ∧ ρ ( x, z ) is also a measuring formula for G ( f ) (againwith measure µ ) and ∃ y τ ′′ ( x, y, z ) is a measuring formula for X (also with measure µ ).Therefore, the dimension and measure of X is ( n + n , k + k , µ/ | S k + k | ).To complete the proof, we only need to show that µ | S k + k | = µ · µ . Note that by assumption, ϕ (¯ y, ¯ v ) has dimension n + k and measure µ · | S k | and˜ η ′ (¯ x, ¯ z ; ¯ b, ¯ h ) has dimension n + k and measure µ · | S k | over any ¯ b, ¯ h such that ϕ (¯ b, ¯ h )holds. Therefore, by the Fubini condition for L -formulas in M , we have that τ (¯ x, ¯ y, ¯ z, ¯ v )has dimension ( n + n ) + ( k + k ) and measure | S k | · | S k | · µ µ .Recall that τ ′ (¯ x, ¯ y, ¯ z, ¯ v ) = W σ ∈ S k k τ (¯ x, ¯ y, σ (¯ z, ¯ v )). By construction, τ is invariantunder permutations in each of the tuples ¯ z and ¯ v . That is, for σ ∈ S k and σ ∈ S k we have that τ (¯ x, ¯ y, σ (¯ z ) , σ (¯ v )) holds if and only if τ (¯ x, ¯ y, ¯ z, ¯ v ) holds. Note also thatif τ (¯ a, ¯ b, ¯ d, ¯ e ) holds with SU(¯ a, ¯ b ) = ω · ( n + n ) + ( k + k ) and ¯ d, ¯ e an enumerationof HB(¯ a, ¯ b ), then ( ¯ d, ¯ e ) is a tuple of distinct elements in H . Therefore, if τ (¯ a ′ , ¯ b ′ , ¯ d ′ , ¯ e ′ )holds with dim(¯ a ′ , ¯ b ′ , ¯ d ′ , ¯ e ′ ) = ( n + n ) + ( k + k ), then the elements in the tuple ( ¯ d ′ , ¯ e ′ )are also pairwise distinct. We conclude that τ ′ (¯ x, ¯ y, ¯ z, ¯ h ) is a disjoint union of (cid:0) k + k k (cid:1) copies of τ (¯ x, ¯ y, ¯ z, ¯ h ). Hence, µ = (cid:18) k + k k (cid:19) · | S k | · | S k | · µ µ = ( k + k )! · µ µ = | S k + k | · µ µ as desired. (cid:3) Claim 5.15. τ ′ (¯ x, ¯ y, ¯ z, ¯ v ) is a measuring formula for the definable set G ( f ) := { (¯ x, f (¯ x )) :¯ x ∈ X } .Proof. We will show that conditions (i)-(iv) from Definition 5.7 hold for τ ′ (¯ x, ¯ y, ¯ z, ¯ v )and G ( f ). (i) Note that G ( f ) is defined over ∅ , and so is the formula τ ′ (¯ x, ¯ y, ¯ z, ¯ v ). (ii) By Lemma 2.18, SU( G ( f )) = ω · ( n + n ) + ( k + k ). Also, note that z, v is atuple of variables of length k + k . By definition of measuring formulas, ϕ (¯ b, H k ) is -STRUCTURES AND GENERALIZED MEASURABLE STRUCTURES 29 finite for any ¯ b , and ˜ η ′ (¯ a, H k ; ¯ b, ¯ h ) is finite for any ¯ a, ¯ b, ¯ h . Thus, τ ′ (¯ a, ¯ b, H k + k ) is finitefor any ¯ a, ¯ b .Let Z be the set defined by ∃ ¯ z, ¯ v ∈ H k + k τ ′ (¯ x, ¯ y, ¯ z, ¯ v ). We need show that SU( G ( f ) △ Z ) <ω · ( n + n ) + ( k + k ), that is, G ( f ) and Z has the same set of generics.Let (¯ a, ¯ b ) ∈ G ( f ) with SU(¯ a, ¯ b ) = ω · ( n + n ) + ( k + k ). Then ¯ b ∈ Y and SU(¯ b ) = ω · n + k . Let ¯ h ∈ H k be an enumeration of HB(¯ b ). Since ϕ (¯ y, ¯ v ) is a measuring formulafor Y , we have that M | = ϕ (¯ b, ¯ h ). Also, we have SU(¯ a/ ¯ b, ¯ h ) = ω · n + k and ¯ a ∈ X ¯ b . Byitem (4) of the listed properties in the proof of Theorem 5.14 the formula ˜ η ′ (¯ x, ¯ y ; ¯ b, ¯ h )is a measuring formula for X ¯ b . Let ¯ h ∈ H k be an enumeration of HB(¯ a/ ¯ b, ¯ h ). Thenwe must have M | = ˜ η ′ (¯ a, ¯ h ; ¯ b, ¯ h ). Therefore, we have M | = ˜ η ′ (¯ a, ¯ h ; ¯ b, ¯ h ) ∧ ϕ ( b, h ), andin particular, (¯ a, ¯ b ) ∈ Z .Now suppose that (¯ a, ¯ b ) ∈ Z is a tuple of maximal SU -rank. Then there are ¯ h ∈ H k and ¯ h ∈ H k such that M | = ˜ η ′ (¯ a, ¯ h ; ¯ b, ¯ h ) ∧ ϕ (¯ b, ¯ h ). Since M | = ∃ ¯ v ∈ H k ϕ (¯ b, ¯ v ), weget SU(¯ b ) ≤ ω · n + k . On the other hand, as ϕ (¯ b, ¯ h ) implies ∀ ¯ x, ¯ z ˜ η ′ (¯ x, ¯ z ; ¯ b, ¯ h ) → _ j ≤ N Ψ p j (¯ x, ¯ z ; ¯ b, ¯ h ) for some types p j of SU -rank ω · n + k . Thus, Ψ p j (¯ a, ¯ e ; ¯ d, ¯ b ) holds for some j ≤ N . So,we get SU(¯ a/ ¯ b, ¯ h ) ≤ ω · n + k . Since ϕ ( y, v ) is a measuring formula for Y , ϕ ( b, H k )is finite, and therefore h ⊆ acl L H ( b ) ∩ H = HB ( b ). Hence,SU(¯ a, ¯ b ) = SU(¯ a/ ¯ b ) ⊕ SU(¯ b ) = SU(¯ a/ ¯ b, ¯ h ) ⊕ SU(¯ b ) ≤ ω · ( n + n ) + ( k + k ) . Since we proved already that Z contains the generics of G ( f ), and (¯ a, ¯ b ) is a tuple ofmaximal rank in Z , we must have that SU(¯ a/ ¯ b ) = ω · n + k and SU(¯ b ) = ω · n + k .Again, since ϕ ( y, v ) is a measuring formula for Y , we have that ¯ b ∈ Y and ¯ h is anenumeration of HB(¯ b ). Therefore, η ′ (¯ x, ¯ z ; ¯ b, ¯ h ) is a measuring formula for X ¯ b . SinceSU(¯ a/ ¯ b ) = ω · n + k and SU( X b △∃ ¯ z η ′ (¯ x, ¯ z ; ¯ b, ¯ h )) < ω · n + k , we get that ¯ a ∈ X ¯ b ,that is, (¯ a, ¯ b ) ∈ G ( f ). (iii) We will show that if (¯ a, ¯ b ) ∈ G ( ¯ f ) and SU(¯ a, ¯ b ) = ω · ( n + n ) + ( k + k ), then M | = τ ′ (¯ a, ¯ b, ¯ h ) if and only if ¯ h is an enumeration of HB(¯ a, ¯ b ). Note that here thetuple h has length k + k .Suppose ¯ h is an enumeration of HB(¯ a, ¯ b ). By additivity of HB-basis, we haveHB(¯ a, ¯ b ) = HB(¯ a/ ¯ b, HB(¯ b )) ∪ HB(¯ b ). Thus, there is σ ∈ S k + k such that σ (¯ h ) = ¯ h , ¯ h where ¯ h is an enumeration of HB(¯ b ) and ¯ h an enumeration of HB(¯ a/ ¯ b, HB(¯ b )). Since ϕ ( y, v ) is a measuring formula for Y , the value of SU( b ) is ω · n + k and h is anenumeration of HB( b ), we have M | = ϕ ( b, h ). Similarly, M | = ˜ η ( a, h ; b, h ). Hence, M | = τ ′ (¯ a, ¯ b, ¯ h ).The other direction is clear because τ ′ is the conjunction of the measuring formulas ϕ ( y, v ) for Y and ˜ η ( x, z, b, h ) for X b . (iv) By assumption, we have dim( ϕ (¯ y, ¯ v )) = n + k and dim(˜ η ′ (¯ x, ¯ z ; ¯ b, ¯ h )) = n + k for any parameter ¯ b, ¯ h satisfying ϕ (¯ b, ¯ h ). Therefore, since the dimension is additive,dim( τ ′ (¯ x, ¯ y, ¯ z, ¯ v )) = ( n + n ) + ( k + k ) . Suppose now that M | = τ ′ (¯ a, ¯ b, ¯ h ) and that dim(¯ a, ¯ b, ¯ h ) = ( n + n ) + ( k + k ).Then there is σ ∈ S k + k such that σ (¯ h ) = ¯ h , ¯ h and M | = ˜ η ′ (¯ a, ¯ h ; ¯ b, ¯ h ) ∧ ϕ (¯ b, ¯ h ). Since dim(˜ η ′ (¯ x, ¯ z ; ¯ b, ¯ h )) = n + k and dim( ϕ (¯ y, ¯ v )) = n + k , we must have dim(¯ b, ¯ h ) = n + k and dim(¯ a, ¯ h / ¯ b, ¯ h ) = n + k . By assumption, there are ¯ b ′ , ¯ h ′ has the same L -type as ¯ b, ¯ h such that SU(¯ b ′ ) = ω · n + k and ¯ h ′ is an enumeration of HB(¯ b ′ ).Let p ( x, y ; b, h ) = tp L (¯ a, ¯ h / ¯ b, ¯ h ) and consider the type p ( x, y ; b ′ , h ′ ). Finally, let¯ a ′ , ¯ h ′ be a realization of p ( x, y ; b ′ , h ′ ). Then M | = ˜ η ′ (¯ a ′ , ¯ h ′ ; ¯ b ′ , ¯ h ′ ) and we also havedim(¯ a ′ , ¯ h ′ / ¯ b ′ , ¯ h ′ ) = n + k . Since ¯ b ′ ∈ Y and ˜ η ′ (¯ x, ¯ z ; ¯ b ′ , ¯ h ′ ) is a measuring formula for X ¯ b ′ , there are ¯ a ′′ , ¯ h ′′ realizing p ( x, z ; b ′ , h ′ ) such that SU(¯ a ′′ / ¯ b ′ , ¯ h ′ ) = ω · n + k and ¯ h ′′ is an enumeration of HB(¯ a ′′ / ¯ b ′ , ¯ h ′ ). Now the tuples ¯ a ′′ , ¯ b ′ and h ′ = σ − (¯ h ′′ , ¯ h ′ ) satisfythe following: • tp L ( a ′′ , b ′ , h ′′ , h ′ ) = tp L ( a ′ , b ′ , h ′ , h ′ ) = tp L ( a, b, h , h ), so we can conclude thattp L ( a ′′ , b ′ , h ′ ) = tp L ( a, b, h ). • By additivity of SU-rank, SU(¯ a ′′ , ¯ b ′ ) = ω · ( n + n ) + ( k + k ) and by theadditivity of the HB-basis, h ′ is an enumeration of HB(¯ a ′′ , ¯ b ′ ). (cid:3) The results of this section can be summarized as: Theorem 5.16. Suppose T is the theory of a coherent measurable structure. Then T ind is the theory of a generalized measurable with measure semi-ring R = (( N × N > × R > ) ∪ { (0 , , k ) : k ∈ N } , ⊕ , ⊙ )The above result has some nice consequences. Remark 5.17. Assume ( M, · ) is a measurable coherent group. Then for every a ∈ M the map that sends x to a · x is a definable bijection and thus preserves the measure,so the (normalized) measure makes the structure ( M, · ) and its expansion ( M, · , H )definably amenable groups. Remark 5.18. Let us consider the special case of measures of L H -definable subsetsof H k with dimension k . By Lemma 2.7 a definable subset Y of H ( M ) k is the set ofsolutions of a formula of the form (¯ x ∈ H k ) ∧ θ (¯ x, ¯ c ), where ¯ c is a tuple from M and θ (¯ x, ¯ y ) is an L -formula. Recall that H with the induced structure from M is a generictrivialization of M (see [4]) and that there is a nice correspondence between its theory T ∗ and the theory T . For example T is supersimple of SU -rank 1 or strongly minimalif and only if T ∗ is supersimple of SU -rank 1 or strongly minimal (see also [4]). If Y isa definable subset of H ( M ) k of dimension k , the density property implies that genericproperties of θ (¯ x, ¯ c ) hold for Y . In our setting, this translates as the measure of Y beingequal to meas( θ (¯ x, ¯ c )), where meas( θ (¯ x, ¯ c )) is the L -measure of the formula θ (¯ x, ¯ c ), asthe following shows: Corollary 5.19. Let Y ⊂ H k be of dimension k and assume that θ (¯ x, ¯ y ) is an L -formulaand ¯ c is a tuple in M such that Y is defined by the expression (¯ x ∈ H k ) ∧ θ (¯ x, ¯ c ) . Thenthe measure of Y is the L -measure of the formula θ (¯ x, ¯ c ) .Proof. Let ¯ x , ¯ z be tuples of variables of length k and let ϕ (¯ x, ¯ z, ¯ c ) be the formula _ σ ∈ S k θ (¯ z, ¯ c ) ∧ (¯ x = σ (¯ z )). Then ϕ (¯ x, ¯ z, ¯ c ) is a measuring formula for Y , and in particularwe have dim( ϕ (¯ x, ¯ z, ¯ c )) = k (which was known) and meas( ϕ (¯ x, ¯ z, ¯ c )) = k ! · meas( θ (¯ x, ¯ c )).Thus, the measure of Y is meas( ϕ (¯ x, ¯ z, ¯ c )) /k ! = meas( θ (¯ x, ¯ c )). (cid:3) -STRUCTURES AND GENERALIZED MEASURABLE STRUCTURES 31 Questions In [1, 10] the authors define the notion of multidimensional asymptotic classes , whichare classes of finite structures whose ultraproducts are generalized measurable struc-tures. If M is an ultraproduct of a one-dimensional asymptotic class, then by Theorem5.16, ( M, H ) is a generalized measurable structure. By the results in [11], ( M, H ) isalso pseudofinite. Question 6.1. Is ( M, H ) elementary equivalent to an ultraproduct of finite structuresin a multidimensional asymptotic class?The answer to this question might shed some light on whether the extra hypothesisthat we used in Section 4 holds in general. References [1] S. Anscombe, D. Macpherson, C. Steinhorn, D. Wolf. Multidimensional asymptotic classes andgeneralised measurable structures . In preparation.[2] A. Berenstein, E. Vassiliev, On lovely pairs of geometric structures. Annals of Pure and AppliedLogic, Volume 161, Issue 7, April 2010, Pages 866-878.[3] A. Berenstein, E. Vassiliev. Geometric structures with a dense independent subset. Selecta Math-ematica, Volume 22, Issue 1, 2016, pp 191-225.[4] A. Berenstein, E. Vassiliev. Generic Trivializations of Geometric Theories , Mathematical LogicQuarterly, Volume 60, Issue 4-5, 2014, pp 243371.[5] Z. Chatzidakis, Model Theory of Difference Fields . In P. Cholak (Ed.), The Notre Dame Lectures(Lecture Notes in Logic, pp. 45-94). Cambridge: Cambridge University Press.[6] Z. Chatzidakis, A. Pillay, Generic structures and simple theories, Annals of Pure and AppliedLogic, Volume 95, Issues 1-3, 1998, pp 71-92.[7] R. Elwes, D. Macpherson. A survey of asymptotic classes and measurable structures. In ModelTheory with Applications to Algebra and Analysis (vol. 2.). London Mathematical Society LectureNote Series 350. Cambridge University Press. 2008.[8] D. Macpherson, C. Steinhorn. One-dimensional asymptotic classes of finite structures. Transactionsof the American Mathematical Society. Volume 360, pages 411-448. 2007.[9] E. Vassiliev. Generic pairs of SU-rank 1 structures , Annals of Pure and Applied Logic, vol.120(2003), 103-149.[10] D. Wolf. Multidimensional asymptotic classes of finite structures . PhD. Thesis. University of Leeds.2016.[11] T. Zou. Pseudofinite H -structures and groups definable in supersimple H -structures . Journal ofSymbolic Logic. Volume 84(3), pp 937-956, 2019.[12] T. Zou. Pseudofinite Difference fields . Submitted. Preprint available at https://arxiv.org/pdf/1806.10026.pdf Alexander Berenstein, Departamento de Matemticas. Universidad de los Andes, Car-rera 1 No. 18A-10, Edificio H, Bogot´a 111711, Colombia.Dar´ıo Garc´ıa, Departamento de Matemticas. Universidad de los Andes, Carrera 1 No.18A-10, Edificio H, Bogot´a 111711, Colombia. E-mail address : [email protected] Tingxiang Zou, Universit´e Claude Bernard - Lyon 1. CNRS 5208, Institut CamilleJordan, 43 Blvd. du 11 Novembre 1918, F.69622 Villeurbanne Cedex, France. Current address : Manchester Building, Einstein Institute of Mathematics, Hebrew University ofJerusalem, 91904 Jerusalem, Israel. E-mail address ::