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arXiv:0901.1584v3 [math.LO] 8 Nov 2011
ON THEORIES OF RANDOM VARIABLES
ITAÏ BEN YAACOV
Abstract. Nous étudions des théories d’espaces de variables aléatoires : en un premier temps, nous considérons les variables aléatoires à valeurs dans l’intervalle [0, 1], puis à valeur dans des structures métriques quelconques, généralisant la procédure d’aléatoirisation de structures classiques due à Keisler. Nous démontrons des résultats de préservation et de non-préservation de propriétés modèle-théoriques par cette construction : (i) L’aléatoirisée d’une structure ou théorie stable est stable. (ii) L’aléatoirisée d’une structure ou théorie simple instable n’est pas simple. Nous démontrons également que dans la structure aléatoirisée, tout type est un type de Lascar. We study theories of spaces of random variables: first, we consider random variables with values in the interval [0, 1], then with values in an arbitrary metric structure, generalising Keisler’s randomisation of classical structures. We prove preservation and non-preservation results for model theoretic properties under this construction: (i) The randomisation of a stable structure is stable. (ii) The randomisation of a simple unstable structure is not simple. We also prove that in the randomised structure, every type is a Lascar type.
Introduction Mathematical structures arising in the theory of probabilities are among the most natural examples for metric structures which admit a model theoretic treatment, albeit not in the strict setting of classical first order logic. Examples include the treatment of adapted spaces by Keisler & Fajardo [FK02], in which no logic of any kind appears explicitly (even though many model theoretic notions, such as types, do appear). Another example, which is the main topic of the present paper, is Keisler’s randomisation construction [Kei99], in which one considers spaces of random variables whose values lie in some given structures. The randomisation construction was originally set up in the formalism of classical first order logic, representing the probability space underlying the randomisation by its probability algebra, namely, the Boolean algebra of events up to null measure (defined abstractly, a probability algebra is a measure algebra of total mass one, see Fremlin [Fre04]). We consider that this formalism was not entirely adequate for the purpose, since the class of probability algebras is not elementary in classical first order logic, a fact which restricts considerably what can be done or proved (for example, the randomised structure interprets an atomless Boolean algebra, and can therefore be neither dependent nor simple). To the best of our knowledge, the first model theoretic treatment of a probabilistic structure in which notions such as stability and model theoretic independence were considered was carried out by the author in [Ben06], for the class of probability algebras, in the formalism of compact abstract theories. While this latter formalism was adequate, in the sense that it did allow one to show that probability algebras are stable and that the model theoretic independence coincides with the probabilistic one, it was quite cumbersome, and soon to become obsolete. Continuous first order logic is a relatively new formalism, at least in its present form, proposed by Alexander Usvyatsov and the author [BU10] for model theoretic treatment of (classes of) complete metric structures. For example, we observe there that the class of probability algebras is elementary, its theory admitting a simple set of axioms, and that the theory of atomless probability algebras admits quantifier elimination, thus simplifying considerably many of the technical considerations contained in [Ben06]. Viewing probability algebras as metric structures in this fashion, rather than as classical structures, allowed Keisler and the author [BK09] to present the randomisation as a metric structure, and we contend that this metric randomisation is the “correct” one. Arguments to this effect include several
Key words and phrases. random variables; continuous logic; metric structures. Research supported by ANR chaire d’excellence junior THEMODMET (ANR-06-CEXC-007) and by Marie Curie re- search network ModNet. The author wishes to thank the referee for many helpful remarks regarding the structure of the articles and references. Revision 1281 of 25th October 2011. 1
2 ITAÏ BEN YAACOV
preservation results which would be false in the formalism of [Kei99]. For example, in [BK09] we prove that if a structure is stable then so is its randomisation, while preservation of dependence was proved by the author in [Ben09a]. Another argument, both æsthetic and practical, is that types in the metric randomisation are very natural objects, namely regular Borel probability measures on the space of types of the original theory, also referred to nowadays as Keisler measures [Kei87], and which turn out to be particularly useful for the study of dependent theories, e.g., in [HPP08]. We still find the current state of knowledge, and existing treatment, of randomisation, wanting on several points. First, since the randomisation of a discrete structure (or theory) necessarily produces a metric one, the question of randomising metric structures arises quite naturally. In fact, it is quite easy to construct the randomisation of a metric structure (or theory) indirectly, by letting its type spaces be the spaces of regular Borel probability measures as mentioned above, a fact which was used in [Ben09a] to point out that the preservation of dependence holds for the randomisation of metric structures as well, even though the latter had not yet been formally defined. However, the point of view of theories as type spaces, while a personal favourite of the author (see for example [Ben03a]), is far from being universally accepted, creating the need for an “ordinary” construction of the randomisation of a metric structure, with a natural language, axioms, and all. A second point is that the treatment of randomisation in [BK09] relies greatly on [Kei99], many times referring to it for proofs, even though some fundamental aspects of the set-up are different, requiring the reader to continually verify that the arguments do transfer. The aim of the present paper is to remedy these shortcomings by providing a self-contained treatment of randomisation in the metric setting, and show (or point out) that the preservation results of [BK09, Ben09a] hold in the metric setting as well. In addition, we turn the preservation of dependence into a dichotomy by showing that if T is not dependent then its randomisation T R^ cannot even be simple, and in fact has T P 2. We also improve a corollary of the preservation of stability of [BK09], namely that in randomised stable structures types over sets are Lascar types, proving the same for arbitrary randomised structures. As a minor point, we simplify the language (and theory), and rather than name in LR^ (the randomisation language) the randomisation JϕK of each L-formula ϕ, we name the function symbols and the randomisations of the relation symbols of L alone. The paper is organised as follows. In Section 1 we consider formal deductions in propositional con- tinuous logic, after Rose, Rosser and Church. These are used in Section 2 to give axioms for the theory of spaces [0, 1]-valued random variables, which play the role played by probability algebras in [BK09]. Model theoretic properties of this theory are deduced from those of the theory of probability algebras, with which it is biïnterpretable. In Section 3 we define and study the randomisations of metric struc- tures, namely spaces of random variables whose values lie in metric structures. We give axioms for the theory of these random structures, prove quantifier elimination in the appropriate language, characterise types and so on. We also prove a version of Łoś’s Theorem for randomisations, in which the ultra-filter is replaced with an arbitrary integration functional. In Section 4 we prove several preservation and non preservation results. In Section 5 we prove that in random structures, types over sets are Lascar types, so in the stable case they are stationary.
- On results of Rose, Rosser and Chang In the late 1950s Rose and Rosser [RR58] proved the completeness of a proof system for Łukasiewicz’s many-valued propositional logic, subsequently improved and simplified by Chang [Cha58b, Cha58a, Cha59]. This logic is very close to propositional continuous logic. Syntactically, the notation is quite different, partially stemming from the fact we identify True with 0 , rather than with 1. Also, the con- nective 12 does not exist in Łukasiewicz’s logic. Semantically, we only allow the standard unit interval [0, 1] as a set of truth values, while some fuzzy logicians allow non-standard extensions thereof (namely, they allow infinitesimal truth values). We should therefore be careful in how we use their results. In these references, Propositional Łukasiewicz Logic is presented using Polish (prefix) notation, without parentheses. A formula is either an atomic proposition, Cϕψ or N ϕ, where ϕ and ψ are simpler formulae. We shall prefer to use the notation of continuous logic, replacing Cϕψ with ψ −. ϕ and N ϕ with ¬ϕ.
Definition 1.1. Let S 0 = {Pi : i ∈ I} be a set distinct symbols, which we view as atomic proposition. Let S be freely generated from S 0 with the formal binary operation −.^ and unary operation ¬. Then S is a Łukasiewicz logic.
Definition 1.2. Let S be a Łukasiewicz logic.
4 ITAÏ BEN YAACOV
Fact 1.6 ([RR58, Cha59]). Let S be a Łukasiewicz logic, and ϕ ∈ S. Then ϕ if and only if ⊢ ϕ.
Proposition 1.7. Let S be a Łukasiewicz logic, and let Σ ⊆ S. Then Σ is consistent if and only if it has a model.
Proof. One direction is by soundness. For the other, assume that Σ has no model. Then by Lemma 1. there are n and ϕi ∈ Σ such that letting ψ = 1 −. nϕ 0 −.... −. nϕm− 1 we have ψ. By Fact 1.6 we have ⊢ ψ, and by Modus Ponens Σ ⊢ 1. By Fact 1.6 we also have ⊢ ϕ −.^1 for every formula ϕ, so Σ ⊢ ϕ and Σ is contradictory. 1.
Unfortunately, this is not quite what we need, and we shall require the following modifications: (i) We wish to allow non-free logics, i.e., logics which are not necessarily freely generated from a set of atomic propositions. In particular, such logics need not be well-founded (i.e., we may have an infinite sequence {ϕn}n∈N such that each ϕi+1 is a “proper sub-formula” of ϕi). (ii) The set of connectives {¬, −.^ } is not full in the sense of [BU10]. We should therefore like to introduce an additional unary connective, denoted 12 , which consists of multiplying the truth value by one half.
Definition 1.8. A continuous propositional logic is a non empty structure (S, ¬, 12 , −.^ ), where −.^ is a binary function symbol and ¬, 12 are unary function symbols. A homomorphism of continuous propositional logics is a map which respects ¬, 12 and −.^. A truth assignment to a continuous propositional logic S is a homomorphism v : S → [0, 1], where [0, 1] is equipped with the natural interpretation of the connectives. Models and logical entailment are defined in the same manner as above. We say that S is free (over S 0 ) if there exists a subset S 0 ⊆ S such that S if freely generated from S 0 by the connectives {¬, 12 , −.^ }. In that case every map v 0 : S 0 → [0, 1] extends to a unique truth assignment.
The new connective 12 requires two more axioms: 1 2 ϕ^ −
. (^) (ϕ −. 1 (A5) 2 ϕ)
(A6) (ϕ −.^12 ϕ) −.^12 ϕ
Formal deductions in the sense of continuous propositional logic are defined as earlier, allowing A1-6 as logical axiom schemes.
Lemma 1.9. For every continuous propositional logic S (not necessarily free), ϕ, ψ ∈ S, Σ ⊆ S and n ∈ N:
(i) ⊢ ϕ −. ϕ. (ii) ⊢ (ϕ −. ψ) −.^ (1 −. n(ψ −. ϕ)). (iii) If Σ, ϕ −. ψ is contradictory then Σ ⊢ ψ −. ϕ.
Proof. (i) In Łukasiewicz logic we have P −. P , and by Fact 1.6, ⊢ P −. P. By substitution of ϕ for P we get a deduction for ϕ −. ϕ in S. (ii) Same argument. (iii) If Σ, ϕ −. ψ is contradictory then it is has no model. By the proof of Proposition 1.7 there is n ∈ N such that Σ ⊢ 1 −. n(ϕ −. ψ). Therefore Σ ⊢ ψ −. ϕ. 1.
Theorem 1.10. Let S be a continuous propositional logic, not necessarily free, and let Σ ⊆ S. Then Σ is consistent if and only if it is satisfiable.
Proof. Let Sf^ be the Łukasiewicz logic freely generated by {Pϕ : ϕ ∈ S}, and let:
Σf 0 ={P¬ϕ −.^ ¬Pϕ, ¬Pϕ −. P¬ϕ : ϕ ∈ S} ∪ {Pϕ−. ψ −.^ (Pϕ −. Pψ ), (Pϕ −. Pψ ) −. Pϕ−. ψ : ϕ, ψ ∈ S} ∪ {P (^1) 2 ϕ^
−. Pϕ−. 1 2 ϕ
, Pϕ−. 1 2 ϕ^
−. P 1
2 ϕ^
: ϕ ∈ S}
Σf^ ={Pϕ : ϕ ∈ Σ} ∪ Σf 0.
Assume that Σf^ has a model vf^. Define v : S → [0, 1] by v(ϕ) = vf^ (Pϕ). Since vf^ Σf 0 , v is a truth assignment in the sense of S, and is clearly a model of Σ. Thus, if Σ has no model, neither does Σf^. By Proposition 1.7 Σf^ is contradictory. Thus, for every ψ ∈ S we have Σf^ ⊢ Pψ. Take any deduction sequence witnessing this, replacing every atomic proposition
ON THEORIES OF RANDOM VARIABLES 5
Pϕ with ϕ. If a formula was obtained from previous ones using Modus Ponens, the same holds after this translation. Premises from Σf^ become translated to one of several cases:
(i) Premises of the form Pϕ for ϕ ∈ Σ are replaced with ϕ ∈ Σ. (ii) Premises of the first two kinds from Σf 0 are replaced with something of the form ϕ −. ϕ, which we know is deducible without premises. (iii) Premises of the last kind from Σf 0 are translated to instances of the axioms schemes A5-6.
We conclude that Σ ⊢ ψ for all ψ ∈ S, and Σ is contradictory. The other direction is by easy soundness. 1. Let 2 −n^ be abbreviation for 12 · · · 12 1 (n times), where 1 is still as per Notation 1.3, so v(2−n) = 2−n for any truth assignment v.
Corollary 1.11. Let S be a continuous propositional logic, not necessarily free, Σ ⊆ S and ϕ ∈ S. Then Σ ϕ if and only if Σ ⊢ ϕ −.^2 −n^ for all n.
Proof. Right to left is clear, so assume that Σ ϕ. Then Σ ∪ { 2 −n^ −. ϕ} is non-satisfiable, and therefore contradictory by Theorem 1.10. By Lemma 1.9: Σ ⊢ ϕ −.^2 −n. 1.
Remark 1.12. With some more effort, one can prove that if S is free and Σ is finite, then Σ ϕ if and only if Σ ⊢ ϕ. This can be shown to fail if we drop either additional hypothesis, and in any case will not be required for our present purposes.
These completeness results are extended to the full continuous first order logic in [BP10]. We conclude with a word regarding the semantics of continuous propositional logics.
Definition 1.13. Let S be a continuous propositional logic. Its Stone space is defined to be the set S^ ˜ = Hom(S, [0, 1]), namely the space of truth assignments to S. We equip S with the induced topology as a subset of [0, 1]S^ (i.e., with the point-wise convergence topology). For each ϕ ∈ S we define a function ϕˆ : S →˜ [0, 1] by ϕˆ(v) = v(ϕ).
Proposition 1.14. Let S be a continuous propositional logic, S˜ its Stone space, and let θS denote the map ϕ 7 → ϕˆ.
(i) The space S˜ is compact and Hausdorff. (ii) θS ∈ Hom
S, C( S˜, [0, 1])
. In particular, each ϕˆ is continuous. (iii) For ϕ, ψ ∈ S we have θS (ϕ) = θS (ψ) if and only if ϕ ≡ ψ. (iv) The image of θS is dense in the uniform convergence topology on C( S˜, [0, 1]).
Moreover, the properties characterise the pair ( S˜, θS ) up to a unique homeomorphism.
Proof. That the image is dense is a direct application of a variant of the Stone-Weierstrass theorem proved in [BU10, Proposition 1.4]. The other properties are immediate from the construction. We are left with showing uniqueness. Indeed, assume that X is a compact Hausdorff space and θ : S → C(X, [0, 1]) satisfies all the properties above. Define ζ : X → S˜ by ζ(x)(ϕ) = θ(ϕ)(x). Thus ζ is the unique map satisfying θS (ϕ)◦ ζ = θ(ϕ), and we need to show that it is a homeomorphism. Continuity is immediate. The image of θ is dense in uniform convergence and therefore separates points, so ζ is injective. Since X is compact and Hausdorff ζ must be a topological embedding. In order to see that ζ is surjective it will be enough to show that its image is dense. So let U ⊆ S˜ be a non empty open set, which must contain a non empty set of the form {v ∈ S˜ : f (v) > 0 } for some f ∈ C( S˜, [0, 1]). For n big
enough there is v 0 ∈ S˜ such that f (v 0 ) > 2 −n+1. By density find ϕ 0 ∈ S such that ‖ ϕˆ 0 − f ‖∞ < 2 −n. and let ϕ = ϕ 0 − 2 −n^ ∈ S. Then {v ∈ S˜ : v(ϕ) > 0 } ⊆ U and v 0 (ϕ) 6 = 0. Since ϕ 6 ≡ 0 there is x ∈ X such that ζ(x)(ϕ) = θ(ϕ)(x) 6 = 0, i.e., ζ(x) ∈ U. This concludes the proof. 1.
- The theory of [0, 1]-valued random variables From this point and through the end of this paper, we switch to the setting of continuous first order logic. This means that structures, formulae, theory and so on, unless explicitly qualified otherwise, should be understood in the sense of [BU10] (or [BBHU08]). Let (Ω, F , μ) be a probability space. In [Ben06] we considered such a space via its probability algebra F^ ¯ , namely the Boolean algebra of events F modulo null measure difference. Equivalently, the probability algebra F¯ can be viewed as the space of { 0 , 1 }-valued random variables (up to equality a.e.). Here we shall consider a very similar object, namely the space of [0, 1]-valued random variables. This space will be
ON THEORIES OF RANDOM VARIABLES 7
(iii) a −.^12 a = 12 a, 12 a −. a = 0 and E( 12 a) = 12 E(a). (iv) Define by induction 20 = 1 (i.e., 20 = ¬ 0 ) and 2 −(n+1)^ = 12 2 −n. Then for all n ∈ N: E(2−n) = 2 −n. (v) a = 0 ⇐⇒ ⊢M a ⇐⇒ M a. (vi) a = b ⇐⇒ a ≡M b.
Proof. (i) From RV4 we have (a −. b) −. a = 0 and using RV3 we obtain d(a, a −. b) = E(a ∧ b). By RV1 E(a ∧ b) ≤ E(b). The rest follows. (ii) This was already observed earlier, using Fact 1.6. (iii) That 12 a = a −.^12 a was observed above (RV5). It follows that a ∧ 12 a = 12 a, so 12 a −. a = 0 by RV1 (with x = 12 a, y = a). Again by RV1 (now with x = a, y = 12 a) we obtain E(a) = 2E
2 a
(iv) Immediate from the previous item. (v) Assume that ⊢M a. Then by RV1 (which implies Modus Ponens) and RV4.1-6 we have a = 0. Thus a = 0 ⇐⇒ ⊢M a. The implication ⊢M a =⇒ M a is by soundness. Finally assume that M a. Then for all n we have ⊢M a −.^2 −n, whereby a −.^2 −n^ = 0. Thus E(a) = E(a ∧ 2 −n) ≤ E(2−n) = 2−n, for arbitrary n. It follows that E(a) = 0, i.e., that a = 0. (vi) Assume that a ≡M b, i.e., that M a −. b and M b −. a. Be the previous item a −. b = b −. a = 0 whereby a = b. 2.
Let M˜ be the Stone space of M, viewed as a continuous propositional logic, and let θM : M → C( M˜, [0, 1]) be as in Proposition 1.14. Recall the notation ˆa = θM(a). By Lemma 2.3(vi) and Proposi- tion 1.14, θM is injective. The space C( M˜, [0, 1]) is naturally equipped with the supremum metric, denoted ‖f − g‖∞. We aim to show now that dM^ is an L^1 distance, i.e., that for an appropriate measure we have dM(a, b) = ‖ˆa−ˆb‖ 1 , which need not be equal to ‖ˆa−ˆb‖∞. Nonetheless, we can relate the two metrics as follows (we essentially say that L∞-convergence of random variables implies L^1 -convergence).
Lemma 2.4. Assume that {an}n∈N ⊆ M is such that {ˆan}n∈N ⊆ C( M˜, [0, 1]) is a Cauchy sequence in
the supremum metric. Then {an}n∈N converges in M and lim ˆan = lim̂ an.
Proof. By assumption, for every k < ω there is Nk such that for all ‖ˆan − ˆam‖∞ ≤ 2 −k^ for all n, m < Nk. Therefore (ˆan −.^ aˆm) −.^2 −k^ = 0, and since θM is injective: an −. am −.^2 −k^ = 0. Thus E(an −. am) = E((an −. am) ∧ 2 −k) ≤ E(2−k) = 2−k. Similarly E(am −. an) ≤ 2 −k, whereby d(an, am) ≤ 2 −k+1. Since M is a (complete) L-structure, it contains a limit a. Now fix n ≥ Nk and let m → ∞. Then am → a, and therefore am −. an −.^2 −k^ → a −. an −.^2 −k. Thus a −. an −.^2 −k^ = 0, and by a similar argument an −. a −.^2 −k^ = 0. We have thus shown that ˆan → ˆa uniformly as desired. 2.
Corollary 2.5. The map θM : M → C( M˜, [0, 1]) is bijective.
Proof. We already know it is injective, and by Proposition 1.14 its image is dense. By the previous lemma its image is complete, so it is onto. 2.
We shall identify M with C( M˜, [0, 1]).
Lemma 2.6. For all a, b ∈ M and r ∈ R+:
(i) If a + b ∈ M (i.e., ‖a + b‖∞ ≤ 1 ) then E(a + b) = E(a) + E(b). (ii) If ra ∈ M (i.e., r‖a‖∞ ≤ 1 ) then E(ra) = rE(a).
Proof. (i) Let c = a + b. Then c −. b = a and b −. c = 0, whereby:
E(c) = E(c −. b) + E(b −.^ (b −. c)) = E(a) + E(b −.^ 0) = E(a) + E(b). (ii) For integer r this follows from the previous item, and the rational case follows. If rn → r then rna → ra in C( M˜, [0, 1]) and a fortiori in M, so the general case follows by continuity of E. 2.
Theorem 2.7. Let M RV , M˜ its Stone space and θM : M → C( M˜, [0, 1]) as in Proposition 1.14.
(i) As a topological space, M˜ is compact and Hausdorff. (ii) The map θM : M → C( M˜, [0, 1]) is bijective and respects the operations ¬, 12 and −.^ (i.e., it is an isomorphism of continuous propositional logics).
8 ITAÏ BEN YAACOV
(iii) There exists a regular Borel probability measure μ on M˜ such that the natural map ρμ : C( M˜, [0, 1]) → L^1 (μ, [0, 1]) is bijective as well, and the composition ρμ ◦ θM : M → L^1 (μ, [0, 1]) is an isomorphism of LRV -structures.
Moreover, these properties characterise ( M˜, μ, θM) up to a unique measure preserving homeomorphism.
Proof. The first two properties are already known. By Lemma 2.6 we can extend E by linearity from C( M˜, [0, 1]) to C( M˜, R), yielding a positive linear functional. By the Riesz Representation Theorem
[Rud66, Theorem 2.14] there exists a unique regular Borel measure μ on M˜ such that E(f ) =
f dμ. Since E(1) = 1, μ is a probability measure. The map M → L^1 (μ, [0, 1]) is isometric and in particular injective. Its image is dense (continuous functions are always dense in the L^1 space of a regular Borel measure). Moreover, since M is a complete metric space the image must be all of L^1 (μ, [0, 1]), whence the last item. The uniqueness of M˜ as a topological space verifying the first two properties follows from Proposi- tion 1.14 and Lemma 2.3.(vi). The Riesz Representation Theorem then yields the uniqueness of μ. 2.
We may refer to ( M˜, μ) (viewed as a topological space equipped with a Borel measure) as the Stone space of M or say that M is based on ( M˜, μ).
Corollary 2.8. Let M be an LRV -structure. Then:
(i) The structure M is a model of RV if and only if it is isomorphic to some L^1 (F , [0, 1]). (ii) A structure of the form L^1 (F , [0, 1]) is a model of ARV if and only if (Ω, F , μ) is an atomless probability space.
Corollary 2.9. Let M RV be based on ( M˜, μ). Then every Borel function M →˜ [0, 1] is equal almost everywhere to a unique continuous function.
2.2. Interpreting random variables in events and vice versa. In the previous section we attached to every probability space (Ω, F , μ) the space L^1 (F , [0, 1]) of [0, 1]-valued random variables and axio- matised the class of metric structures arising in this manner. While we cannot quite recover the original space Ω from L^1 (F , [0, 1]) we do consider that L^1 (F , [0, 1]) retains all the pertinent information An alternative approach to coding a probability space in a metric structure goes through its probability algebra, namely the space of { 0 , 1 }-valued random variables. It can be constructed directly as the Boolean algebra quotient F¯ = F /F 0 where F 0 is the ideal of null measure sets. In addition to the Boolean algebra structure, it is equipped with the induced measure function μ : F¯ → [0, 1] and the metric d(a, b) = μ(a△b) (in fact, the measure μ is superfluous and can be recovered as μ(x) = d(x, 0)). The metric is always complete, so a probability algebra is a structure in the language LP r = { 0 , 1 , ∩, ∪, ·c, μ}. Let us define the theory P r to consist of the following axioms, quantified universally:
(Bool) The theory of Boolean algebras: (x ∩ y)c^ = xc^ ∪ yc,...
(Pr1) μ(1) = 1
(Pr2) μ(x) + μ(y) = μ(x ∪ y) + μ(x ∩ y)
(Pr3) d(x, y) = μ(x△y).
The theory AP r (atomless probability algebras) consists of P A 0 along with:
sup x
inf y
∣μ(y^ ∧^ x)^ −^
μ(x) 2
(APr) ∣ = 0
Fact 2.10. The class of probability algebras is elementary, axiomatised by P r. The class of atomless probability algebras is elementary as well, axiomatised by AP r. Moreover, the theory AP r eliminates quantifiers (it is the model completion of P r). It is ℵ 0 -categorical (there is a unique complete separable atomless probability algebra), and admits no compact model, whereby it is complete. It is ℵ 0 -stable and its notion of independence coincides with probabilistic independence. All types over sets (in the real sort) are stationary.
Proof. Most of this is shown in [BU10, Example 4.3]. The fact regarding stability and independence were shown in [Ben06] in the setting of compact abstract theories. The arguments carry nonetheless to models of AP r in continuous logic. 2.
10 ITAÏ BEN YAACOV
Proof. For the first item, the sequence (fr)r∈D is decreasing so τr (fs)s∈D = fr. It follows that |ϕn(c, fr)r∈D −
c f^ |^ <^2
−n (^) and ϕ(x, fr )r∈D = ∫ c f^. We now show the second item. To see that f 7 → f˜ is injective assume that f˜ = ˜g. By the previous item this means that
c f^ =^
c g^ for every^ c^ ∈^ F¯ , whereby f = g. To see it is surjective let ϕ(x, A) be any instance of ϕ. Define:
br = τr (A) ∈ F¯ r ∈ D,
fn =
k< 2 n
2 −n (^1) bk/ 2 n ∈ M n ∈ N.
One readily checks that d(fn, fm) < 2 −^ min(n,m), so the sequence fn converges to a limit g ∈ M with d(fn, g) ≤ 2 −n. For every event c ∈ F¯ we have ϕn(c, A) =
c fn. It follows that^ |ϕn(c, A)^ −^
c g| ≤^2
−n
and therefore ϕ(c, A) =
c g. In other words,^ ˜g^ is a canonical parameter for^ ϕ(x, A). Let us now prove the third item. In order to prove that ( f ,˜ ˜g) 7 → f˜ −. g is definable it is enough to
show that we can define the predicate ϕ
x, f˜ −. g
uniformly from f˜ and ˜g. Indeed:
ϕ
x, f˜ −. g
x
(f −. g) = sup y
[∫
x∩y
f −.
x∩y
g
]
= sup y
[
ϕ(x ∩ y, f˜) −. ϕ(x ∩ y, g˜)
]
Similarly: ∫
x
x
¬f = ¬
x
f,
x
1 2 f^ =^
1 2
x
f.
It follows that all the connectives which one can construct from these primitives are definable, and in
particular (x, y) 7 → |x − y|. Thus the distance d(f, g) = ϕ
1 , |˜f − g|
is definable. We leave the moreover part to the reader. 2.
The intrinsic distance on the imaginary sort F¯ϕ is by definition:
dϕ(f, g) = sup b∈ F¯
b
(f − g)
∣ = max
‖f −. g‖ 1 , ‖g −. f ‖ 1
The distance dϕ is easily verified to be uniformly equivalent to the L^1 metric on the space of [0, 1]-valued random variables. This is a special case of the general fact that any two definable distance functions on a sort are uniformly equivalent. At the cost of additional technical complexity we could have arranged to recover L^1 (F , [0, 1]) on an imaginary sort in which the intrinsic distance is already the one coming from L^1. Indeed, we could have defined a formula ψ(y, z, xr)r∈D such that ψ(b, c, fr)r∈D =
b f^ +^
crb ¬f^ , obtaining further down the road:
dψ (f, g) = sup b,c∈ F¯
b
(f − g) +
crb
(g − f )
∣ =^ ‖f^ −^ g‖^1.
2.3. Additional properties of RV and ARV. Models of RV admits quantifier-free definable continu- ous functional calculus.
Lemma 2.13. If θ : [0, 1]ℓ^ → [0, 1] is a continuous function, then the function f¯ 7 → θ ◦ ( f¯) is uniformly quantifier-free definable in models of RV. By “quantifier-free definable” we mean that for every definable predicate P (¯y, z), the definable predicate P (¯y, θ ◦ (¯x)) is definable with the same quantifier complexity. Specifically, d(y, θ ◦ (¯x)) is quantifier-free definable.
Proof. We can uniformly approximate θ by a sequence of terms τn(¯x) in ¬, 12 , −.^ , in which case P (¯y, θ ◦ (¯x)) = lim P (¯y, τn(¯x)) uniformly. 2.
For example, the predicates E(xp) or E(|x − y|p) are definable for every p ∈ [1, ∞[, and thus the Lp^ distance ‖x − y‖p = E(|x − y|p)^1 /p^ is definable as well, all the definitions being quantifier-free and uniform. For A ⊆ L^1 (F , [0, 1]), let σ(A) ⊆ F denote the minimal σ-sub-algebra by which every member of A is measurable, i.e., such that A ⊆ L^1 (σ(A), [0, 1]) (For this to be entirely well-defined we may require σ(A) to contain the null measure ideal of F .)
ON THEORIES OF RANDOM VARIABLES 11
Lemma 2.14. Let M be a model of RV , say M = L^1 (F , [0, 1]). Then for every σ-sub-algebra F 1 ⊆ F , the space L^1 (F ′, [0, 1]) is a sub-structure of M. Conversely, every sub-structure N ⊆ M arises in this manner as L^1 (σ(N ), [0, 1]).
Proof. The first assertion is clear, so we prove the converse. It is also clear that N ⊆ L^1 (σ(N ), [0, 1]). Let f ∈ N , and define mf˙ = f ∔... ∔ f (m times). Then mf˙ ∈ N , and as m → ∞ we have mf ˙ → (^1) {f > 0 } in L^1 , so (^1) {f > 0 } ∈ N. Since N is complete and closed under ¬ and −.^ , it follows that (^1) A ∈ N for every A ∈ σ(N ). Considering finite sums of the form ( 12 )k^1 A 0 ∔ · · · ∔ ( 12 )k (^1) An− 1 we see that every simple function in L^1 (σ(N ), [0, 1]) whose range consists solely of dyadic fractions belongs to N. Using the completeness of N one last time we may conclude that L^1 (σ(N ), [0, 1]) ⊆ N. 2.
Lemma 2.15. Let M and N be two models of RV , say M = L^1 (F , [0, 1]), N = L^1 (Ω′, [0, 1]). Then two ℓ-tuples f¯ ∈ M ℓ^ and ¯g ∈ N ℓ^ have the same quantifier-free type in LRV if and only if they have the same joint distribution as random variables.
Proof. Assume that f¯ ≡qf^ ¯g. By the previous Lemma we have E(θ ◦ ( f¯ )) = E(θ ◦ (¯g)) for every continuous function θ : [0, 1]ℓ^ → [0, 1], which is enough in order to conclude that f¯ and g¯ have the same joint distribution. Conversely, assume that f¯ and g¯ have the same joint distribution. Then E(τ ( f¯ )) = E(τ (¯g)) for every term τ (¯x). It follows that f¯ ≡qf^ g¯. 2.
Let F¯a denote the set of atoms in F¯ , which we may enumerate as {Ai : i ∈ I}. Then I is necessarily countable and every f ∈ L^1 (F , [0, 1]) can be written uniquely as f 0 +
i∈I αi^1 Ai^ , where^ f^0 is over the atomless part and αi ∈ [0, 1].
Lemma 2.16. The set F¯a ∪ { 0 } is uniformly definable in F¯. In L^1 (F , [0, 1]), both the sets F¯a ∪ { 0 } (i.e., { (^1) A : A ∈ F¯a} ∪ { 0 }) and {α (^1) A : α ∈ [0, 1], A ∈ F¯a} are uniformly definable.
Proof. For the first assertion let ϕ(x) be the LP r -formula supy
μ(x ∩ y) ∧ μ(x r y)
. If A is an atom or zero then clearly ϕ(A) = 0. If A is an event which is not an atom then the nearest atom to A is the biggest atom in A (or any of them if there are several of largest measure, or 0 if A contains no atoms). Let us construct a partition of A into two events A 1 and A 2 by assigning the atoms in A (if any) sequentially to A 1 or to A 2 , whichever has so far the lesser measure, and by splitting the atomless part of A equally between A 1 and A 2. If B ⊆ A is an atom of greatest measure (or zero if there are none) then |μ(A 1 ) − μ(A 2 )| ≤ μ(B) and:
ϕ(A) ≥ μ(A 1 ) ∧ μ(A 2 ) ≥ 12 μ(A) − 12 μ(B) = 12 μ(A r B) = 12 d
A, F¯a ∪ { 0 }
Thus F¯a ∪ { 0 } is definable. For the second assertion, F¯a ∪{ 0 } is relatively definable in F¯ which is in turn definable in L^1 (F , [0, 1]), so F¯a ∪ { 0 } is definable in L^1 (F , [0, 1]). We may therefore quantify over F¯a ∪ { 0 }, and define:
ψn(x) = inf A∈ F¯a∪{ 0 }
k≤ 2 n
d
x, 2 kn 1 A
Then lim ψn defines the distance to the last set. 2.
If follows that for each n, the set of events which can be written as the union of at most n atoms is definable, as is the set of all finite sums
i<n α^1 Ai^ where each^ Ai^ is an atom (or zero). These definitions cannot be uniform in n, though. Indeed, an easy ultra-product argument shows that the set of all atomic events (i.e., which are unions of atoms) cannot be definable or even type-definable, and similarly for the set of all random variables whose support is atomic. The atoms of a probability space always belong to the algebraic closure of the empty set (to the definable closure if no other atom has the same measure). They are therefore rather uninteresting from a model theoretic point of view, and we shall mostly consider atomless probability spaces.
Theorem 2.17. (i) The theory ARV is complete and ℵ 0 -categorical. (ii) The theory ARV eliminates quantifiers. (iii) The universal part of ARV is RV , and ARV is the model completion of RV. (iv) If M = L^1 (F , [0, 1]) ARV and A ⊆ M then dcl(A) = acl(A) = L^1 (σ(A), [0, 1]) ⊆ M. (v) Two tuples f¯ and ¯g have the same type over a set A (all in a model of ARV ) if and only if they have the same joint conditional distribution over σ(A).
ON THEORIES OF RANDOM VARIABLES 13
∏ Let^ Ω^ be an arbitrary set and let^ M^ =^ MΩ^ =^ {Mω}ω∈Ω^ be a family of^ L-structures. The product M =
ω∈Ω Mω^ consists of all functions^ a^ : Ω^ →^
Mω which satisfy a(ω) ∈ Mω for all ω ∈ Ω. Function symbols and terms of L are interpreted naturally on
M. For an L-formula ϕ(¯x) we define 〈〈ϕ(¯a)〉〉 ∈ Ω[0,1], 〈〈ϕ(¯a)〉〉 : ω 7 → ϕMω^ (¯a(ω)).
Definition 3.4. Let Ω be a set, MΩ = {Mω}ω∈Ω a family of L-structures. Let also M ⊆
M ,
A ⊆ [0, 1]Ω^ and E : A → [0, 1]. We say that (M, A , E) is a randomisation based on MΩ if
(i) The triplet (Ω, A , E) is an integration space. (ii) The subset M ⊆
M is non empty, closed under function symbols, and 〈〈P (¯a)〉〉 ∈ A for every n-ary predicate symbol P ∈ L and ¯a ∈ Mn.
We equip M with the pseudo-metric d(a, b) = E〈〈d(a, b)〉〉 and A with the L^1 pseudo-metric d(X, Y ) = E
|X − Y |
We may choose to consider E as part of the structure on A , in which case the randomisation is denoted by the pair (M, A ) alone. If (Ω, F , μ) is a probability space, every X ∈ A is F -measurable and E[X] =
X dμ then we say that (M, A ) is based on the random family M(Ω,F ,μ) (and then we almost always omit E from the notation).
The randomisation signature LR^ is defined as follows:
- The sorts of LR^ include the sorts of L, referred to as main sorts, plus a new auxiliary sort.
- Every function symbol of L is present in LR, between the corresponding main sorts. It is equipped with the same uniform continuity moduli as in L.
- For every predicate symbol P of L, LR^ contains a function symbol JP K from the corresponding main sorts into the auxiliary sort. It is equipped with the same uniform continuity moduli as P in L.
- The auxiliary sort is equipped with the signature LRV.
A randomisation (M, A ) admits a natural interpretation as an LR-pre-structure (M, A ). The corres-
ponding structure will be denoted (M̂ , Â ), and the canonical map [·] : (M, A ) → (M̂ , Â ). We also say that the randomisation (M, A ) is a representation of the structure (M̂ , Â ).
Example 3.5. A special case of a randomisation is when M =
M (i.e., the set of all sections from Ω into MΩ), A = [0, 1]Ω, U is an ultra-filter on Ω, and EU : [0, 1]Ω^ → [0, 1] is the integration functional corresponding to limits modulo U, i.e., EU (X) = limω→U X(ω). In this case  = [0, 1] and M̂ can be identified with the ultra-product
U M^.
Definition 3.6. We say that a randomisation (M, A ) is full if for every a, b ∈ M and X ∈ A , there is a function c ∈ M satisfying:
c(ω) =
a(ω) X(ω) = 1, b(ω) X(ω) = 0, anything otherwise.
We shall sometimes write c = 〈X, a, b〉, even though there is no uniqueness here. We say that (M, A ) is atomless if A is a pre-model of ARV (i.e., if (Ω, A , E) is atomless).
Example 3.7 (Randomisation of a single structure). Let M be a structure, (Ω, F , μ) an atomless prob- ability space. Let Mc ⊆ M Ω^ consist of all functions a : Ω → M which take at most countably many values in M , each on a measurable set. Define Ac ⊆ [0, 1]Ω^ similarly, equipping it with integration with respect to μ. Then (Mc, Ac) is a full atomless randomisation. Assume now that (Ω, F , μ) is merely a finitely additive probability space, namely that F is a mere Boolean algebra and μ is finitely additive. Let Mf ⊆ M Ω^ and Af ⊆ [0, 1]Ω^ consist of functions which take at most finitely many values, each on a measurable set. Again, (Mf , Af ) is an atomless, full randomisation. If (Ω, F , μ) is a true (i.e., σ-additive) probability space then both constructions are possible and (Mf , Af ) ⊆ (Mc, Ac). It is not difficult to check that they have the same completion (M̂ f , Âf ) =
(M̂ c, Âc). In particular, Âf = Âc = L^1 (F , [0, 1]). Moreover, the resulting structure only depends on A = L^1 (F , [0, 1]), and we denote it by (MA^ , A ) (or just MA^ ).
14 ITAÏ BEN YAACOV
3.2. The randomisation theory. Our first task is to axiomatise the class of LR-structures which can be obtained from full atomless randomisations (and in particular show that it is elementary). We shall use x, y,... to denote variables of L, x, y,... to denote the corresponding variables in the main sort of LR and U, V,... to denote variables in the auxiliary sort of LR. For simplicity of notation, an LR-structure (M, A ) may be denoted by M alone. In this case, the auxiliary sort will be denoted by A M^ and we may write somewhat informally M = (M, A M). When A M^ RV we shall refer to the probability algebra of A M^ as F M^ (so A M^ = L^1 (F M, [0, 1])). The “base theory” for randomisation, which will be denoted by T 0 Ra , consists of the theory RV for the auxiliary sort along with the following additional axioms (we recall that a −. b −. c = (a −. b) −. c): ( δf,i(ε) −.^ Jd(x, y)K
Jd(f (¯x′, x, y¯′), f (¯x′, y, y¯′))K −. ε
(R1f ) = 0 ( δP,i(ε) −.^ Jd(x, y)K
JP (¯x′, x, y¯′)K −.^ JP (¯x′, y, ¯y′)K −. ε
(R1P ) = 0
(R2) d(x, y) = EJd(x, y)K
sup U∈F
inf z
E
[(
Jd(x, z)K ∧ U
Jd(y, z)K ∧ ¬U
)]
(R3)
In axiom R1, δs,i denotes the uniform continuity modulus of the symbol s with respect to its ith argument, with |x¯′| = i and |y¯′| = ns − i − 1. In axiom R3, F denotes the probability algebra of the auxiliary sort, over which, modulo RV , we may quantify. The role of axiom R1 is to ensure that the values of JP (¯a)K(ω), f (¯a)(ω) only depends on a¯(ω) and respect the uniform continuity moduli prescribed by L. Axiom R2 is straightforward, requiring the distance in the main sort of be the expectation of the random variable associated to L-distance. Axiom R3 is a gluing property, corresponding to fullness of a randomisation. It can be informally stated as
(∀xy)(∀U ∈ F )(∃z)
Jd(x, z)K ∧ U = Jd(y, z)K ∧ ¬U = 0
(R3’) ,
where the existential quantifier is understood to hold in the approximate sense. We prove in Lemma 3. below that it actually holds in the precise sense.
Lemma 3.8. Let (M, A ) be a randomisation. Then (M, A ) is a pre-model of RV (in the auxiliary sort) and of R1,2. If (M, A ) is full then (M, A ) is a pre-model of T 0 Ra.
Proof. All we have to show is that if (M, A ) is full then (M, A ) verifies R3, or equivalently, (M̂ , Â ) does. However, we chose to write R3 using a quantifier over a definable set, a construct which need not have the apparent semantics in a pre-structure such as (M, A ), and we find ourselves forced to work
with (M̂ , Â ). (Indeed, since A is a mere pre-model of RV , the algebra of characteristic functions in A may well be trivial.)
Let F̂ denote the probability algebra of  and let A ∈ F̂ , a, b ∈ M̂. First, choose X ∈ A and a′, b′^ ∈ M such that [a′] and [b′] are very close to (^1) A, a and b, respectively. Define (recalling that for t ∈ [0, 1] and n ∈ N, nt˙ = (nt) ∧ 1 ):
Y = ˙2(X −.^1 /4) ∈ A , c = 〈Y, a′, b′〉 ∈ M (by fullness), W =
Jd(a′, c)K ∧ Y
Jd(b′, c)K ∧ ¬Y
∈ A.
For every ω ∈ Ω we have Y (ω) ∈ { 0 , 1 } =⇒ W (ω) = 0, or in other words, W (ω) 6 = 0 =⇒ 0 < Y (ω) < 1 =⇒ 1 / 4 < X(ω) < 3 / 4. Thus W ≤ ( ˙ 4 X)∧( ˙ 4 ¬X). Having chosen our approximations good enough (we allow ourselves to skip the detailed epsilon chase here), we see that [W ] ≤ ( ˙4[X]) ∧ ( ˙ 4 ¬[X]) is arbitrarily close to 0 and [Y ] close to (^1) A. We conclude that
Jd(a, [c])K ∧ A
Jd(b, [c])K ∧ ¬A
can be arbitrarily
close to 0 in  , which is what we needed to prove. 3.
In order to prove a converse we need to construct, for every model M T 0 Ra , a corresponding randomisation.
Definition 3.9. Assume (M, A ) T 0 Ra. Let (Ω, μ) = (ΩM, μM) be the Stone space of A as per Theorem 2.7. Then we say that (M, A ) is based on (Ω, μ).
We recall that Ω is a compact Hausdorff topological space, μ is a regular Borel probability measure and we may identify A = C(Ω, [0, 1]) = L^1 (μ, [0, 1]). Under this identification
Ω X dμ^ =^ E(X)^ for all X ∈ A.
16 ITAÏ BEN YAACOV
Definition 3.12. Let (M, A ) T 0 Ra , t : Mn^ → A a function. We say that t is local if it is always true that:
t(... , 〈A, a, b〉,.. .) = t(... , a,.. .) ∧ A + t(... , b,.. .) ∧ ¬A.
For a function t : Mn+1^ → A we define infy t(¯x, y) : Mn^ → A by infy t(¯a, y) = inf{t(¯a, b) : b ∈ M} ∈ A.
Lemma 3.13. Let t(¯x, y) be a uniformly definable local function in models of T 0 Ra from the main sort into the auxiliary sort. Then the function s(¯x) = infy t(¯x, y) is uniformly definable and local as well, and T 0 Ra implies that:
infz d
infy t(¯x, y), t(¯x, z)
Moreover, for every ¯a in a model of T 0 Ra and ε > 0 there is b such that:
t(¯a, b) ≤ infy t(¯a, y) + ε
(Similarly for supy t.)
Proof. It follows directly from the definition that if t is local then so is infy t (no definability is needed here). We start by proving the moreover part. Let (M, A ) T 0 Ra , ¯a ∈ Mn. Following the discussion of the completeness of the lattice structure on A there is a sequence {cn}n∈N such that infy t(¯a, y) = infn t(¯a, cn). Let us define a sequence {bn} by:
b 0 = c 0 , bn+1 =
{t(¯a, bn) − t(¯a, cn+1) > ε}, cn+1, bn
In other words, when passing from bn to bn+1 we use cn+1 only where this means a decrease of more than ε, and elsewhere keep bn. Clearly
n μ{t(¯a,^ bn)^ −^ t(¯a,^ cn+1)^ >^ ε}^ ≤^1 /ε.^ By construction,^ d(bn,^ bn+1)^ ≤^ μ{t(¯a,^ bn)^ − t(¯a, cn+1) > ε}, so the sequence {bn} converges to some b. Since t is local, we have t(¯a, b) ≤ t(¯a, cn) + ε, whence t(¯a, b) ≤ infy t(¯a, y) + ε, as desired. We can now prove the first assertion. Indeed, it follows from the moreover part that the graph of infy t is uniformly definable as:
X = infy t(¯a, y) ⇐⇒
supz E
X −. t(¯a, z)
infz E
t(¯a, z) −. X
Once we know that infy f is definable, the sentence in the second assertion is expressible, and holds true by the moreover part. 3.
We now proceed to define by induction, for every L-formula ϕ(¯x), a T 0 Ra -definable local function Jϕ(¯x)K to the auxiliary sort, in the following natural manner:
- Atomic formulae: JP (¯τ )K = JP K ◦ (¯τ ) is a term, the composition of the function symbol JP K with the L-terms τ¯ , which are also LR-terms. These are local by Theorem 3.11.
- Connectives: Jϕ −. ψK = JϕK −.^ JψK, and so on. Locality is clear.
- Quantifiers: Jinfy ϕ(¯x, y)K = infyJϕ(¯x, y)K, Jsupy ϕ(¯x, y)K = supyJϕ(¯x, y)K. Locality follows from Lemma 3.13. Our somewhat minimalist approach differs from that of Keisler, who introduces a function symbol Jϕ(¯x)K for every L-formula ϕ (see [Kei99, BK09]). Keisler’s Boolean Axioms and Fullness Axiom are valid in our setting by definition of JϕK (using Lemma 3.13 for fullness). Keisler’s Distance Axiom for the main sort is our R2. While not entirely equivalent, Keisler’s Event Axiom corresponds to our axiom R3. (More precisely, Keisler’s Event Axiom is equivalent to R3 plus supx,y d(x, y) = 1. We do not find it necessary or desirable to assume the latter.) Other axioms related to the auxiliary sort, with the exception of atomlessness, are coded in RV. We shall add atomlessness later on, when it is needed for Theorem 3.32. We are left with the Validity Axioms which we also claim follow from T 0 Ra.
Theorem 3.14. Let (M, A ) be a model of T 0 Ra which we identify as usual with its canonical repres- entation, based on (Ω, μ). Then for every formula ϕ(¯x) and tuple ¯a of the appropriate length we have 〈〈ϕ(¯a)〉〉 = Jϕ(¯a)K as functions on Ω (and not merely up to a null measure set).
ON THEORIES OF RANDOM VARIABLES 17
Proof. We prove by induction on ϕ. If ϕ is atomic this is known by construction and the induction step for connectives is immediate. We are left with the case of a formula infx ϕ(x, y¯). First of all, by construction, we have:
〈〈infx ϕ(x, a¯)〉〉 = infs
〈〈ϕ(b, a¯)〉〉 : b ∈ M
Jinfx ϕ(x, ¯a)K = infxJϕ(x, a¯)K = infL
Jϕ(b, a¯)K : b ∈ M
Here infs^ means the simple, or point-wise, infimum of functions on Ω. By definition Jinfx ϕ(x, ¯a)K ≤ Jϕ(b, ¯a)K for all b, and by the induction hypothesis for ϕ we have Jinfx ϕ(x, ¯a)K ≤ 〈〈ϕ(b, a¯)〉〉. It follows that Jinfx ϕ(x, ¯a)K ≤ 〈〈infx ϕ(x, ¯a)〉〉. Conversely, by Lemma 3.13, for every ε > 0 there exists b such that Jinfx ϕ(x, a¯)K + ε ≥ Jϕ(b, ¯a)K. Using the induction hypothesis again we obtain:
Jinfx ϕ(x, ¯a)K + ε ≥ 〈〈ϕ(b, ¯a)〉〉 ≥ 〈〈infx ϕ(x, ¯a)〉〉.
Equality follows. 3.
Corollary 3.15. Let M T 0 Ra and assume its canonical representation is based on the family M = {Mω}ω∈Ω. Then for every L-sentence ϕ:
M JϕK = 0 ⇐⇒ Mω ϕ for all ω ∈ Ω.
Proof. Immediate from the fact that JϕK = 〈〈ϕ〉〉 on Ω. 3.
Definition 3.16. Let T be a set of L-sentences. We define its randomisation T Ra^ to be the LR-theory consisting of the base theory along with the translation of T (Keisler’s Transfer Axioms):
T Ra^ = T 0 Ra ∪ {JϕK = 0}ϕ∈T.
Corollary 3.17. Let T be arbitrary set of sentences, ϕ a sentence. Then T ⊢ ϕ ⇐⇒ T Ra^ ⊢ JϕK = 0.
Proof. Immediate. 3.
Corollary 3.18 (Keisler’s Validity Axiom). Assume ϕ is a valid L-sentence. Then T 0 Ra ⊢ JϕK = 0.
3.4. A variant of Łoś’s Theorem.
Theorem 3.19 (Łoś’s Theorem for randomisation). Let MΩ be a family of structures, M =
M , and
let E be an integration functional on A = [0, 1]Ω. Let (M̂ , Â ) denote the structure associated to the randomisation (M, A ). Then (M, A ) is full and for every formula ϕ(¯x) and every ¯a ∈ Mn: [ 〈〈ϕ(¯a)〉〉
]
= Jϕ([¯a])K.
Proof. Fullness is immediate. We claim that
[
〈〈infy ϕ(¯a, y)〉〉
]
= infb∈M
[
〈〈ϕ(¯a, b)〉〉
]
for every formula
ϕ(¯x, y) and every ¯a ∈ Mn, where the infimum on the right hand side is in the sense of the lattice Â. Indeed, the inequality ≤ is immediate. For ≥ observe that using the Axiom of Choice, for every ε > 0 we can find b ∈ M such that 〈〈infy ϕ(¯a, y)〉〉 + ε ≥ 〈〈ϕ(¯a, b)〉〉, whereby
[
〈〈infy ϕ(¯a, y)〉〉
]
[
〈〈ϕ(¯a, b)〉〉
]
We now prove the main assertion. First of all, we may replace ϕ with an equivalent formula ψ. Indeed, on the left hand side we have immediately 〈〈ϕ(¯a)〉〉 = 〈〈ψ(¯a)〉〉. For the right hand side, we have |JϕK−JψK| = J|ϕ−ψ|K, whereby T 0 Ra ⊢ JϕK = JψK. We may therefore assume that ϕ is in prenex form. We now proceed by induction on the number of quantifiers. If ϕ is quantifier-free then
[
〈〈ϕ(¯a)〉〉
]
= Jϕ([¯a])K by construction. For the induction step, recall that
Jinfy ϕ([¯a], y)K = infyJϕ([¯a], y)K = inf b∈̂ M
Jϕ([¯a], b)K = inf b∈M
Jϕ([¯a], [b])K.
We conclude using the claim and the induction hypothesis. 3.
Let us go back to the ultra-product example (Example 3.5), where M =
M and M̂ =
U M^. By construction EJinfy ϕ([¯a], y)K = infb∈M EJϕ([¯a], [b])K. One also always has EJ¬ϕ([¯a])K = ¬EJϕ([¯a])K, EJ 12 ϕ([¯a])K = 12 EJϕ([¯a])K. Since E = EU is given by an ultra-filter, we have moreover EJϕ([¯a])−. ψ([¯a])K = EJϕ([¯a])K −. EJψ([¯a])K. Thus the truth value of ϕ([¯a]) in the ultra-product is precisely EJϕ([¯a])K in the sense of the randomised structure. Now the last item of Theorem 3.19 yields the classical version of Łoś’s Theorem:
ϕ([¯a]) = EU
[
〈〈ϕ(¯a)〉〉
]
= lim U ϕ(a(ω)).
ON THEORIES OF RANDOM VARIABLES 19
(ii) For every a¯ in
M ′^ and every ϕ(¯x), if 〈〈ϕ(¯a)〉〉 ∈ A 0 ′ then
EJϕ([¯a])K M̂ ′ = E′ 0 〈〈ϕ(¯a)〉〉.
Proof. For the first item it is easy to check that [σ] is indeed an embedding. In order to see that [σ] is a J·K-embedding let ¯a ∈ Mn^ and let ϕ(¯x) be a formula. Then 〈〈ϕ(¯a)〉〉 = Jϕ(¯a)K ∈ A ⊆ [0, 1]Ω^ by Theorem 3.14, so [ σJϕ(¯a)K
]
[
σ〈〈ϕ(¯a)〉〉
]
[
〈〈ϕ(σ¯a)〉〉
]
= Jϕ([σ¯a])K.
The second item is an immediate consequence of Theorem 3.19. 3.
3.5. Quantifier elimination and types. Let T 0 R consist of T 0 Ra along with the atomlessness axiom ARV. In other words, T 0 R consists of the theory ARV for the auxiliary sort plus axioms R1-3. Similarly, we define T R^ = T Ra^ + ARV = T 0 R ∪ {JϕK = 0}ϕ∈T.
Example 3.28. Let M T and let (Ω, F , μ) be any atomless probability space. Let (M, A ) be an associated structure to the constant random family M(Ω,F ,μ) = {M}ω∈Ω. Then (M, A ) T Ra^ by Corollary 3.24 and A is atomless, whereby (M, A ) T R.
Lemma 3.29. Every model (M, A ) T Ra^ admits a J·K-embedding σ : (M, A ) → (M^1 , A 1 ) T R. In particular, T Ra^ and T R^ are companions (which, as in classical logic, means that every model of one embeds in a model of the other, or equivalently, that the two theories have the same universal consequences sup¯x ϕ for quantifier-free ϕ).
Proof. Let ([0, 1], B, λ) denote the Lebesgue measure on [0, 1]. Apply Corollary 3.27 to (Ω′, F ′, μ′) = (Ω, F , μ) × ([0, 1], B, λ) and M′ ω,r = Mω. The resulting embedding σ : (M, A ) → (M^1 , A 1 ) is a
J·K-embedding and A 1 is atomless. If ϕ ∈ T is a sentence then JϕKM 1 = σJϕKM^ = σ0 = 0. Thus (M^1 , A 1 ) T R, as desired. 3.
Let us now fix an L-theory T. As usual, Sn(T ) (or sometimes Sx¯(T )) denotes the space of n-types of T. Similarly, Sn(T R) (or Sx¯(T R)) denotes the space of n-types of the LR-theory T R. Let us fix some additional notation. For a compact Hausdorff space X, let R(X) denote the space of regular Borel probability measures on X. For ϕ ∈ C(X, C) and μ ∈ R(X) let 〈ϕ, μ〉 =
ϕ dμ and equip R(X) with the weak topology, namely μs → μ if 〈ϕ, μs〉 → 〈ϕ, μ〉 for all ϕ. It is a classical (and easy) fact that this renders R(X) a compact Hausdorff space as well. Let p(¯x) ∈ Sn(T R). It is not difficult to verify (e.g., using the Riesz Representation Theorem) that there exists a unique regular Borel probability measure νp ∈ R(Sn(T )) characterised by the identity EJϕ(¯x)Kp^ = 〈ϕ, νp〉 for every L-formula ϕ(¯x). The map p 7 → νp is continuous by definition of the topology on R(Sn(T )). We next claim that p 7 → νp is surjective. Indeed, let μ ∈ R(Sn(T )). For each p ∈ Sn(T ) choose a model Mp and a realisation ¯ap ∈ M (^) pn of p (we do not assume that T is complete so Mp may have to vary with p). Let (M, A ) be a structure associated to the random family M = M(Sn (T ),μ) = {Mp}p∈Sn(T ). Let ¯a ∈
M be given by ¯a(p) = ¯ap. By Corollary 3.24, for every formula ϕ(¯x): EJϕ([¯a])K = E[〈〈ϕ(¯a)〉〉] = 〈ϕ, μ〉.
In particular, if ϕ ∈ T is a sentence then EJϕK = 0, so (M, A ) T Ra. By Lemma 3.29 we can embed (M, A ) in a model (M^1 , A 1 ) T R, and if p = tpM 1 (¯a) then νp = μ. We argued above for types in finitely many variables, but in exactly the same manner we associate to each p ∈ SI (T R) a regular Borel probability measure νp ∈ R(SI (T )) and this map is surjective, for an arbitrary index set I. For quantifier elimination we shall require the following fact from [BV75].
Fact 3.30. Let S be any set, (Ω, F , μ) an atomless probability space. For each x ∈ S let us be given a weight wx ≥ 0 and an event Cx ∈ F. For T ⊆ S let wT =
x∈T wx,^ CT^ =^
x∈T Cx.^ Then the following are equivalent:
(i) For all T ⊆ S: μ(CT ) ≥ wT. (ii) There exists a disjoint family {Dx}x∈S such that Dx ⊆ Cx and μ(Dx) = wx.
If wS = 1 then {Dx}x∈S is a partition of Ω (up to null measure).
Lemma 3.31. Let (M, A ) T 0 R be ℵ 0 -saturated, a¯ ∈ Mn, and let ν¯a be an abbreviation for νtp(¯a). Let θ : Sn+1(L) → Sn(L) be the restriction to the first n variables. Then:
(i) For every b ∈ M, ν¯a is the image measure of ν¯a,b under θ.
20 ITAÏ BEN YAACOV
(ii) Conversely, let η ∈ R(Sn+1(L)) by such that its image measure under θ is ν¯a. Then there is b ∈ M such that η = ν¯a,b.
Proof. The first item is immediate. For the second, it is enough to show that for every finite family ϕi(¯x, y), i < ℓ, and for every ε > 0 , there is b ∈ M such that
∣〈ϕi, η〉 − EJϕi(¯a, b)K
∣ (^) < ε for i < ℓ. Let S = {sj }j<k be a partition of [0, 1]ℓ^ into finitely many Borel subsets, diam(si) < ε. For j < k let wj = η{ ϕ¯ ∈ sj }. Choose also ¯tj ∈ sj and let ψj =
i<ℓ |ϕi^ −^ tj,i|. Notice that ∣ ∣ ∣ ∣ ∣ ∣
〈ϕi, η〉 −
j<k
wj tj,i
j<k
wj diam(sj ) < ε.
Let Cj ∈ F be the event
Jinfy ψj (¯a, y)K < ε}. Following the notations of Fact 3.30, we claim that μ(CT ) ≥ wT for all T ⊆ k. Indeed, notice that { ϕ¯ ∈ sj } ⊆ θ−^1 {ψj < ε}, whereby:
wT =
j∈T
η{ ϕ¯ ∈ sj } = η
j∈T
{ ϕ¯ ∈ sj }
(^) ≤ η
j∈T
θ−^1 {ψj < ε}
= ν¯a
j∈T
{ψj < ε}
(^) = μ(CT ).
By Fact 3.30 there are events Dj ⊆ Cj such that wT = μ(DT ) for all T ⊆ k. Since the total weight is one, {Dj }j<k is a partition. By Lemma 3.13 and saturation of M there are bj ∈ M such that Jinfy ψj (¯a, y)K = Jψj (¯a, bj )K. Notice that:
J|ϕi(¯a, bj ) − tj,i|K (^1) Dj ≤ Jinf y ψj (¯a, y)K (^1) Cj < ε.
Let b =
D 0 , b 0 , 〈D 1 , b 1 ,.. .〉
, i.e., b(ω) = bj (ω) when ω ∈ Dj. Now: ∣ ∣ ∣ ∣ ∣ ∣ ∑
j<k
wj tj,i − EJϕi(¯a, b)K
j<k
∣wj tj,i − E
Jϕi(¯a, b)K (^1) Dj
j<k
∣E
[(
tj,i − Jϕi(¯a, bj )K
(^1) Dj
]∣
≤ E
j<k
∣tj,i − Jϕi(¯a, bj )K
∣ (^1) Dj
(^) < ε.
Thus
∣〈ϕi, η〉 − EJϕi(¯a, b)K
∣ (^) < 2 ε, which is good enough. 3.
Theorem 3.32. (i) The theories of the form T R^ (and in particular T 0 R ) eliminate quantifiers in the main sort down to formulae of the form EJϕ(¯x)K. (ii) The map p 7 → νp defined by 〈ϕ, νp〉 = EJϕKp^ induces a homeomorphism Sx¯(T R) ≃ R(Sx¯(T )). (iii) Let f : n → m be any map. Let f ∗^ : Sm(T ) → Sn(T ) be the map tp(a 0 ,... , am− 1 ) 7 → tp(af (0),... , af (n−1)) and similarly f ∗,R^ : Sm(T R) → Sn(T R). Let f˜ ∗^ : R(Sm(T )) → R(Sn(T )) be the image measure map corresponding to f˜ ∗. Then the following diagram commutes:
Sm(T R)
Sn(T R)
R(Sm(T ))
R(Sn(T ))
.............. ............... ............... ............... ............... ............... ....................................
f ∗,R
.............. ............... ............... ............... ............... ............... ....................................
f^ ˜ ∗
..................................................................................................................................................................................... ...............................
.......................................................................................................................................................................................... ...............................
(iv) The completions of T R^ are in bijection with regular Borel probability measures on the space of completions of T. In particular, if T is complete then so is T R.
Proof. The first item follows from Lemma 3.31 via a standard back-and-forth argument. For the second item, we have already seen that the map p 7 → νp is continuous and surjective. From the first item it follows that it is injective. Since both spaces are compact and Hausdorff, it is a homeomorphism. The third item is easily verified. The last item is a special case of the second item for 0 -types. 3.
