BBOOLEAN TYPES IN DEPENDENT THEORIES
ITAY KAPLAN, ORI SEGEL AND SAHARON SHELAH
Abstract.
The notion of a complete type can be generalized in a natural manner to allowassigning a value in an arbitrary Boolean algebra B to each formula. We show some basicresults regarding the effect of the properties of B on the behavior of such types, and showthey are particularity well behaved in the case of NIP theories. In particular, we generalizethe third author’s result about counting types, as well as the notion of a smooth type andextending a type to a smooth one. We then show that Keisler measures are tied to certainBoolean types and show that some of the results can thus be transferred to measures — inparticular, giving an alternative proof of the fact that every measure in a dependent theorycan be extended to a smooth one. We also study the stable case. We consider this paper asan invitation for more research into the topic of Boolean types. Introduction
A complete type over a set A in the variable x is a maximal consistent set of formulas from L x ( A ), the set of formulas in x with parameters from A . Of course, one can think of L x ( A ) asa Boolean algebra by identifying formulas which define the same set in C (the monster model).Viewed this way, a type can also be defined as a homomorphism of Boolean algebras from L x ( A ) to the Boolean algebra 2. The idea behind this work is to generalize this definitionby allowing arbitrary an Boolean algebra B in the range. We call these homomorphisms B -types over A (see Definition 2.3). Without any assumptions, Boolean types may behave verywildly, but it turns out that if the ambient theory T is dependent (NIP) then there are somerestrictions on their behavior which gives some credence to the claim that this is the rightcontext to study such types in full generality. Mathematics Subject Classification. a r X i v : . [ m a t h . L O ] A ug OOLEAN TYPES IN DEPENDENT THEORIES 2
Let us consider an example. Suppose that T is the theory of the random graph in thelanguage { R } and that B is any Boolean algebra. Let M | = T be of size ≥ |B| and let h : M → B be a surjective map. Let p : L x ( M ) → B be the unique homomorphism definedby mapping formulas of the form x R a to h ( a ) and formulas of the form x = a to 0 (sucha homomorphism exists by quantifier elimination and Sikorski’s extension theorem, see Fact2.1 below). Thus for any B there is a type whose image has size size |B| .However, when T is NIP (dependent) this fails. For example, suppose that T = DLO (the theory of ( Q , < )), and suppose that B is a Boolean algebra with c.c.c (the countablechain condition: if (cid:104) a i | i < ℵ (cid:105) are non-zero elements then for some i, j < ℵ , a i ∧ a j (cid:54) = 0),for example the algebra of measurable sets up to measure 0 in some (real) probability space.Let M be any model and let p : L x ( M ) → B be any B -type. Suppose that the image of p has size (cid:0) ℵ (cid:1) + . By quantifier elimination, the image of p is the algebra generated byelements of the form p ( x < a ) for a ∈ M (since x = a is equivalent to ¬ ( x < a ∧ a < x )).It follows that |{ p ( x < a ) | a ∈ M }| = (cid:0) ℵ (cid:1) + , so we can find (cid:68) a i (cid:12)(cid:12)(cid:12) i < (cid:0) ℵ (cid:1) + (cid:69) such that p ( x < a i ) (cid:54) = p ( x < a j ) for i (cid:54) = j . By Erd˝os–Rado, we may assume that (cid:104) a i | i < ω (cid:105) is eitherincreasing of decreasing. Assume the former. Then p ( x < a i ) < B p ( x < a j ) so p ( a i ≤ x ) · p ( x < a j ) = p ( a i ≤ x < a j ) (cid:54) = 0 for all i < j < ω and thus { p ( a i ≤ x < a i +1 ) | i < ω } is aset of size ℵ of nonzero mutually disjoint elements from B , contradiction. This boundednessof the image generalizes to any NIP theory T , see Proposition 2.12 below.Let us consider another example. In the classical settings, any type can be realized inan elementary extension. Once a type is realized, it has a unique extension to any model.Boolean types that have unique extensions are called smooth . Going back to the theory ofthe random graph, let us assume that B is the algebra of Borel subsets of 2 κ up to measure 0(with the measure µ being the product measure, see [Fre03, 254J]), and assume that | M | ≥ κ . Let h : M → κ be surjective. Let p : L x ( M ) → B defined by p ( x R a ) = U h ( a ) = { η ∈ κ | η ( h ( a )) = 1 } so that its measure is 1 / p ( x = a ) = 0 for any a ∈ M (againsuch a B -type exists). Note that if N (cid:31) M and q is a B -type extending p , then q ( x = a ) = 0for any a ∈ N since otherwise it has positive measure, which leads to a contradiction (sincefor any conjunction ϕ of atomic formulas or their negations over M which a realizes satisfies µ ( q ( ϕ )) ≥ µ ( q ( x = a )) and the right hand side tends to 0 as the number of distinct conditions OOLEAN TYPES IN DEPENDENT THEORIES 3 in ϕ grows by the choice of p ). Thus again by Sikorski, we have a lot of freedom in extending q to any N (cid:48) (cid:31) N . Hence no smooth extension of p exists. Again, when T is NIP and theBoolean algebra is nice enough, every type has a smooth extension, see e.g. Corollary 2.37below.1.1. Structure of the paper.
In Section 2 we prove all main results on Boolean types.In Subsection 2.1 we give the basic definitions. In particular we define two kinds of mapsbetween Boolean types: those induced by elementary maps (like in the classical setting) andthose induced by embeddings of the Boolean algebra itself. In Subsection 2.2 we then givebounds to the number of Boolean types up to conjugation (in particular generalizing a resultof the third author [She12, Theorem 5.21]) or just image conjugation. In Subsection 2.3we define and discuss smooth Boolean types as well as a stronger notion, that of a realizedBoolean type. We prove that under NIP, if the algebra is complete, smooth types exist.In Section 3 we relate Boolean types to Keisler measures, i.e., finitely additive measureson definable sets. In Subsection 3.2 we apply the results of Subsection 2.2 to count Keislermeasures up to conjugation (we also give a more direct proof, using the VC-theorem). InSubsection 3.3 we give an alternative proof to the well-known fact that every Keisler measureextends to a smooth one [Sim14, Proposition 7.9], using the results of Subsection 2.3.In Section 4 we analyze the case where the theory (or just one formula) is stable, as well asthe totally transcendental case, showing that in this case Boolean types are locally averagesof types (and in the t.t. case this is true for complete types as well).Throughout, let T be a complete first order theory. Most of the time, we will only dealwith dependent T . We use standard notations, e.g., C is a monster model for T . As usual,all sets and models are subsets or elementary substructures of C of cardinality < | C | . Acknowledgments.
We would like to thank Artem Chernikov for giving some comments onprivate communication. In particular for pointing out the alternative proof of Proposition3.12. 2.
Boolean types
Basic Definition.
Let us start by recounting some basic notation for Boolean algebras.
OOLEAN TYPES IN DEPENDENT THEORIES 4
Let B a Boolean algebra, and denote by 0 and 1 the distinguished elements of B corre-sponding to ⊥ and (cid:62) for formulas. We denote by B + the set of all nonzero elements of B . Let a , b ∈ B . We denote by − a , a + b and a · b the complement, sum and product –corresponding to ¬ , ∨ and ∧ for formulas, respectively (for example, − a − b = a · ( − b ). We say a , b are disjoint if a · b = 0. We also write ( − · a for − a and 1 · a for a .Every Boolean algebra has a canonical order relation: a ≤ b iff a · b = a or equivalently a + b = b . This corresponds to → for formulas. Recall that 0 and 1 are a minimum andmaximum with respect to this order.If the supremum of a set A ⊆ B exists we denote it by (cid:80) A , likewise, if the infimum existswe denote it by (cid:81) A . An algebra is complete if both always exist.A ( κ -)complete subalgebra of B is some subalgebra B (cid:48) ⊆ B such that if A ⊆ B (cid:48) (and | A | < κ ), and if (cid:80) A exists in B , then (cid:80) A ∈ B (cid:48) .Since we will deal with homomorphisms of Bolean algebras, we will also need the followingfacts (Sikorski’s extension theorem): Fact 2.1. [Kop89, Theorem 5.5 and Theorem 5.9]
Assume B is a complete Boolean algebraand A is any Boolean algebra. Assume A ⊆ A a subset and f : A → B is a function.Then there is a homomorphism g : A → B extending f iff, for any a , ..., a n − ∈ A and ε , ..., ε n − ∈ {± } such that (cid:81) i Assume B is a complete Boolean al-gebra and A is any Boolean algebra. Assume A (cid:48) ⊆ A a subalegbra, and f : A (cid:48) → B is ahomomorphism.Assume further that a ∈ A , b ∈ B . Then there exists a homomorphism g : A → B extending f such that g ( a ) = b iff (cid:80) a (cid:48) ∈A (cid:48) , a (cid:48) ≤ a f ( a (cid:48) ) ≤ b ≤ (cid:81) a (cid:48) ∈A (cid:48) , a ≤ a (cid:48) f ( a (cid:48) ) . Definition 2.3. Suppose B is Boolean algebra. For a set A , a (complete) B -type in x over A is a Boolean algebra homomorphism from the algebra of formulas L x ( A ) consisting offormulas in x over A up to equivalence in C to B . By slight abuse of notation, we will use aformula to refer to its equivalence class in L x ( A ). The set of all complete B -types in x over OOLEAN TYPES IN DEPENDENT THEORIES 5 A is denoted by S x B ( A ). The set S n B ( A ) for n a natural number or an ordinal, will be the setof all complete B -types over A in some n fixed variables.The group of elementary permutations π : A → A acts on S x B ( A ) by ( π ∗ p ) ( ϕ ( x, a )) = p (cid:0) ϕ (cid:0) x, π − ( a ) (cid:1)(cid:1) . Say that p , p ∈ S x B ( A ) are elementarily conjugates over A if they are inthe same orbit.We say that two B -types are image conjugates if there is some partial isomorphism ofBoolean algebras σ whose domain contains the image of p such that p = σ ◦ p .Finally, we say that p , p ∈ S x B ( A ) are conjugates over A if there is some π : A → A asabove, and some Boolean isomorphism σ as above such that p = σ ◦ ( π ∗ p ). Remark . Note that σ ◦ ( π ∗ p ) = π ∗ ( σ ◦ p ). Thus, as image conjugation and elementaryconjugation are clearly equivalence relations, so is conjugation. Remark . Note that when B = 2 (that is { , } ), a complete B -type is the same as acomplete type.Also, the two notions of elementarily conjugation and conjugation identify in this case. Remark . Note that L x ( A ) is isomorphic to the quotient of the Lindenbaum-Tarski algebra L (the algebra of L formulas in x up to logical equivalence, see [MB89, Chapter 26]) by thefilter F generated by all sentences ϕ that hold in C (where parameters are considered to beconstants).Indeed, C (cid:15) ψ ( x ) ↔ ψ ( x ) iff C (cid:15) ∀ x ( ψ ( x ) ↔ ψ ( x ));and since (cid:96) ∀ x ( ψ ( x ) ↔ ψ ( x )) → ( ψ ( x ) ↔ ψ ( x ))we get ∀ x ( ψ ( x ) ↔ ψ ( x )) ≤ ψ ( x ) ↔ ψ ( x ) in L .On the other hand if ϕ → ( ψ ( x ) ↔ ψ ( x )) for a sentence ϕ such that C (cid:15) ϕ , certainly C (cid:15) ψ ( x ) ↔ ψ ( x ).This means that homomorphisms from L sending F to 1 are canonically isomorphic tohomomorphisms from L x ( A ).Thus we can define S x B ( A ) to be (cid:8) p ∈ Hom ( L , B ) | F ⊆ p − (1) (cid:9) without changing any-thing. Example 2.7. One might wonder if there are any natural examples of Boolean types. OOLEAN TYPES IN DEPENDENT THEORIES 6 Let (cid:104)B i (cid:105) i ∈ I a sequence of Boolean algebras. Then the product algebra (cid:81) i ∈ I B i is defined inthe usual sense of products of algebraic structures — its elements are choice functions, andoperations are preformed coordinatewise.By the universal property of product algebras (see [Kop89, Proposition 6.3]), S x (cid:81) i ∈ I B i ( A ) ∼ = (cid:89) i ∈ I S x B i ( A ) . In particular, for any cardinal λ , S x λ ( A ) ∼ = ( S x ( A )) λ (= ( S x ( A )) λ ) naturally.Thus we can consider a sequence of λ complete types to be the same thing, for all intentsas purposes, as a B -type for B = 2 λ .This case will give us an idea about the behavior of general Boolean types.2.2. Counting Boolean types. Counting Boolean types up to conjugation. We first concern ourselves with the questionof the number of Boolean types up to conjugation. Remark . Note that ( S x ( A )) λ can be embedded in S | x |· λ ( A ) – choose for each p ∈ S x ( A )some b p ∈ C realizing it; and send each (cid:104) p i (cid:105) i<λ to tp (cid:0) (cid:104) b p i (cid:105) i<λ /A (cid:1) . Obviously, this is injective.Assume we took p , p ∈ S x λ ( A ), took the corresponding p , p ∈ ( S x ( A )) λ (as in Example2.7) and took their images q , q ∈ S | x |· λ ( A ) under this embedding . If q , q are conjugatesas witnessed by π , so are p , p (in each coordinate) and thus also p , p .Indeed for any ϕ ( x, a ) and i < λ , p (cid:0) ϕ (cid:0) x, π − ( a ) (cid:1)(cid:1) i = 1 ⇐⇒ C (cid:15) ϕ (cid:16) b ( p ) i , π − ( a ) (cid:17) ⇐⇒ C (cid:15) ϕ (cid:16) b ( p ) i , a (cid:17) ⇐⇒ ( p ( ϕ ( x, a ))) i = 1Thus p (cid:0) ϕ (cid:0) x, π − ( a ) (cid:1)(cid:1) = p ( ϕ ( x, a )) ⇒ p = π ∗ p . Corollary 2.9. The number of types in S x λ ( A ) up to elementary conjugation is at most thenumber of types in S λ | x | ( A ) up to conjugation. Definition 2.10. Fix some complete Boolean algebra B and regular cardinal κ . B is κ -c.c ifthere is no antichain (a set of pairwise disjoint elements from B + ) in B of size κ . Let A ⊆ C . OOLEAN TYPES IN DEPENDENT THEORIES 7 Definition 2.11. Suppose p ∈ S x B ( A ), and ϕ ( x, y ) is some formula. Then p (cid:22) ϕ , or p | ϕ , isthe restriction of p to the definable sets of the form ϕ ( x, a ) for a some tuple (in the length of y ) from A . Proposition 2.12. Assume T has N IP and B is κ -c.c. For any p ∈ S x B ( A ) and any formula, ϕ ( x, y ) , the image of p | ϕ has cardinality ≤ <κ .Proof. Recall that a subset A of B is independent if every nontrivial finite product from itis non-empty: for every a , . . . , a n and b , . . . , b m such that a i (cid:54) = b j for all i, j , the product a · . . . · a n · − b · . . . · − b m is not 0. By [She80, Kop89, Theorem 10.1], if λ is a cardinal suchthat λ is regular and µ <κ < λ for all µ < λ , then every subset X ⊆ B of cardinality λ has anindependent subset Y ⊆ X of cardinality λ .Let λ = (2 <κ ) + . It is easy to see that for all µ < λ , µ <κ < λ – since κ is regular, for every β < κ every function from | β | to sup (cid:8) | i | (cid:9) i<κ = 2 <κ is contained in some 2 | α | for α < κ .Thus (cid:0) <κ (cid:1) | β | ≤ (cid:88) α<κ (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) | α | (cid:17) | β | (cid:12)(cid:12)(cid:12)(cid:12) = (cid:88) α<κ | α || β | ≤ κ · sup (cid:110) | α || β | | α < κ (cid:111) = κ · sup (cid:110) | α | | α < κ (cid:111) = 2 <κ , so µ <κ = sup (cid:110) µ | β | (cid:111) β<κ ≤ sup (cid:110)(cid:0) <κ (cid:1) | β | (cid:111) β<κ ≤ <κ < λ Since T is NIP, ϕ ( x, y ) has dual VC-dimension n < ω : for any (cid:104) a i (cid:105) i Remark . If ϕ has IP and B is complete, then there is a B -type p such that | Im ( p | ϕ ) | = |B| .Indeed from compactness there is a sequence of parameters (cid:104) b a | a ∈ B(cid:105) witnessing IP, and f defined as f ( ϕ ( x, b a )) = a can be extended to a B -type by Fact 2.1.Assume B has κ -c.c.. Let A be the set of subalgebras B of B of size at most µ = 2 <κ + | x | + | T | . Proposition 2.12 says that for p ∈ S x B ( A ), the algebra B p which is the image of p ,is in A . For each B ∈ A , choose some enumeration (cid:104) a B,i | i < µ (cid:105) (maybe with repetitions) of B + . Let us say that B is conjugate to B if there is a (unique) isomorphism σ : B → B taking a B ,i to a B ,i .Given B ∈ A , for each i < µ , choose an ultrafilter D B,i (of B ) which contains a B,i (equiva-lently a homomorphism from B to 2 such that D B,i ( a B,i ) = 1, see [Kop89, Proposition 2.15]).We ask that if B and B are conjugates, say via σ , then σ ( D B ,i ) = D B ,i – or in the lan-guage of homomorphisms, D B ,i = D B ,i ◦ σ (that is we choose just one sequence of ultrafiltersper conjugation class and construct the others from it). Let D B = (cid:104) D B,i (cid:105) i<µ : B → µ , theproduct homormophism.For i < µ and p ∈ S x B ( A ), let (cid:101) q p ∈ S x µ ( A ) be the 2 µ -type D B p ◦ p . Finally, let q p = f ( (cid:101) q p )in S (cid:104) x i | i<µ (cid:105) ( A ) where f : S x µ ( A ) → S (cid:104) x i | i<µ (cid:105) ( A ) is as in Remark 2.8. Proposition 2.14. (Assuming B has κ -c.c. and T has N IP ) Suppose p , p ∈ S x B ( A ) and B = B p = B p and that (cid:101) q p = (cid:101) q p . Then p = p .Proof. If not, then for some ϕ ( x, a ), p ( ϕ ( x, a )) (cid:54) = p ( ϕ ( x, a )).WLOG p ( ϕ ( x, a )) (cid:2) p ( ϕ ( x, a )), and let b = p ( ϕ ( x, a )) − p ( ϕ ( x, a )), so b ∈ B + . Forsome i < µ , b = a B,i . It follows that p ( ϕ ( x, a )) − p ( ϕ ( x, a )) ∈ D B,i , hence (cid:101) q p ( ϕ ( x, a )) i = 1and (cid:101) q p ( ϕ ( x, a )) i = 0 that is (cid:101) q p (cid:54) = (cid:101) q p . (cid:3) Corollary 2.15. (Assuming B has κ -c.c. and T has N IP ) Suppose p , p ∈ S x B ( A ) and B = B p = B p . If q p and q p are conjugates over A , then p and p are elementarilyconjugates over A .Proof. Suppose that π ∗ q p = q p for an elementary permutation π : A → A . Then also, byRemark 2.8, π ∗ (cid:101) q p = (cid:101) q p . Note that π ∗ (cid:101) q p = (cid:101) q π ∗ p .But B π ∗ p = B p (since π just permutes the domain), thus we can apply Proposition 2.14to π ∗ p , p . (cid:3) OOLEAN TYPES IN DEPENDENT THEORIES 9 Corollary 2.16. (Assuming B has κ -c.c. and T has N IP ) If p , p ∈ S x B ( A ) , B p and B p are conjugates and q p and q p are conjugates over A , then p and p are conjugates over A .Proof. Suppose σ : B p → B p takes a B p ,i to a B p ,i . Then σ ◦ p and p satisfy the conditionof Corollary 2.15: note that B σ ◦ p = Im ( σ ◦ p ) = σ ( Im ( p )) = σ ( B p ) = B p .Further (cid:101) q p = D B p ◦ p = D B p ◦ σ ◦ p = (cid:101) q σ ◦ p And thus q σ ◦ p = q p and q p are conjugates.So for some elementary permutation π : A → A , π ∗ ( σ ◦ p ) = p (cid:3) Corollary 2.17. Assume B has κ -c.c. and T has N IP , and let µ = 2 <κ + | T | + | x | .The number of types in S x B ( A ) up to conjugation is bounded by the number of types in S | x |· µ ( A ) (equivalently S µ ( A ) ) up to conjugation + µ . Hence, if λ, κ are cardinals and α and ordinal such that λ = λ <λ = ℵ α = κ + α ≥ κ ≥ (cid:105) ω + µ + , and if A = M is a saturatedmodel of size λ , this number is bounded by < κ + | α | µ + 2 µ = 2 < κ + | α | µ .Proof. Given p , map it to the pair consisting of q p and the quantifier free type of the algebra B p , enumerated by (cid:10) a B p ,i | i < µ (cid:11) (whose number is bounded by 2 µ , since the language ofBoolean algebras is finite). By Corollary 2.16, this map is injective. The second part of thestatement follows immediately from Theorem 5.21 in [She12], since µ ≥ | T | . (cid:3) Counting types up to image conjugation. One may wonder whether we can get a mean-ingful bound for | S x B ( A ) | (without taking elementary conjugation into consideration). For ex-ample, the universal property of the product gives us a simple equation: | S x µ ( A ) | = | S x ( A ) | µ .Note that if σ : B ∼ −→ B (cid:48) is an isomorphism between two different subalgebras of B and p ∈ S x B ( A ) is such that Im ( p ) = B , then σ ◦ p ∈ S x B ( A ) is different from p . This means thatif B has many copies of small subalgebras then we necessarily have at least as many B types.Thus if we want to give a bound that is independent of the size of B , we must restrict ourselvesto counting up to image conjugation. Note that 2 has only a single subalgebra (itself) and asingle partial isomorphism (the identity), so for 2-types (i.e., classical types) counting typesup to image conjugation is the same as just counting types. OOLEAN TYPES IN DEPENDENT THEORIES 10 Corollary 2.18. Assume B has κ -c.c and T has N IP , and assume | A | ≥ ℵ . We havethe following bounds on λ , the number of types in S x B ( A ) up to image conjugation, where µ = 2 <κ + | T | + | x | .If T is stable, λ ≤ | A | µ ;If T is NIP, λ ≤ ( ded | A | ) µ , where ded θ is the supremum on the number of cuts on alinearly ordered set of cardinality ≤ θ .If T has IP and x is finite then λ could be maximal, i.e., sup {| S x B ( A ) | | | A | ≤ κ } = |B| κ for all κ ≥ |B| + | T | .Proof. The proof of Corollary 2.16 shows also that if p , p ∈ S x B ( A ), B p and B p are conju-gates and q p = q p , then p and p are image conjugates.We conclude that the number of B types over A up to image conjugation is at most | S µ ( A ) | .Let φ ( x, y ) some partitioned formula, and denote by S φ ( A ) the set of φ -types, that is, amaximal consistent set of formulas of the form φ ( x, b ) or ¬ φ ( x, b ) where b is a y -tuple in A .By a standard argument, for θ = sup {| S φ ( A ) | | φ ( x, y ) } , λ ≤ θ µ .According to [Sim14, Proposition 2.69], if T has NIP then θ ≤ ded | A | .Further, by the preceding remarks there, if T is stable then θ ≤ | A | thus λ ≤ | A | µ for stable T . Assume that T has IP. Note that 2 κ ≤ | B | κ ≤ κ κ = 2 κ . If φ ( x, y ) has IP then there issome model M | = T of size κ such that | S φ ( M ) | = 2 | M | . Note that the algebra 2 is embeddedin B so S x ( M ) ⊆ S x B ( M ). Finally any two of these types are not image conjugates sincethere is only one embedding of 2 to B . (cid:3) Smooth Boolean types.Definition 2.19. Let A ⊆ B , p ∈ S x B ( A ) and q ∈ S x B ( B ); then q extends p if it extends it asa function, that is for any formula ϕ ( x, a ) over A , p ( ϕ ( x, a )) = q ( ϕ ( x, a )) (technically, theimages of the equivalence classes are the same).We say that p is smooth if for every such B there exists a unique B -type q over B extending p . Remark . If B = 2 and M is a model, a type is smooth iff it is realized, that is equal totp ( a/M ), for some tuple a in M . OOLEAN TYPES IN DEPENDENT THEORIES 11 Remark . As we remarked, p ∈ S x κ ( A ) is essentially equivalent to a sequence (cid:104) p i (cid:105) i<κ ofcomplete types; it is obvious that q extends p iff q i extends p i for all i ; and thus p is smoothiff p i is smooth for each i .Assume | x | < ω (i.e., x is a finite tuple). Then this case gives rise naturally to the followingdefinition: Definition 2.22. A B -type p ∈ S x B ( M ) over a model M is called realized if (cid:80) a ∈ M p ( x = a ) = 1.(Here and later, when we write a ∈ M in the sum, we mean a ∈ M x .) Remark . If B = 2, (cid:80) a ∈ M p ( x = a ) = 1 iff exists a ∈ M such that p ( x = a ) = 1. Thereforethis definition agrees with the classical one for complete types.The following claim explains why this definition makes sense: Claim . Assume p ∈ S x B ( M ) is realized. Then p is smooth. Proof. Let b a = p ( x = a ) ∈ B for each x -tuple a in M . Assume q ∈ S x B ( N ) extends p . Let ϕ ( x, c ) ∈ L x ( N ) be some formula.Then by [Kop89, Lemma 1.33], q ( ϕ ( x, c )) = q ( ϕ ( x, c )) · (cid:88) a ∈ M b a = (cid:88) a ∈ M q ( ϕ ( x, c )) · b a = (cid:88) a ∈ M q ( ϕ ( x, c )) · q ( x = a ) = (cid:88) a ∈ M q ( x = a ∧ ϕ ( x, c ))But for any a ∈ M , if C (cid:15) ϕ ( a, c ) then ( x = a ∧ ϕ ( x, c )) = ( x = a ) (as definable sets) thus q ( x = a ∧ ϕ ( x, c )) = q ( x = a ) = b a , while if C (cid:15) ¬ ϕ ( a, c ) then ( x = a ∧ ϕ ( x, c )) = ⊥ thus q ( x = a ∧ ϕ ( x, c )) = 0.Thus we get necessarily q ( ϕ ( x, c )) = (cid:88) a ∈ ϕ ( M,c ) b a That is, we get q is uniquely determined. (cid:3) One may naturally ask if every smooth type is realized. We start with the following result: Claim . Assume p ∈ S x B ( M ) is smooth for B a complete Boolean algebra. Then (cid:80) a ∈ M p ( x = a )is maximal : for any extension q ∈ S x B ( N ) of p , (cid:88) a ∈ N q ( x = a ) = (cid:88) a ∈ M p ( x = a ) . OOLEAN TYPES IN DEPENDENT THEORIES 12 Proof. Assume otherwise. Since for any a ∈ M , q ( x = a ) = p ( x = a ), we have in particularsome a ∈ N \ M be such that q ( x = a ) > q (cid:48) ∈ S x B ( N ) extending p such that q (cid:48) ( x = a ) = 0. Indeed,for any consistent ϕ ( x, b ) ∈ L x ( M ) it cannot be ϕ ( x, b ) → x = a , as that would imply that a is definable over M ; but M ≺ N thus its definable closure is itself.Thus by Fact 2.2, there exists q (cid:48) as required.This means that if q ( x = a ) > q and q (cid:48) are distinct extensions of p , thus p is notsmooth. (cid:3) The property in Claim 2.25 has an alternative formulation which is somewhat easier toreason about: Claim . Assume B is complete. A type p ∈ S x B ( M ) has the property that (cid:80) a ∈ M p ( x = a ) isnot maximal iff there exists a subalgebra B (cid:48) of B such that Im ( p ) ⊆ B (cid:48) and an atom a ∈ B (cid:48) such that a ≤ − (cid:80) a ∈ M p ( x = a ). Proof. Assume q extends p and (cid:80) a ∈ N q ( x = a ) > (cid:80) a ∈ M p ( x = a ) and let c ∈ N \ M such that q ( x = c ) > 0. Let B (cid:48) = Im ( q ) and a = q ( x = c ).Then a must be an atom in B (cid:48) , since if b ≤ a and b ∈ B (cid:48) , let ϕ ( x, b ) such that q ( ϕ ( x, b )) = b . If C (cid:15) ϕ ( c, b ) then ϕ ( x, b ) ∧ x = c is the same as x = c and thus b = b · a = q ( ϕ ( x, b ) ∧ x = c ) = q ( x = c ) = a ; similarly if C (cid:50) ϕ ( c, b ) then ϕ ( x, b ) ∧ x = c is ⊥ thus we likewise get b = 0. Finally since a is disjoint from p ( x = a ) for any a ∈ M , a ≤ − (cid:80) a ∈ M p ( x = a ).On the other hand, let B (cid:48) and a ∈ B (cid:48) an atom. Let D a : B (cid:48) → a represented as a homomorphism (i.e. D a ( b ) = 1 ⇐⇒ a ≤ b ) and let p (cid:48) = D a ◦ p which is a complete type over M . p (cid:48) is not realized in M : as a ≤ − (cid:80) a ∈ M p ( x = a ), D a (cid:18) (cid:80) a ∈ M p ( x = a ) (cid:19) = 0 thus p (cid:48) ( x = a ) = D a ( p ( x = a )) ≤ D a (cid:18) (cid:80) a ∈ M p ( x = a ) (cid:19) = 0 for any a in M .Let c realize p (cid:48) outside of M , and let N containing c and M . Then for any ϕ ( x, b ) ∈ L x ( M ),if ϕ ( x, b ) → x = c then ϕ ( x, b ) = ⊥ (since c cannot be definable over M ) thus p ( ϕ ( x, b )) = 0 ≤ a ; OOLEAN TYPES IN DEPENDENT THEORIES 13 and if x = c → ϕ ( x, b ) then C (cid:15) ϕ ( c, b ) thus D a ( p ( ϕ ( x, b ))) = p (cid:48) ( ϕ ( x, b )) = 1 ⇒ a ≤ p ( ϕ ( x, b )) . Thus by Fact 2.2, p can be extended to a type q over N which satisfies q ( x = c ) = a thus (cid:88) a ∈ N q ( x = a ) > (cid:88) a ∈ M p ( x = a )as required. (cid:3) Remark . If p ∈ S x B ( M ) is onto and B atomless and complete, then (cid:80) a ∈ M p ( x = a ) ismaximal and in fact p ( x = a ) = 0 for all a . Example 2.28. Let L = { E B } B ∈B ( R ) (one unary predicate E B for every Borel set in R ), T = T h L ( R ) (with the obvious interpretations) and B the algebra B ( R ) /I where I = { B ∈ B ( R ) | µ ( B ) = 0 } where µ is the Lebesgue measure; B is a σ -complete and c.c.c. –thus complete – as well as atomless.Then T proves that every Boolean combination of the unary predicates is equivalent tosingle unary predicate, and by a standard argument eliminates quantifiers. Thus for any M ⊆ R , L x ( M ) is isomorphic to B ( R ) with E B ( x ) (cid:44) → B ( x = a is equivalent to E { a } ( x )).Let p : L x ( M ) → B the projection.We get that for any q : L x ( N ) → B extending p , q is a surjection to an atomless Booleanalgebra; therefore it sends atoms to 0, that is q ( x = a ) = 0 for all a ∈ N . Thus since x = y isthe only atomic formula involving both x and a parameter, and since T eliminates quantifiers, q is uniquely determined.Thus p is smooth, but not realized by the previous remark. Definition 2.29. Recall that a Boolean algebra B is called ( κ, -distributive if, for anysequence of pairs (cid:104) ( a i , b i ) (cid:105) i<κ in B such that the following two elements exist:1. (cid:81) i<κ ( a i + b i ) and2. (cid:81) i ∈ S a i (cid:81) i/ ∈ S b i for any S ⊆ κ ,the equality (cid:81) i<κ ( a i + b i ) = (cid:80) (cid:40) (cid:81) i ∈ S a i (cid:81) i/ ∈ S b i | S ⊆ κ (cid:41) holds in B . OOLEAN TYPES IN DEPENDENT THEORIES 14 Corollary 2.30. Assume T has N IP and M a model of T . If B is complete, has κ -c.c. andis (2 <κ + | T | , -distributive, then p ∈ S x B ( M ) is smooth iff p is realized iff (cid:80) a ∈ M p ( x = a ) ismaximal.Proof. By [Kop89, Proposition 14.8] and Proposition 2.12 under these assumptions on B thereis an atomic B (cid:48) between Im ( p ) and B .Thus by the previous two claims, if p is smooth then there is no atom under (cid:18) − (cid:80) a ∈ M p ( x = a ) (cid:19) in B (cid:48) – that is (cid:18) − (cid:80) a ∈ M p ( x = a ) (cid:19) = 0 ⇒ (cid:80) a ∈ M p ( x = a ) = 1. (cid:3) The following proposition gives us a way to extend types in a way that maximizes theirimages, in the following precise sense: Proposition 2.31. Assume B is complete. Let p ∈ S x B ( M ) a κ -c.c. B -type over M . Thenthere exists N ⊇ M and a type q over N extending p such that for every N (cid:48) ⊇ N , q (cid:48) extending q and ϕ ( x, y ) , Im ( q | ϕ ) = Im ( q (cid:48) | ϕ ) ; that is q has a maximal image, with respect to set inclusionfor extensions of q , for each formula.Proof. Let (cid:104) ϕ i ( x, y ) | i < | T |(cid:105) an enumeration of all partitioned formulas (recall that we areassuming that x is a finite tuple) and let (cid:104) b α | α < |B|(cid:105) be an enumeration of the elements of B . We construct recursively two increasing sequences with respect to the lexicographic orderon ( | T | + 1) × ( |B| + 1):1. An increasing sequence of models (cid:104) M i,α | α ≤ |B| , i ≤ | T |(cid:105) .2. An increasing (with respect to extension) sequence (cid:104) p i,α | α ≤ |B| , i ≤ | T |(cid:105) such that p i,α ∈ S x B ( M i,α ).The construction is as follows:For (0 , M , = M, p , = p Fix i and assume we have M i,α , p i,α for α < |B| ; if there exist M (cid:48) and p (cid:48) over M (cid:48) suchthat M i,α ⊆ M (cid:48) , p (cid:48) extends p and b α ∈ Im ( p (cid:48) | ϕ i ), let M i,α +1 = M (cid:48) , p i,α +1 = p (cid:48) ; otherwiselet M i,α +1 = M i,α , p i,α +1 = p i,α . Assume we have M i,α , p i,α for all α < δ ≤ |B| a limitordinal; then define M i,δ = (cid:83) α<δ M i,α and p i,δ = (cid:83) α<δ p i,α . Note that since ( p i,α ) α<δ is a chain ofhomomorphisms this is a well defined homomorphism. OOLEAN TYPES IN DEPENDENT THEORIES 15 Assume we have M i, |B| , p i, |B| for i < | T | and let M i +1 , , p i +1 , = M i, |B| , p i, |B| . Finallyassume we have M i, |B| , p i, |B| for all i < j ≤ | T | a limit ordinal. Then define M j, = (cid:83) i Suppose B is complete. Then every B -type p ∈ S x B ( M ) can be extended toa B -type q such that (cid:80) a q ( x = a ) is maximal.Proof. Im ( p | x = y ) \ { } is maximal iff (cid:80) a ∈ M p ( x = a ) is maximal: indeed assume q over N extends p . Then (cid:80) a ∈ M p ( x = a ) < (cid:80) a ∈ N q ( x = a ) implies in particular Im ( p | x = y ) \ { } (cid:54) = Im ( q | x = y ) \ { } , while if Im ( p | x = y ) \ { } (cid:54) = Im ( q | x = y ) \ { } then for some a such that q ( x = a ) / ∈ Im ( p | x = y ) ∪ { } we get in particular (cid:32) (cid:88) a ∈ M p ( x = a ) (cid:33) · q ( x = a ) = (cid:88) a ∈ M p ( x = a ) q ( x = a ) = (cid:88) a ∈ M q ( x = a ) q ( x = a ) = (cid:88) a ∈ M q ( x = a ∧ x = a ) = (cid:88) a ∈ M < q ( x = a ) ⇒ q ( x = a ) (cid:2) (cid:88) a ∈ M p ( x = a ) ⇒ (cid:88) a ∈ M p ( x = a ) (cid:54) = (cid:88) a ∈ N q ( x = a )We conclude from Proposition 2.31 that every B type p over M has an extension q over N such that (cid:80) a ∈ N q ( x = a ) is maximal. (cid:3) Corollary 2.34. Assume T has N IP . Suppose B is complete, has κ -c.c. and is (2 <κ + | T | , distributive. Then every B -type p can be extended to a realized type. Corollary 2.30 gives us a satisfactory characterization of smooth B -types in the case ofBoolean algebras which are close to atomic ones, in the sense they have many large atomicsubalegbras. OOLEAN TYPES IN DEPENDENT THEORIES 16 A similar approach can still be useful for any algebra. We no longer assume that x is afinite tuple. We start with a useful claim: Claim . Assume A , B are Boolean algebras where B is complete, A (cid:48) ⊆ A a subalgebra,and p : A (cid:48) → B and p , p : A → B homomorphisms such that p ⊆ p , p and p (cid:54) = p . Thenthere exist distinct extensions q , q : A → B of p and a ∈ A such that q ( a ) < q ( a ) and σ ◦ q (cid:54) = q for any automorphism σ of B . Proof. Let a ∈ A \ A (cid:48) be such that p ( a ) (cid:54) = p ( a ).Let b = (cid:80) a (cid:48) ≤ a , a (cid:48) ∈A (cid:48) p ( a (cid:48) ) and b = (cid:81) a (cid:48) ≥ a , a (cid:48) ∈A (cid:48) p ( a (cid:48) ). Then b ≤ p ( a ) (cid:54) = p ( a ) ≤ b thus b < b .By Fact 2.2 there are extensions q i of p such that q i ( a ) = b i for 0 ≤ i < 2. Assume thereis a homomorphism σ : B → B such that σ ◦ q = q .Then for any p ( a (cid:48) ) ∈ Im ( p ), σ ( p ( a (cid:48) )) = σ ( q ( a (cid:48) )) = q ( a (cid:48) ) = p ( a (cid:48) ) thus σ | Im ( p ) = id Im ( p ) .Further σ ( b ) = σ ( q ( a )) = q ( a ) = b . We conclude that b = σ ( b ) ≤ σ ( b ). On theother hand, for any a (cid:48) ≥ a , a (cid:48) ∈ A (cid:48) , b = q ( a ) ≤ q ( a (cid:48) ) = p ( a (cid:48) ) thus σ ( b ) ≤ σ ( p ( a (cid:48) )) = p ( a (cid:48) ) thus σ ( b ) ≤ (cid:81) a (cid:48) ≥ a,a (cid:48) ∈ A (cid:48) p ( a (cid:48) ) = b .We get σ ( b ) = b = σ ( b ), thus σ is not injective and in particular, not an automorphism. (cid:3) Proposition 2.36. Assume B is complete, and assume that N is a model of T and q is a B -type over N that has the property in the conclusion of Proposition 2.31: it has maximalimage with respect to every formula. Then for any formula ϕ ( x, y ) , if there exists extensions q , q of q to the same model such that q | ϕ (cid:54) = q | ϕ , then ϕ is independent.In particular, if T has NIP, such a q is smooth.Proof. Assume otherwise, and let n the such that n is greater than the dual VC-dimensionof ϕ ( x, y ) (see e.g., [Sim14, Lemma 6.3]).Let N (cid:48) be an elementary extension of N , q i ∈ S x B ( N (cid:48) ) extending q and a ∈ N (cid:48) such that q i ( ϕ ( x, a )) = a i . By the previous claim we may assume a < a and let b = a − a > b (cid:48) ≤ b there exists some q (cid:48) ∈ S x B ( N (cid:48) ) extending q such that q (cid:48) ( ϕ ( x, a )) = a + b (cid:48) , thus by assumption (on q ) there exists a (cid:48) ∈ N such that q ( ϕ ( x, a (cid:48) )) = a + b (cid:48) . OOLEAN TYPES IN DEPENDENT THEORIES 17 Assume first b ∗ ≤ b for some atom b ∗ of B . Let x (cid:48) ⊆ x be a finite tuple containing allvariables from x appearing in ϕ ( x, y ). Since Im ( q | x (cid:48) = z ) is maximal, we get by Claim 2.26that b ∗ (cid:2) − (cid:80) c ∈ N | x (cid:48) | q ( x (cid:48) = c ); but if b ∗ (cid:2) q ( x (cid:48) = c ) for all c ∈ N | x (cid:48) | then since b ∗ is an atomwe get b ∗ ≤ − q ( x (cid:48) = c ) for all c ∈ N | x (cid:48) | that is b ∗ ≤ (cid:89) c ∈ N | x (cid:48) | − q (cid:0) x (cid:48) = c (cid:1) = − (cid:88) c ∈ N | x (cid:48) | q (cid:0) x (cid:48) = c (cid:1) . Let then c ∈ N | x (cid:48) | be such that q ( x (cid:48) = c ) ≥ b ∗ , in particular q ( x (cid:48) = c ) · ( a − a ) ≥ b ∗ > ϕ ( c, a ) holds, then x (cid:48) = c ∧ ¬ ϕ ( x, a ) is false in N thus necessarily0 = q (cid:0) x (cid:48) = c ∧ ¬ ϕ (cid:0) x (cid:48) , a (cid:1)(cid:1) = q (cid:0) x (cid:48) = c (cid:1) · − a but q (cid:0) x (cid:48) = c (cid:1) · − a ≥ q (cid:0) x (cid:48) = c (cid:1) · ( a − a ) > ¬ ϕ ( c, a ) holds then x (cid:48) = c ∧ ϕ ( x (cid:48) , a ) is ⊥ thus 0 = q ( x (cid:48) = c ∧ ϕ ( x, a )) = q ( x (cid:48) = c ) · a which is again impossible.Thus there is no atom of B under b , therefore by trivial induction there exist disjoint b η > η ∈ {± } n such that (cid:80) η ∈{± } n b η = b . Let b i = (cid:80) η ∈{± } n ,η ( i )=1 b η ≤ b . Then b − b i = (cid:80) η ∈{± } n ,η ( i )= − b η thus for any such η , b (cid:81) i Connecting Keisler Measures and Boolean Types. Remark . Suppose the algebra B comes with some extra structure, for instance, supposeit is a measure algebra ( B , λ ) (see Definition 3.3). Then we can extend the definitions ofconjugate types (Definition 2.3) or conjugate subalgebras to this context, which means thatpartial isomorphisms are now required to keep the extra structure, and in the case of measure,to preserve it. Then the proofs leading to Corollary 2.16 still work, but now we have to takeinto account the number of isomorphism types of the new structure.In the case of a measure, the number of possible isomorphism types is ≤ µ + (cid:0) ℵ (cid:1) µ = 2 µ (the isomorphism type of B p consists of the quantifier free type of B p as in the proof ofCorollary 2.17 and the list of values λ (cid:0) a B p ,i (cid:1) ). Finally, note that if λ is a probability measure,then B is c.c.c (i.e., ω -c.c.), so we can take κ = ω and µ = 2 ℵ + | T | + | x | . Definition 3.2. A Keisler measure in x over a set A is a finitely additive probability measureon L x ( A ).Two Keisler measures λ, λ (cid:48) in x over A are conjugates if exists an elementary π : A → A such that λ (cid:0) ϕ (cid:0) x, π − ( a ) (cid:1)(cid:1) = λ (cid:48) ( ϕ ( x, a )) for any formula ϕ ( x, a ). Definition 3.3. A measure algebra is a σ -complete Boolean algebra B (not necessarily analgebra of sets) equipped with a σ -additive measure that is positive on every element otherthan 0 B .A probability algebra is a measure algebra that assigns measure 1 to 1 B . Example 3.4. For every probability space, the algebra of measurable subsets up to measure0 is a probability algebra. OOLEAN TYPES IN DEPENDENT THEORIES 19 Definition 3.5. Let κ an infinite cardinal. Then ( U κ , ν κ ) is the probability algebra of Borelsubsets of 2 κ up to measure 0, with ν κ the usual product measure (see [Fre03, 254J]). Remark . Since every probability algebra has c.c.c., every supremum or infimum is effec-tively countable. Thus in particular the σ -complete subalgebra generated by some subset isthe same as the complete subalgebra generated by the same set.We will show that we can attach to a Keisler measure a B -type for a measure algebra B . Remark . Given a Keisler measure λ over A , we can consider it as a measure on clopensets of S x ( A ) and then extend it uniquely to a regular σ -additive measure on the Borel setsof S x ( A ) (see [Sim14, Section 7.1]).Let ( B , λ ) be the probability algebra of Borel subsets of S x ( A ) up to λ measure 0 and let ψ be the projection from the algebra of Borel subsets onto B . Since the clopen sets are abasis for the topology, the complete subalgebra of B generated by the clopen sets is B itself,and there are at most | T | + | A | + | x | clopen sets. Fact 3.8. Let κ an infinite cardinal. If B is a probability algebra, and there is B ⊆ B suchthat | B | ≤ κ and the smallest ( σ -) complete subalgebra of B containing B is B itself then thereis a measure preserving homomorphism f from B to ( U κ , ν κ ) (see [Fre04, Lemma 332N] , andsee also the proposition on page 126 and 331F there).Further, every measure algebra homomorphism is an embedding. Proposition 3.9. Assume A ⊆ U κ a complete subalgebra that can be completely generated bya set S such that | S | < κ . Assume further that f : A → U κ is measure preserving (thus anembedding).Then exists a measure preserving automorphism σ of U κ extending f .Proof. By following the proof for [Fre04, Theorem 331I] ( U κ satisfies the requirements for thetheorem by [Fre04, Theorem 331K]), we find that a recursive construction of an automorphismcan start from any partial isomorphism, as long as the domain of said partial isomorphism isa complete subalgebra (in Fremlin’s terminology, closed subalgebra) that can be completelygenerated by less than κ elements. (cid:3) Proposition 3.10. Let κ ≥ | A | + | T | + | x | be some cardinal. There is an injection from theset of Kiesler measures over a set A to the set of ( U κ , ν κ ) -types over A . Further, this injection OOLEAN TYPES IN DEPENDENT THEORIES 20 respects conjugation; that is, if the images of λ, λ (cid:48) are conjugate with π : A → A , then so are λ, λ (cid:48) .Proof. Let f and ψ be as in Fact 3.8, using Remark 3.7.Let p λ : L x ( A ) → U κ be f ◦ ψ | L x ( A ) (where L x ( A ) is thought of as the algebra of clopensubsets of S x ( A )). λ (cid:44) → p λ would be our injection.By choice of f , λ ( ϕ ) = ν κ ( p λ ( ϕ )), which means that this map is indeed injective. Itfollows that if p λ and p λ are conjugate (as in Remark 3.1, i.e., as ( U κ , ν κ )-types) then λ and λ are conjugate:Suppose that π : A → A is an elementary map and that σ : B p λ → B p λ is a measurepreserving isomorphism such that σ ◦ ( π ∗ p λ ) = p λ . Then π ∗ λ ( ϕ ( x, a )) = λ (cid:0) ϕ (cid:0) x, π − ( a ) (cid:1)(cid:1) = ν κ (cid:0) p λ (cid:0) ϕ (cid:0) x, π − ( a ) (cid:1)(cid:1)(cid:1) = ν κ (cid:0) σ (cid:0) p λ (cid:0) ϕ (cid:0) x, π − ( a ) (cid:1)(cid:1)(cid:1)(cid:1) = ν κ ( p λ ( ϕ ( x, a )))= λ ( ϕ ( x, a )) . (cid:3) Counting Keisler Measures. By applying Proposition 3.10, Remark 3.1 and Corollary2.17 we get that: Corollary 3.11. Assume T has N IP . The number of Keisler measures in x up to conjugationover a set A is bounded by the number of types in S | x |· µ ( A ) up to conjugation + µ where µ = 2 ℵ + | T | + | x | .Hence, if again λ, κ are cardinals and α an ordinal such that λ = λ <λ = ℵ α = κ + α ≥ κ ≥ (cid:105) ω + µ + , and A = M is a saturated model of size λ , this number is bounded by < κ + | α | µ . However, Keisler measures have been studied extensively in the context of NIP (see forexample [Sim14, Chapter 7]), and there are results which give a better bound. Indeed: Proposition 3.12. Assume T has N IP . Then there is an injection from the set of Keislermeasures in x over a set A to the set of µ -types over A where µ = | T | + | x | .Further, if the images of λ, λ (cid:48) are elementarily conjugate, λ and λ (cid:48) are conjugates.Proof. Recall that given complete types p , ..., p n − , and a formula φ ( x, b ), Av ( p , .., p n − ; φ ( x, b ))is |{ i Fix a Keisler measure λ . By Proposition 7.11 from [Sim14], (for X = (cid:62) ) for any m < ω and partitioned formula φ ( x, y ) there exist n m,φ < ω and types (cid:68) p m,φi ( x ) (cid:69) i Corollary 3.13. Assume T has N IP . The number of Keisler measures up to conjugationover a set A is bounded by the number of types in S | x |· µ ( A ) up to conjugation where µ = | T | + | x | (with the same explicit bound as in Corollary 3.11, but replacing µ = 2 ℵ + | T | + | x | with µ = | T | + | x | ).Remark . This bound improves upon the one in Corollary 3.11, since µ is now potentiallysmaller (for small | T | and | x | ). OOLEAN TYPES IN DEPENDENT THEORIES 22 Smooth Keisler Measures. Proposition 3.10 has a significant advantage over Propo-sition 3.12, in the fact that certain additional properties of the measure are preserved. Since U κ has many desirable properties, this can prove quite useful. For example: Lemma 3.15. Assume λ i is a Keisler measure over A i for i ∈ { , } such that A ⊆ A and λ extends λ ; let κ i = | A i | + | T | + | x | and let f i : B i → U κ + i be an embedding of measurealgebras, where B i is the algebra of Borel subsets of S x ( A i ) up to λ i -measure (see Remark3.7).Fix an embedding ι of U κ +1 in U κ +2 (one exists by Fact 3.8).Then there is a measure algebra homomorphism g : B → U κ +2 that extends ι ◦ f and suchthat ν κ +2 ◦ f = ν κ +2 ◦ g .Proof. Note first that since λ extends λ , B can be embedded into B naturally with thepreimage of the projection map from S x ( A ) to S x ( A ).Note that ( ι ◦ f ) ( B ) is generated as a ( σ -)complete algebra by κ elements (as the imageof such an algebra) and that ( ι ◦ f ) ( B ) is a complete subalgebra of U κ +2 , generated as acomplete subalgebra by the images of the clopen sets (see [Fre04, Proposition 324L]).By Proposition 3.9 and as λ extends λ , there is an automorphism σ of (cid:16) U κ +2 , ν κ +2 (cid:17) ex-tending f ◦ ( ι ◦ f ) − .But now we get σ − ◦ f | B = ι ◦ f | B , thus g = σ − ◦ f is as required. (cid:3) Corollary 3.16. If p ∈ S x U κ + ( M ) is smooth, then so is ν κ + ◦ p for κ ≥ | M | + | T | + | x | .Proof. Assume λ = ν κ + ◦ p is not smooth. So there is N ⊇ M and distinct λ , λ over N extending λ , and hence there is one of cardinality at most | M | by L¨owenheim Skolem(restrict λ , λ to a smaller model containing M and some b , for which exists ϕ ( x, b ) suchthat λ ( ϕ ( x, b )) (cid:54) = λ ( ϕ ( x, b ))).Let B be the measure algebra of Borel sets in S x ( M ) up to λ -measure 0 and I the idealof sets of λ -measure 0 in the algebra of Borel subsets of S x ( M ) (see Remark 3.7). Then since ν κ + ( a ) = 0 ⇐⇒ a = 0, I ∩ L x ( M ) = ker ( p ) thus L x ( M ) / ker ( p ) is naturally embedded in B .Write p = (cid:101) p ◦ π for π : L x ( M ) → L x ( M ) / ker ( p ) the projection.Then (cid:101) p : L x ( M ) / ker ( p ) → U κ + is a measure preserving function from a subalgebra of B , and the complete (in Fremlin’s terminology, order-closed) subalgebra of B generated by OOLEAN TYPES IN DEPENDENT THEORIES 23 L x ( M ) / ker ( p ) is B itself. Thus by [Fre04, Proposition 324O and Proposition 323J] (cid:101) p has aunique extension to a measure preserving p : B → U κ + , that is λ = ν κ + ◦ p (here we treat λ as a measure on S x ( M )).For each i ∈ { , } , we do the following. Let B i be the measure algebra of Borel subsetsof S x ( N ) up to λ i -measure 0, which naturally embeds B as a subalegbra via the preimagemap of the projections S x ( N ) → S x ( M ) (this uses the fact that λ i extends λ ). Take anembedding f i : B i → U κ + for λ i guaranteed by Fact 3.8, and using Lemma 3.15 we can find ameasure preserving g i : B i → U κ + extending p such that λ i = ν κ + ◦ g i .Let π i : L x ( N ) → B i be the projection, note that it extends π . Then for p i = g i ◦ π i ∈ S x U κ + ( N ) we find λ i = ν κ + ◦ p i (when we consider λ i as Keisler measures), and since π i extends π and g i extend p (thus (cid:101) p ) we find p , p extend p , but they are distinct since λ and λ aredistinct.We conclude that p is not smooth. (cid:3) On the other hand: Proposition 3.17. Assume λ is a smooth Keisler measure in x over M . Let p ∈ S x U κ + ( M ) such that λ = ν κ + ◦ p for κ ≥ | M | + | T | + | x | . Then p is smooth.Proof. Let p , p ∈ S x U κ + ( N ) be distinct types extending p (for N ⊇ M ). Again, without lossof generality | N | ≤ κ . By Claim 2.35 we can choose p and p such that for no automorphism σ of U κ + , σ ◦ p = p .Let λ (cid:48) the unique extension of λ to N ; in particular λ (cid:48) = ν κ + ◦ p i for i = 1 , 2. Sinceker ( p i ) = { ϕ ( x, b ) | λ (cid:48) ( ϕ ( x, b )) = 0 } , B = L x ( M ) / ker ( p i ) is independent of i . Let B (cid:48) theprobability algebra of Borel sets in S x ( M ) up to measure 0.We conclude that both p i ’s can be written as f i ◦ π where π is the projection from L x ( N )to the algebra B , and f i : B → Im ( f i ) are measure preserving embeddings (note Im ( f i ) is acomplete subalgebra, see [Fre04, Proposition 324L]), and we can extend them uniquely to B (cid:48) by [Fre04, Proposition 324O and Proposition 323J], like in the proof of Corollary 3.16.We get f ◦ f − is a partial measure preserving isomorphism from Im ( f ) into U κ + (whichcan, by Proposition 3.9, be extended to an automorphism) and f ◦ f − ◦ p = f ◦ π = p ,contradiction. (cid:3) OOLEAN TYPES IN DEPENDENT THEORIES 24 By [Fre04, Theorem 322Ca-c], every probability algebra is complete as a Boolean algebra(Dedekind complete, in his terminology).Thus by choosing a sufficiently large κ (at least (cid:0) ℵ (cid:1) + ) in Proposition 3.10 and usingCorollary 2.37 we get, recovering [Sim14, Proposition 7.9]: Corollary 3.18. Every Keisler measure over an NIP theory can be extended to a smoothmeasure.Proof. Take a Keisler measure λ over a model M in x . Let κ = (cid:0) ℵ + | T | + | x | + | M | (cid:1) + .Then by Proposition 3.10 there exists a type p ∈ S x U κ ( M ) such that λ = ν κ ◦ p .By Corollary 2.37 there exists a model N ⊇ M of cardinality at most 2 ℵ + | T | + | x | + | M | and a smooth extension q ∈ S x U κ ( N ) of p .Thus letting λ (cid:48) = ν κ ◦ q , we find λ (cid:48) extends λ thus by Corollary 3.16 we are done. (cid:3) Remark . By [Sim14, Lemma 7.8] and the remark following it, if N ⊇ M is an extensionand λ over N is a smooth extension of µ over M , then there is some M ⊆ N (cid:48) ⊆ N such that λ | N (cid:48) is still smooth and | N (cid:48) | ≤ | M | + | T | .Indeed the lemma and the remark state that λ is smooth iff for every ε > φ ( x, y )exist ψ i ( y ) , θ i ( x ) , θ i ( x ) ( i = 1 , ..., n ) over N such that:1. (cid:18) n (cid:87) i =1 ψ i ( y ) (cid:19) = (cid:62) .2. if b is a y -tuple in C and ψ i ( b ) holds then θ i ( x ) → φ ( x, b ) → θ i ( x ).3. λ (cid:0) θ i ( x ) (cid:1) − λ (cid:0) θ i ( x ) (cid:1) < ε .Thus we can take an elementary substructure N (cid:48) of N containing M and all parameters inthe formulas θ ji , ψ i mentioned in the lemma for each ε = n and φ ( x ; y ); then all the formulasare over N (cid:48) and retain the required properties, ensuring that λ | N (cid:48) is still smooth.The proof of [Sim14, Lemma 7.8] relies on the following fact: Fact 3.20. [Sim14, Lemma 7.4] Let φ ( x, b ) a formula ( b ∈ C ), let λ a Keisler measure in x over M . Let: r = sup { λ ( ϕ ( x, a )) | ϕ ( x, a ) ∈ L x ( M ) , C (cid:15) ϕ ( x, a ) → φ ( x, b ) } r = inf { λ ( ϕ ( x, a )) | ϕ ( x, a ) ∈ L x ( M ) , C (cid:15) φ ( x, b ) → ϕ ( x, a ) } OOLEAN TYPES IN DEPENDENT THEORIES 25 Then there exists an extension λ (cid:48) of λ to C such that λ ( φ ( x, b )) = r iff r ≤ r ≤ r .Remark . For any measure algebra B and F ⊆ B there exists a countable A ⊆ F such A and F have the same set of lower bounds, hence (cid:81) F and (cid:81) A exist and are equal (since B hasc.c.c. and is complete; see for example [Fre04, Proposition 316E]); and if F is closed underfinite products we can choose A to be a chain. Thus we conclude ν κ ( (cid:81) F ) = inf { ν κ ( a ) } a ∈ F .A similar argument shows ν κ ( (cid:80) F ) = sup { ν κ ( a ) } a ∈ F if F is closed under finite sums.Let p ∈ S x U κ ( M ), φ ( x, y ) a partitioned formula and b ∈ C a y -tuple. Then { p ( ϕ ( x, a )) | ϕ ( x, a ) ∈ L x ( M ) , φ ( x, b ) → ϕ ( x, a ) } and { p ( ϕ ( x, a )) − p ( ϕ ( x, a )) | ϕ i ( x, a i ) ∈ L x ( M ) , ϕ ( x, a ) → φ ( x, b ) → ϕ ( x, a ) } are closed under finite products, while { p ( ϕ ( x, a )) | ϕ ( x, a ) ∈ L x ( M ) , ϕ ( x, a ) → φ ( x, b ) } is closed under finite sums. Remark . We can also get [Sim14, Lemma 7.4] by combining Proposition 3.10, Remark3.21 and Fact 2.2.Indeed we need only find for any r ≤ r ≤ r some b ≤ (cid:89) ψ ( x,b ): ϕ ( x,a ) → ψ ( x,b ) p ( ψ ( x, b )) − (cid:88) ψ ( x,b ): ψ ( x,b ) → ϕ ( x,a ) p ( ψ ( x, b )) (for p some type corresponding to λ ) such that ν κ ( b ) = r − r ; and it is easy to see that when κ ≥ ℵ , for every a ∈ U + κ and every 0 ≤ s ≤ ν κ ( a ) exists b ≤ a such ν κ ( b ) = s (indeed thesame holds for any non-atomic measure alegbra, but here we can see it directly).4. Analysis of the stable case In this section, we analyze the stable any totally transcendental (t.t.) cases. We show thatin the stable case, local Boolean types are essentially averages of classical ϕ -types, and thatin the t.t. case the same is true for complete Boolean types. We start with the t.t. case, sincethe general stable case is similar (and easier, since local ranks are bounded by ω ). OOLEAN TYPES IN DEPENDENT THEORIES 26 The t.t. case.Definition 4.1. For q ∈ S x B ( A ), let supp ( q ) = { p ( x ) ∈ S x ( A ) | p (cid:96) θ ⇒ q ( θ ) > } .Note that when q ∈ S x B ( A ), supp ( q ) is a closed subset of S x ( A ). Also note that if Γ is acollection of formulas, closed under finite conjunctions, such that q ( θ ) > θ ∈ Γ thenthere is a type p ∈ supp ( q ) such that p (cid:96) Γ (because Γ ∪ {¬ ψ | q ( ψ ) = 0 } is consistent).We will use some basic facts from stability theory, namely: Fact 4.2. [Pil96, Section 3] For a topological space X , let X (cid:48) = X \ { x ∈ X | x is isolated } .Assume that T is t.t. and let X = S x ( A ) for some A ⊆ M | = T . For α ∈ ord , let X ( α ) bethe Cantor-Bendixon analysis of X ( X ( α +1) = (cid:0) X ( α ) (cid:1) (cid:48) and X ( α ) = (cid:84) β<α X ( β ) when α is alimit). Then for some (limit) α < | T | + , X ( α ) = ∅ . (Note that this definition is a bit differentthan the one in [Pil96, Section 3] , where one considers global types.)Remark . Note that for any topological space X , if Y ⊆ X then Y (cid:48) ⊆ X (cid:48) . Lemma 4.4. Suppose that T is t.t.. Suppose that B is any complete Boolean algebra. Let A ⊆ M | = T and let q ∈ S x B ( A ) . Let X = supp ( q ) . Let U be the set of all isolated types r ∈ X . For each r ∈ U , let θ r ( x ) be an isolating formula for r . Then: (1) { q ( θ r ) | r ∈ U } is an antichain. (2) For any ψ ( x ) ∈ L x ( A ) , q ( ψ ) ≥ (cid:80) r ∈ U q ( θ r ) · r ( ψ ) (where we treat r as a -type).Proof. (1) Note that 0 < q ( θ r ) for all r ∈ U since r ∈ supp ( q ). Suppose that r (cid:54) = r ∈ U and 0 < b ≤ q ( θ r ) , q ( θ r ). Hence 0 < b ≤ q ( θ r ∧ θ r ) and thus for some r ∈ supp ( q ), r (cid:96) θ r ∧ θ r hence r = r = r by assumption.(2) It is enough to show that if ψ ( x ) ∈ r for some r ∈ U then q ( ψ ) ≥ q ( θ r ). If ψ ( x ) ∈ r ,it follows that q ( θ r ∧ ¬ ψ ) = 0 so q ( θ r ) = q ( ψ ∧ θ r ) ≤ q ( ψ ). (cid:3) Theorem 4.5. Suppose that T is totally transcendental, A is some set, B is a completeBoolean algebra and that q ∈ S x B ( A ) is a B -type. Then there is a maximal antichain { b r | r ∈ U } where U ⊆ supp ( q ) such that for all ψ ( x ) ∈ L x ( A ) , q ( ψ ) = (cid:80) r ∈ U b r · r ( ψ ) .Proof. For any 0 < b ∈ B , let B| b be the relative algebra: its universe is { a ∈ B | a ≤ b } , with+ , · and 0 inherited from B , and with 1 B| b = b and ( − a ) B| b being b − a . Note that B| b is OOLEAN TYPES IN DEPENDENT THEORIES 27 complete. Letting X = S x ( A ), we try to construct a sequence (cid:104) b α , q α , U α , ¯ c α | α < α ∗ (cid:105) forsome α ∗ ≤ | T | + such that: • < b α ≤ b β for β < α ; b = 1 and more generally b α = − (cid:80) β<α,r ∈ U β c β,r ; q α ∈ S x B α where B α = B| b α ; q α ( ψ ) = q ( ψ ) · b α for any ψ ∈ L x ( A ); U α ⊆ supp ( q α ) ⊆ X ( α ) ;¯ c α = (cid:104) c α,r | r ∈ U α (cid:105) is an antichain contained in B α ; q α ( ψ ) ≥ (cid:80) r ∈ U α c α,r · r ( ψ ) for any ψ ∈ L x ( A ).Given (cid:104) b β , q β , U β , ¯ c β | β < α (cid:105) , if (cid:80) β<α,r ∈ U β c β,r = 1 we stop and let α ∗ = α . Otherwise, let b α , q α as above and let U α ⊆ supp ( q α ) the set of isolated types in supp ( q α ) (so for α = 0, q = q and b = 1). For r ∈ U α , let c α,r = q α ( θ r ) where θ r isolates r . By Lemma 4.4 (1), { c α,r | r ∈ U α } is an antichain in B α . Note that q α ( ψ ) ≥ (cid:80) r ∈ U α c α,r · r ( ψ ) by Lemma 4.4 (2).Now prove by induction on α that supp ( q α ) ⊆ X ( α ) (this follows from Remark 4.3 and thefact that supp ( q α +1 ) ⊆ supp ( q α ) (cid:48) ) and that { c β,r | r ∈ U β , β < α } is an antichain in B .Finally, since for some β < | T | + , X ( β ) = ∅ , it follows that α ∗ ≤ β (otherwise supp ( q β ) = ∅ and so b β = q β ( x = x ) = 0, contradiction). Hence for all ψ ∈ L x ( A ), q ( ψ ) ≥ (cid:80) α<α ∗ q α ( ψ ) ≥ (cid:80) r ∈ U α ,α<α ∗ c α,r · r ( ψ ) and { c α,r | α < α ∗ , r ∈ U α } is a maximal antichain in B . Since this isalso true for ¬ ψ , we have equality and we are done. (cid:3) The stable case. Fix a partitioned formula ϕ ( x, y ) in some theory T , and let A ⊆ C .As in [Pil96, Section 2], by a ϕ -formula over A , we will mean a formula ψ ( x ) ∈ L x ( A ) whichis equivalent to a Boolean combination of instances of ϕ over A (over a model M , a ϕ -formulais just a Boolean combination of instances of ϕ over M ). Let L ϕ,x ( A ) be the Boolean algebraof ϕ -formulas over A up to equivalence in C . Let S xϕ ( A ) be the set of all complete ϕ -typesover A in x , i.e., maximal consistent sets of ϕ -formulas over A . Definition 4.6. (local Boolean type) Suppose that B is a Boolean algebra and ϕ ( x, y ) is a par-titioned formula. A B , ϕ -type over a set A is a homomorphism from L ϕ,x ( A ) to B . Denote theset of B , ϕ -types by S x B ,ϕ ( A ). For q ∈ S x B ,ϕ let supp ϕ ( q ) = (cid:8) p ( x ) ∈ S xϕ ( A ) | p (cid:96) θ ⇒ q ( θ ) > (cid:9) .Similarly to the previous section, we have: Fact 4.7. [Pil96, Section 3] Assume that ϕ ( x, y ) is stable in some complete theory T , A ⊆ M | = T and let X = S xϕ ( A ) . Then for some n < ω , X ( n +1) = ∅ . OOLEAN TYPES IN DEPENDENT THEORIES 28 Theorem 4.8. Suppose that ϕ ( x, y ) is stable, A is some set, B is a complete Boolean algebraand that q ∈ S x B ,ϕ ( A ) is a B , ϕ -type. Then there is a maximal antichain { b r | r ∈ U } where U ⊆ supp ϕ ( q ) such that for all ψ ( x ) ∈ L ϕ,x ( A ) , q ( ψ ) = (cid:80) r ∈ U b r · r ( ψ ) .Proof. The proof is exactly as the proof of 4.5, working with X = S xϕ ( A ) and with localBoolean types, Replacing Fact 4.2 with 4.7. We leave the details to the reader. (cid:3) Remark. When T is stable and q ∈ S x B ( A ), this essentially means that q | ϕ factors through2 | U | (cid:44) → (cid:81) r ∈ U B| b r ∼ = B , see [Kop89, Proposition 6.4]. In particular we get again, more directly,that for B which is κ -c.c., | Im ( q | ϕ ) | ≤ <κ (see Proposition 2.12). When T is t.t., we getsimilarly that | Im ( q ) | ≤ <κ .4.3. Non-forking. Using (the proof of) Theorem 4.8, one can recover the theory of forkingin stable theories. Definition 4.9. Let B be any Boolean algebra and let T be any theory. Let A ⊆ B be anysets . Say that a B -type or a B , ϕ -type q forks over A if for some formula θ ( x ) over B whichforks over A , q ( θ ) > 0. (For the definition of forking see e.g., [TZ12, Definition 7.1.7]). Fact 4.10. (E.g., [Pil96, Section 2] ) If M ≺ N , ϕ ( x, y ) is stable, then any ϕ -type p ∈ S xϕ ( M ) has a unique non-forking extension p | N to S ϕ ( N ) . The same is true assuming elimination ofimaginaries when M is replaced by an algebraically closed set A .Remark . Suppose that p ∈ S x B ( B ) does not fork over A ⊆ B . Then there is a globalnon-forking (over A ) extension q ∈ S x B ( C ) (and the same is true for local Boolean types).This follows by the fact that forking formulas over A form an ideal and using Fact 2.1. When B is a model M then p does not fork over M and if T is stable (or even simple) then this istrue in general.Suppose that ϕ ( x, y ) is stable and p ∈ S B ,ϕ ( M ) for some model M | = T . Then we canfind an explicit extension: by Theorem 4.8, we can write p as the sum (cid:80) r ∈ U b r · r for somemaximal antichain U ⊆ supp ϕ ( p ) and we let q = (cid:80) r ∈ U b r · r | C (where r | C is the unique globalnon-forking extension of r ). A similar statement holds in the t.t. case.Next we would like to prove that there is a unique non-forking extension. OOLEAN TYPES IN DEPENDENT THEORIES 29 Lemma 4.12. Suppose that ϕ ( x, y ) is stable and B is any complete Boolean algebra. Let M ≺ N | = T and let q ∈ S B ,ϕ ( N ) be non-forking over M . Let X = supp ϕ ( q | M ) ( q | M is therestriction of q to L ϕ,x ( M ) ). Let U be the set of all isolated ϕ -types r ∈ X . For each r ∈ U ,let θ r ( x ) be an isolating formula for r (so it is a ϕ -formula). Then • For any ψ ( x ) ∈ L ϕ,x ( N ) , q ( ψ ) ≥ (cid:80) r ∈ U q ( θ r ) · r | N ( ψ ) (where r | N is the uniquenon-forking extension of r to N ).Proof. It is enough to show that q ( ψ ) ≥ q ( θ r ) when ψ ∈ r | N . Suppose that ψ ( x ) ∈ r | N but q ( θ r ∧ ¬ ψ ) > 0. Then there is some r (cid:48) ∈ supp ϕ ( q ) such that r (cid:48) contains θ r ∧ ¬ ψ . But then r (cid:48) does not fork over M (because q does not fork over M ) and r (cid:48) | M contains θ r and is in X and thus r (cid:48) | M = r | M and by Fact 4.10, r (cid:48) = r . (cid:3) Theorem 4.13. Suppose that ϕ ( x, y ) is stable. Suppose that n < ω . Then, whenever B is a complete Boolean algebra, M ≺ N and q ∈ S B ,ϕ ( N ) does not fork over M , there is amaximal antichain { b r | r ∈ U } where U ⊆ supp ϕ ( q | M ) such that for all ψ ( x ) ∈ L ϕ,x ( N ) , q ( ψ ) = (cid:80) r ∈ U b r · r | N ( ψ ) . In particular, q is the unique non-forking extension of q | M .Proof. The proof follows the same lines as in the proof of Theorem 4.8 (and Theorem 4.5),using Lemma 4.12 instead of Lemma 4.4. (cid:3) Remark . As in the classical case, we can extend these results (existence and uniquenessof non-forking extensions) for an arbitrary algebraically closed set A , assuming eliminationof imaginaries.4.4. Connection to Keisler measures. Using the general results on Boolean types we canrecover some results on Keisler measures. The following result appeared in [Pil, Fact 1.1],[AC, Fact 2.2] for models. Corollary 4.15. Suppose that ϕ ( x, y ) is stable and that µ is a Keisler measure on L ϕ,x ( A ) for some set A . Then there is a countable family (cid:104) p i | i < ω (cid:105) of complete ϕ -types over A and positive real numbers (cid:104) α i | i < ω (cid:105) such that (cid:80) i<ω α i = 1 and for any ψ ( x ) ∈ L ϕ,x ( A ) , µ ( ψ ) = (cid:80) α i p i ( ψ ) .Similarly, if T is t.t. and µ is a Keisler measure on L x ( A ) then there is a countablefamily (cid:104) p i | i < ω (cid:105) of complete types over A and positive real numbers (cid:104) α i | i < ω (cid:105) such that (cid:80) i<ω α i = 1 and for any ψ ( x ) ∈ L x ( A ) , µ ( ψ ) = (cid:80) α i p i ( ψ ) . OOLEAN TYPES IN DEPENDENT THEORIES 30 Proof. Given µ , let B be the Boolean algebra of Borel subsets of S xϕ ( A ) up to µ -measure0 (recall Remark 3.7) and let q ∈ S x B ,ϕ ( A ) be the natural homomorphism from ϕ -formulasover A (up to equivalence over C ) to B . Now apply Theorem 4.8 to q and B to obtaina maximal antichain { b r | r ∈ U } where U ⊆ supp ϕ ( q ) such that for all ψ ( x ) ∈ L x ( A ), q ( ψ ) = (cid:80) r ∈ U b r · r ( ψ ). Note that U must be countable as B is c.c.c. Letting α r = µ ( b r ) weare done. The second statement follows similarly from Theorem 4.5. (cid:3) Definition 4.16. A Keisler measure µ on L x ( N ) does not fork over M ≺ N if whenever µ ( θ ) > θ does not for over M . Corollary 4.17. Suppose that ϕ ( x, y ) is stable and that µ is a Keisler measure on L ϕ ( M ) for some model M . Then µ has a unique global non-forking extension to C . More gener-ally, this holds when replacing M by any algebraically closed set A , assuming elimination ofimaginaries.Proof. We use a local version of Proposition 3.10: given µ , we can find p ∈ S xϕ, U κ ( M ) such that µ = ν κ ◦ p for κ = | T | + | M | . By Remark 4.11 there is a non-forking extension q ∈ S xϕ, U κ ( N )and then we can define µ (cid:48) = µ ◦ q . For uniqueness, suppose that λ , λ are two non-forkingmeasures over N (cid:31) M extending µ . We may assume | N | = | M | and let κ be as above. Let q , q ∈ S xϕ, U κ + ( N ) be corresponding ϕ, U κ + -types. By a local version of Lemma 3.15 we mayassume that both q , q extend p . Thus we are done by Theorem 4.13. The more generalstatement follows similarly by Remark 4.14. (cid:3) References [AC] Kyle Gannon Artem Chernikov. Definable convolution and idempotent keisler measures. preprint . arXiv:2004.10378 . 29[Fre03] D. H. Fremlin. Measure theory. Vol. 2 . Torres Fremlin, Colchester, 2003. Broad foundations, Correctedsecond printing of the 2001 original. 2, 19[Fre04] D. H. Fremlin. Measure theory. Vol. 3 . Torres Fremlin, Colchester, 2004. Measure algebras, Correctedsecond printing of the 2002 original. 19, 22, 23, 24, 25[Kop89] Sabine Koppelberg. Handbook of Boolean algebras. Vol. 1 . North-Holland Publishing Co., Amsterdam,1989. Edited by J. Donald Monk and Robert Bonnet. 4, 6, 7, 8, 11, 14, 28[MB89] J. Donald Monk and Robert Bonnet, editors. Handbook of Boolean algebras. Vol. 2 . North-HollandPublishing Co., Amsterdam, 1989. 5[Pil] Anand Pillay. Domination and regularity. preprint . arXiv:1806.08806 . 29 OOLEAN TYPES IN DEPENDENT THEORIES 31 [Pil96] Anand Pillay. Geometric stability theory , volume 32 of Oxford Logic Guides . The Clarendon Press,Oxford University Press, New York, 1996. Oxford Science Publications. 26, 27, 28[She80] Saharon Shelah. Remarks on Boolean algebras. Algebra Universalis , 11(1):77–89, 1980. 7[She12] Saharon Shelah. Dependent dreams: recounting types. 2012. 3, 9[Sim14] Pierre Simon. A Guide to NIP Theorie . 2014. to appear in Lecture Notes in Logic. 3, 10, 16, 19, 20,21, 24, 25[TZ12] Katrin Tent and Martin Ziegler. A course in model theory , volume 40 of Lecture Notes in Logic .Association for Symbolic Logic, La Jolla, CA; Cambridge University Press, Cambridge, 2012. 28 Itay Kaplan, The Hebrew University of Jerusalem, Einstein Institute of Mathematics, Ed-mond J. Safra Campus, Givat Ram, Jerusalem 91904, Israel E-mail address : [email protected] URL : math.huji.ac.il/~kaplan Ori Segel, The Hebrew University of Jerusalem, Einstein Institute of Mathematics, EdmondJ. Safra Campus, Givat Ram, Jerusalem 91904, Israel E-mail address : [email protected] Saharon Shelah, The Hebrew University of Jerusalem, Einstein Institute of Mathematics,Edmond J. Safra Campus, Givat Ram, Jerusalem 91904, IsraelSaharon Shelah, Department of Mathematics, Hill Center-Busch Campus, Rutgers, The StateUniversity of New Jersey, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019 USA E-mail address : [email protected] URL ::