Models, coproducts and exchangeability: Notes on states on Baire functions

Definition 3.1

Let A be a σ-complete Riesz MV-algebra. For any cardinal κ, a κ-dimensional observable on A is any σ-homomorphism of Riesz MV-algebras from IRL(Y) to A, with |Y| = κ. When we need to make explicit reference to the dimension of the observable we shall write IRL(κ) instead of IRL(Y).

A crucial point in [6] is Theorem 2.3, which is a representation theorem. Let us prove here the analogous result for observables defined over IRL(Y), with Y an arbitrary set. We give the whole proof for sake of completeness, but for the most part it can be deduced from [6].

Theorem 3.2

Let X, Y be nonempty sets, 𝓣 ⊆ [0, 1]^Xbe a Riesz tribe and f : X → [0, 1]^Ya measurable function w.r.t. 𝓢(𝓣)={A ⊆ X|_χA ∈ 𝓣} and 𝓑𝓐([0, 1]^Y). Then the function

is a κ-dimensional observable on 𝓣, where |Y| = κ.

Conversely, for any κ-dimensional observable 𝓧: IRL(Y) → 𝓣, there exists a unique f : X → [0, 1]^Ysuch that 𝓧 = 𝓧_f.

P r o o f. The fact that 𝓧_f is a homomorphism of Riesz MV- algebras is straightforward. Thus, we have to prove that it preserves countable joins. Let {g_n}_n∈ℕ be a sequence in IRL(Y). We recall that countable suprema and infima are defined pointwise in IRL(Y) and in any tribe, therefore

Finally, let us prove that for any a ∈ IRL(Y), a o f ∈ 𝓣. By [20] Lemma 11.8(ii)], 𝓣 is the tribe of all 𝓢(𝓣)-measurable functions, whence it is enough to prove that a o f is 𝓢(𝓣)- measurable for any a ∈ IRL(Y). This fact is easily deduced: For any E ∈ 𝓑𝓐([0, 1]), (a o f)⁻¹(E) = f⁻¹(a⁻¹(E)) ∈ 𝓢(𝓣) since f is measurable and a ∈ IRL(Y).

Conversely, given an observable 𝓧, we need to prove that there exists a function f that satisfies the claim. Let κ = |Y| and let f be the function defined by f = (f_i)_i∈κ : X → [0, 1]^Y with f_i = 𝓧(π_i) ∈ 𝓣. Using the fact that the projections form a generating set for IRL(Y), it is straightforward that 𝓧 = 𝓧_f and 𝓧_f(IRL(Y)) ⊆ 𝓣. Thus, we only need to prove that f is measurable w.r.t. 𝓢(𝓣). To do so, take E ∈ 𝓑𝓐([0, 1]^Y) and let us prove that χ_f⁻¹(E) ∈ 𝓣 : since χ_f⁻¹(E)(x) = 1 if, and only if, f(x) ∈ E if, and only if, χ_E(f(x)) = 1, it follows that χ_f⁻¹(E) = 𝓧_f(χ_E) = 𝓧(χ_E), which belongs to 𝓣 by definition of 𝓧.

Finally, let g : X → [0, 1]^Y be another function that satisfies the claim. Then, π_i o g = 𝓧(π_i) = π_i o f, from which we deduce that for any x ∈ X, g(x) = (π_i(g(x))_i∈κ = (π_i(f(x)))_i∈κ = f(x). ◻

Note that we have endowed [0, 1]^Y with the σ-algebra 𝓑𝓐([0, 1]^Y). When |Y| ≤ ω, 𝓑𝓐([0, 1]^Y) = 𝓑𝓞([0, 1]^Y) and we recover the countable case of [6].

When 𝓣 carries a σ-state s, to the pair (𝓣, s) is naturally associated the probability space (X, 𝓢(𝓣), μ_s) as defined in Section 2. Thus, to each observable 𝓧 it is associated the measurable function f : X → [0, 1]^Y given by Theorem 3.2, which is a random variable, and the correspondence is one-one. See also the discussion after [6: Theorem 2.3].

Let us now generalize [6: Theorem 2.6] to the case of any arbitrary Y. The observables that arise from the following result will be subsequently called joint observables.

THEOREM 3.3

(Joint observable theorem). Given the Riesz tribe 𝓣 ⊆ [0, 1]^Xand κ one- dimensional observables over 𝓣, namely 𝓧_i: IRL(X_i) → 𝓣, with i ∈ κ and X_isingletons, forX=∪iXithe joint function 𝓙_κ: IRL(X) → 𝓣 defined by 𝓙_κ(a) = a o f, with f = (𝓧_i(id))_i∈κ, is a κ-dimensional observable over 𝓣, where id: [0, 1]^X_i → [0, 1]^X_iis the identity function. Moreover, for any i ∈ κ and any a ∈ IRL(X_i), 𝓙_κ(a o π_i) = 𝓧_i(a).

P r o o f. Recalling that IRL(X) is the free X-generated algebra in RMV_σ, it follows that the assignment π_i ⟼ f_i = 𝓧_i(id) ∈ 𝓣 extends to a σ-homomorphism 𝓧: IRL(X) → 𝓣. Applying now Theorem 3.2 to such an 𝓧, we can say that 𝓧 = 𝓧_g, with g = (g_i)_i∈κ given by g_i = 𝓧(π_i). By definition of 𝓧, g_i = f_i and 𝓧_g is exactly the map 𝓙_κ, which is, therefore, an observable.

Finally, 𝓙_κ(a o π_i) = (a o π_i) o f = a o f_i = 𝓧_i(a). ◻

Following [6], we call process any indexed set {𝓧_i}_i∈I of one-dimensional observables, with I being an arbitrary set. Unless otherwise specified, for any observable 𝓧_i : IRL(Y) → 𝓣 with 𝓣 ⊆ [0, 1]^X, we will denote by f_i the unique measurable function given in Theorem 3.2, that is, the unique function such that 𝓧_i(a) = a o f_i for any a ∈ IRL(Y).

As mentioned in the Introduction, the main goal of this section is to define and analyze, from the point of view of non-classical logic, the pivotal notions that are involved in de Finetti's theorem on exchangeable probabilities. To do so, let us start with analyzing basic notions of probability theory such as distributions and independence.

Definition 3.4

Let 𝓧: IRL(Y) → 𝓣 be an observable on 𝓣 ⊆ [0, 1]^X, given by 𝓧(a) = a o f, and assume that 𝓣 carries the state s: 𝓣 → [0, 1]. Its distribution is the state:

Remark 3.5

With the same notations as above, the function f : X → [0, 1]^Y is a classical random variable between the spaces (X, 𝓢(𝓣), μ_s) and ([0, 1]^Y, 𝓑𝓐([0, 1]^Y).

For any E ∈ 𝓑𝓐([0, 1]^Y), it follows that 𝓧(χ_E) = χ_E o f = χ_f⁻¹(E) and its distribution state is given by s_𝓧(χ_E) = s(𝓧(χ_E)) = s(χ_f⁻¹ (E)). Thus, if we consider the measure μ_sx associated to s_𝓧 by Theorem 2.3, μ_s𝓧 (E) = μ_s(f⁻¹(E)).

Consequently, μ_s𝓧 is the pushforward measure of μ_s under f, and F(E): = μ_s(f⁻¹(E)) is the probability of the event (f ∈ E) = {x ∈ X | f(x) ∈ E}, that is, the distribution law of f.

Lemma 3.6

Let {𝓧_i}_i∈Ibe an indexed set of one-dimensional observables, with I being an arbitrary set. For any natural number k, let F_kbe the (classical) joint distribution function of f_i1 , … , f_ik. Then, for any characteristic function χ_E, with E ∈ 𝓑𝓐([0, 1]^k), we have F_k(E) = s_{𝓙k (χE}), wheres_𝓙kdenotes the distribution state of the joint observable of 𝓧_i1 , … , 𝓧_ik.

P r o o f. For each f_ij: X → [0, 1], its distribution is the function F_ij: 𝓑𝓐([0, 1]) → [0, 1] that maps E↦μs(fij−1(E))=s(χfii−1(E)). Let us denote by f_k : X → [0, 1]^k the function x ↦ (f_i1 (x), … , f_ik(x)). Then 𝓙_k(a) = a o f_k and the (classical) joint distribution of these random variables is given by F_k: 𝓑𝓐([0, 1]^k) → [0, 1], B↦μs(fk−1(B)). Hence, for E ∈ 𝓑𝓐([0, 1]^k), sJk(χE)=s(χfk−1(E))=Fk(E), settling the claim. ◻

Lemma 3.7

Let (𝓣, s) be a probability Riesz tribe. Any function f ∈ 𝓣 induces a one- dimensional observable 𝓧_f: IRL(1) → 𝓣whose distribution state corresponds to the distribution law of f on 𝓑𝓞([0, 1])

P r o o f. It follows from the fact that any f ∈ 𝓣 is 𝓢(𝓣)-measurable. ◻

Using this notion of distribution, we now define exchangeable processes. We also recall that the definition of a sequence of exchangeable random variables can be found at the end of Section 2.

Let {𝓧_i}_i∈I be an indexed set of one-dimensional observables, with I being an arbitrary set. For any finite k ∈ ℕ and two sets of indexes i₁, … , i_k, j₁, … , j_k, let us denote by Jki the joint observable of 𝓧_i1 , … 𝓧_ik and by Jki the joint observable of 𝓧_j1 , … 𝓧_jk.

Definition 3.8

The process {𝓧_i}_i∈I, posed in the probability Riesz tribe (𝓣, s), is called

1. weakly exchangeable if the sequence {f_i}_i∈I of corresponding random variables (such that 𝓧_i = 𝓧_fi) is exchangeable.

2. strongly exchangeable if for any finite k ∈ ℕ and two sets of indexes i1,…,ik,j1,…,jk,Jki and Jki have the same distribution state.

Thus, the process {𝓧_i}_i∈I is weakly exchangeable when, for any k ∈ ℕ and any two sets of indexes i₁, … , i_k, j₁, … , j_k, the distribution laws F_i,k of (f_i1 , … , f_ik) and F_j,k of (f_j1 , … , f_jk) coincide. Strong exchangeability requires the analogous property directly on the joint observable, rather than the joint random variable associated to it. The next lemma shows that indeed strong exchangeability implies weak exchangeability.

Lemma 3.9

If {X_i}_i∈Iis a strongly exchangeable process, then it is weakly exchangeable.

proof. Recalling the definition of exchangeable random variables from Section 2, by Corollar 3.6, for any E ∈ 𝓑𝓐([0,1]^k), Fi, k(E) = sJki(χE) and Fj, k(E) = sJkj(χE). By hypothesis on {X_i}_i∈I, it follows that sJki(χE)  = sJkj(χE), settling the claim.

Moving now on the notion of independence, we generalize the definition given for processes in [6: Section 5], which is itself adapted from the definition given in [17].

Definition 3.10

Let I be a set of indexes, {(𝒯_i, s_i)}_i∈I and (𝒯, s) be probability Riesz tribes. The collection {(𝒯_i, s_i)}_i∈I is said to be (𝒯, s)-independent if for any k ∈ ℕ and any i₁,…,i_k there exists an k-additive operator (as defined in Section 2)

such that for all a_i1 ∈ 𝓣_i1,…,a_ik ∈ 𝓣_ik we have

If all β are surjective, the collection is called surjectively independent.

In [6], the definition was given for a stochastic process {X_i}_i∈I posed in a probability Riesz tribe (T, s). The process was called independent if any finite subset {Im(X_i1),…,Im(X_ik)} of the sequence of tribes Im(X_i) is Im(ℐ_k)-independent, where ℐ_k is their joint observable.

Note that, if any of the k-additive operators β of the definition of an independent process is surjective, for any f ∈ Im(ℐ_k) there exists f_i1 ∈ Im(X_i1),…,f_ik ∈ Im(X_ik) such that β(f_i1,…f_ik) = f. Consequently, for any a ∈ IRL(k) there exists a_i1,…,a_ik ∈ IRL(1) such that Jki(a) = β(X_i1(a₁),…,X_ik(a_k)).

Lemma 3.11

Let {X_i}_i∈Ibe a one-dimensional process posed on a Riesz tribe (𝓣, s), with 𝓣 ⊆ [0,1]^X. If the process is independent and identically distributed with allβ’s being the product of functions, then it is weakly exchangeable.

proof. By[6: Proposition 5.9], the associate sequence {f_i}_i∈I is independent; by Remark 3.5 for any i ∈ I, the distribution law of f_i is Fi:=μs∘fi−1 and by hypothesis for any i,j ∈ I we have s ⚬ X_i = s ⚬ X_j. Consequently, again by Remark 3.5, for any E ∈ 𝓑𝓐([0,1]), F_i(E) = μsfi−1(E)=sXiχE=sXjχE=μsfj−1(E)=Fj(E). Hence, the classical process {f_i}_{i ∈ I} is independent and identically distributed, and therefore exchangeable. Indeed, for any k ∈ ℕ and any two sets of indexes i₁,…,i_k, j₁,…,j_k, let F_i,k be the distribution law of fik=fi1,…,fik and let F_{j, k} be the distribution law of fjk=fj1,…,fjk. For any generator ∏n=1kEn, of 𝓑𝓐([0, 1]^k), since the f_i’s are independent and identically distributed,

Hence, the X_i’s are weakly exchangeable.

One open problem is to understand when the conclusion of Lemma 3.11 can be extended to strong exchangeability. Furthermore, for a countable and weakly exchangeable process, the classical version of de Finetti’s exchangeability implies that the distribution law of the associated sequence of random variables is conditionally i.i.d. (see [16: Chapter 11]). It is still unclear how and if we can give a suitable logical interpretation for this notion, in terms of observables. To tackle this issue, in the next section we define a counterpart for the notion of product measure.

Proposition 4.1

Let {A_i}_i∈Ibe a collection ofσ-semisimple algebras inRMV_σ, indexed in a set I. For any i ∈ I, let A_i ≃ IRL(X_i)/I_i, where we assume the setsX_ito be pairwise disjoint. Then the free product⊕i Aiexists inRMV_σand

with J   =   〈 ∪   i I i 〉 σ , the σ-ideal generated by∪i Ii.

proof. Let us denote by A the algebra IRL(∪i Xi) / J∈ RMVσ. For any i ∈ I, define the map

Each α_i is well defined because IRL(X_i) embeds in IRL(∪i Xi)  and I_i ⊆ J.

Let us divide the proof in three steps.

1. The maps α_i are embeddings.

   Let f / I_i be an element such that f/J = 0. Let us prove that f∈I_i.

   By hypothesis, f / J = 0 implies f∈J. By [7: Proposition 2.6] there exists a countable set of elements G = {fn| n ∈ ℕ} ∈ ∪i Ii such that f ≤ τ(f₁,…,f_n,…), where τ is a term build using only ⊕ and ⋁. We note that the variables that appear in f belong to X_i. Moreover, any f_n belongs to a proper ideal I_j, j∈I.

   Let F_i = {g ∈ G|g ∈ I_i} be the subset of the f_n’s that belong to I_i. For any g ∈ G \ F_i there exists j ≠ i and a point in y_j∈[0,1]^X_j, such that g ∈ I_j and g(y_j) = 0. To see this, assume that for some g∈ G \ F_i, g ∈ I_j, we have 𝕍(g) = ∅. Hence we deduce that 𝕀(𝕍(g)) = IRL(X_j). But [7: Lemma 4.15] implies 𝕀(𝕍(g)) = 〈g〉_σ ⊆ I_j, and I_j is proper ideal, which is a contradiction.

   Take now y∈[0,1]^X_i such that g(y) = 0 for any g ∈ F_i. Denote by z the point of [0, 1]∪i Xi such that z coincides with y in the X_i-coordinates and it coincides with y_j in the X_j-coordinates, for j≠i, and it is arbitrary in the coordinates eventually not taken into account. Note that this is well defined as the sets of variables are disjoint. Then τ(f₁,…,f_n,…)(z) = 0 and therefore f(z) = 0.

   We have proved that g(y)= 0 for any g∈F_i implies f(z)= 0, and consequently (since the only variables appearing in f belong to X_i) f(y)= 0. Thus, 𝕍(f) ⊇ 𝕍(F_i). By [7: Theorem 4.7 and Lemma 4.15] and the hypothesis on A_i we deduce that 〈f〉_σ = 𝕀(𝕍(f)) ⊆ 𝕀(𝕍(F_i)) ⊆ 𝕀(𝕍(I_i)) = I_i. Thus, we have proved that f∈I_i and α_i is one-one.

2. ∪ i   α i ( A i ) generates A.

   Each IRL(X_i) is generated by {π_x|x ∈ X_i}, the coordinate projections. Analogously, {π_x | x ∈ ∪i Xi} is a set of generators for IRL(∪i Xi). Therefore, the set P = {πx/J|x∈∪iXi} generates A. By the definition of α_i, P coincides with ∪i{αi(πx/Ii)|x∈Xi}⊆∪iXi, which settles the claim.

3. (A,{α_i}_{i ∈ I}) satisfies the universal property.

   Let E ∈ RMV_σ be an algebra such that we have homomorphisms η_i: A_i→E for any i∈I. We have to prove that there exists η: A→E such that η o α_i = η_i for any i∈I.

   Let Q  =  {πx| x ∈∪iXi} be the set of generators for IRL(∪i Xi). Define ϱ: Q→E as ϱ(π_x) = η_i(π_x/I_i), when x∈X_i. Note that ϱ is well defined since the X_i’s are pairwise disjoint. Let ϱ¯:IRL(∪i Xi)→E the unique σ-homomorphism that extends ϱ to the free algebra. Note that, by definition, and because {π_x|x∈X_i} generates IRL(Xi)⊆ IRL(∪i Xi), for any i∈I and any f∈IRL(X_i), ϱ̅(f) = η_i(f/I_i). Take now η: A→E to be defined as η(f/J) = ϱ̅(f).

Let us prove that η is well defined. Take f,g∈ IRL (∪i Xi) such that f/J = g/J. Then, denoted by d Chang’s distance, d(f,g)∈J, whence there exists a countable set {hn}n ∈ ∪i Ii such that d(f,g)≤τ(h₁,…,h_n,…), where in τ only ⊕ and ⋁ appear. We have

Since each h_n belongs to ∪i Ii,  ϱ¯(hn)=0 and τ(ϱ̅(h₁),…,ϱ̅(h_n),…) = 0. Consequently, d(ϱ̅(f),ϱ̅(g)) = 0 and ϱ̅(f) = ϱ̅(g).

Finally, for any i∈I and any f∈IRL(X_i), (η ∘ α_i)(f/I_i) = η(α_i(f/I_i)) = η(f/J) = ϱ̅(f) = η_i(f/I_i), settling the final part of the claim.

Definition 4.2

We shall call an algebra A∈RMV_σ countably presented if A≃IRL(X)/I where X is a countable set of generators and I is principal. We shall denote by RMVσω the full subcategory of RMV_σ whose objects are countably presented and σ-complete Riesz MV-algebras.

Remark 4.3

For any algebra A∈RMV_σ, it is an easy exercise to show that countably generated σ-ideals in A are principal. Indeed, for a sequence {a_n}_n∈ℕ in A, we have 〈{an}n∈ℕ〉σ = 〈Vn∈ℕ an〉σ.

Corollary 4.4

The full subcategory R M V σ ω of RMV_σ whose objects are countably presented and σ-complete Riesz MV-algebras is closed under countable free products. The full subcategory R M V σ f p of RMV_σ whose objects are finitely presented and σ-complete Riesz MV-algebras is closed under finite free products.

proof. It follows from [7: Lemma 4.15] that both countably presented algebras and finitely presented algebras are σ-semisimple. Moreover, if we deal with a countable set of algebras A_i ≃ IRL(X_i)/I_i, and for some f_i∈IRL(X_i) we have Ii = 〈fi〉σ, J = 〈Vi fi〉σ by Remark 4.3.

Thus, IRL(∪i Xi)/ J is a countably presented algebra. Analogously, a finite free product will be finitely presented.

The next proposition, given in [7], is crucial for what follows.

Proposition 4.5

Let A be a σ-semisimple σ-complete Riesz MV-algebra and assume that IRL(X)/Iis a presentation for A. Then A is isomorphic to the Riesz tribeIRL(X)|_𝕍(I) = {g ∈ [0,1]^𝕍(I)} for some f∈IRL(X)}.

Given a tribe 𝓣 ⊆ [0,1]^𝓣, we recall that S(𝓣) = {A⊆𝓣|χA:[0,1]^𝓣→{0,1},χA∈𝓣} denotes its Boolean center, as defined in Section 2. For any P⊆[0,1]^X we can define the tribe of restrictions IRL(X)|_P. In this case, for brevity and if it is clear from the context, we shall denote the Boolean center S(IRL(X)|_P) by S(P).

Lemma 4.6

For any P⊆[0,1]^X, denoted by 𝓑𝓐(P) = {A∩P|A∈𝓑𝓐([0,1]^X)}, the algebra of restrictionsIRL(X)|_P = {f|_P|f∈IRL(X)} is the Riesz tribe of all 𝓑𝓐(P)-measurable functions.

proof. As remarked before, the algebra of restrictions IRL(X)|_P = {f|P|f∈IRL(X)} is a Riesz tribe and by [20: Lemma 11.8] it is the algebra of all S(P)-measurable functions, where S(P) = {B⊆P|_χB∈IRL(X)|_P}.

Thus, we only need to prove that the σ-algebras 𝓑𝓐(P) and S(P) coincide.

For any B⊆P, B∈S(P) if, and only if, χB∈IRL(X)|_P. The latter is equivalent to saying that there exists p∈IRL(X) such that χB = p|P. Take A⊆[0,1]^X to be the Baire set 𝕍(1−p). It is easily seen that B=A∩P.

Conversely, for any set of type A∩P, with A∈𝓑𝓐([0,1]^X), χA∩P:P→{0,1} coincide with χA|_P. Since A is Baire-measurable, it follows that χA∈IRL(X) and A∩P∈S(P).

Building on the previous results, let us describe a somewhat canonical way to define a state on the coproduct starting from algebras endowed with states.

Take a sequence of pairs (IRL(X_i)/I_i, s_i)_i∈I. With the same notation as before, let IRL(X)/J be the free product of the sequence. We have X = ∪i Xi, the X_i are chosen to be pairwise disjoint, IRL(X)/J ≃ IRL(X)|_𝕍(J). When it is clear from the context, we denote by π_xi the x_i-coordinate projection in either X-coordinates or X_i-coordinates, that is, π_xi:[0,1]^X→[0,1] and π_xi[0,1]^X_i→[0,1], with x_i∈X_i⊆X.

Following the notation set out in [7] and briefly recalled in Section 2, we denote by Baire^ω the full subcategory of IRL whose objects are Baire subsets of hypercubes of type [0,1]^X, for a countable X. Note that the duality given in [7] for finitely presented algebras is straightforwardly extended to a duality between Baire^ω and RMVσω. More precisely, the duality is between Baire^ω and the category of presentations for algebras in RMVσω. Nonetheless, this category of presentations is equivalent to RMVσω, see [4 Remark 4.7]MSC. Thus, the coproduct of a sequence IRL(X_i)}/I_i ∈ RMVσωis reflected in a product in the dual category Baire^ω.

Hence, to a sequence of algebras IRL(Xi)/Xi∈RMVσω we can naturally associate the sequence of measure spaces (𝕍(I_i), 𝒮(𝕍(I_i)), and to their coproduct we can associate the space (𝕍(J), 𝒮(𝕍(J))).

Lemma 4.7

Let I be a countable set and let {X_i}_i∈Ibe a pairwise disjoint sequence of sets of variables. For any i ∈ I, let f_i ∈ IRL(X_i) be an IRL-polynomial, letX=∪iXiand let J be the σ-ideal generated by the sequence {f_i}_i∈Iin [0, 1]^X. Then 𝕍(J) is the product inBaire^ωof the objects 𝕍(f_i), where the projection from 𝕍(J) to 𝕍(f_i) is induced by the inclusion ofX_iin X.

p r o o f. By definition, a point x ∈ [0, 1]^X belongs to 𝕍(J) if, and only if, f_i(x) = 0 for all i ∈ I. Since we have chosen the X_i’s to be pairwise disjoint, it follows that x ∈ 𝕍(J) if, and only if, (π_xi(x))_{xi ∈ Xi} ∈ 𝕍(f_i). Thus, 𝕍(J) is actually the set of those points in [0, 1]^X such that the ‘X_i-coordinates’ form a point of 𝕍(f_i), that is, their cartesian product.

For brevity, let us denote 𝕍(J) by V and 𝕍(f_i) by V_i. For any i ∈ I, let η_i: V ⊆ [0, 1]^X → V_i ⊆ [0, 1]^X_i be defined by η_i(x) = (π_xi(x))_{xi ∈ Xi}, where π_xi[0, 1]^X_i → [0, 1] are the coordinate projections that generate IRL(X_i). This is well defined because the inclusion X_i ⊆ X allows to think of IRL(X_i) as embedded in IRL(X). Hence, by definition, each η_i is an arrow in Baire^ω since it is a tuple of IRL-polynomials.

Take now any other W ∈ Baire^ω, W ⊆ [0, 1]^Y such that for any i ∈ I there exists ν_i:W → V_i. Thus, by definition of arrows, ν_i(w) = (ν_xi(w))_{xi ∈ Xi} for some ν_xi ∈ IRL(Y).

Define ν: W → V by ν(w)=(νx(w))x∈∪iXi. We note that ν is an IRL-map, moreover it is well defined since the X_i-coordinates of the point ν(w) belong to V_i, making ν(w) a point in V. Furthermore, for any i ∈ I, η_i(ν(w)) = (π_xi(ν(w)))_{xi ∈ Xi} = (ν_xi(w))_{xi ∈ Xi} = ν_i(w). Consequently, (V, η_i) is the product of the V_i in Baire^ω.

For any collection of σ-algebras {𝓑_i}_i∈I, we denote by ×iBi the product σ-algebra generated by ∏iAi,Ai∈Bi. When I is an infinite countable set, we require that A_i coincides with the whole space for all but a finite number of indexes.

Lemma 4.8

With the same notations of the previous lemma, the Boolean center 𝒮(V) of IRL(X)|_Vis the product σ-algebra of the Boolean centers 𝒮(V_i) of IRL(X_i)|V_i.

p r o o f. We shall use the characterization of the Boolean center given in Lemma 4.6, as well as the notation fixed there.

Let us first remark that, since each X_i is countable, [0, 1]^X_i is a compact and metric space, whence it is separable. Thus, we can apply [16: Lemma 1.2] and deduce that BA([0, 1]X)=×iBA([0, 1]Xi). Therefore sets of type ∏iBi, with B_i ∈ 𝓑𝓐([0, 1]^X_i) are generators for 𝓑𝓐([0, 1]^X). Furthermore, since V ∈ 𝓑𝓐([0, 1]^X) it is easily seen that (∏iBi)∩V is a set of generators for 𝓑𝓐(V), where ∏iBi is defined as before.

The conclusion is now straightforward since, by Lemma 4.7, V=∏iVi. Indeed, take A_i = B_i ∩ V_i with B_i ∈ 𝓑𝓐([0, 1]^X_i). From the previous remarks we have that the product ∏iAi is a generator of ×iS(Vi), the product ∏iBi is a generator for 𝓑𝓐([0, 1]^X) and the intersection ∏iBi∩V is a generator for 𝓑𝓐(V).

Since ∏iAi=∏i(Bi∩Vi)=∏iBi∩∏iVi=∏iBi∩V, we can see that the sets of generators of XiS(Vi) and S(V) are the same, and the two σ algebras coincide.

We can now build on the previous lemmas, using the same notation set out before. Assume that each IRL(X_i)/I_i/I_i ≃ IRL(X_i)|_Vi carries a state s_i. Then, we can naturally associate the measure space (V_i,S(V_i),μ_i), where μ_i is the probability measure defined by μ_i(A)= s_i(χ_A). Take now the usual product measure μ of the μi and define s: IRL(X)_𝕍(J) →[0, 1] as the integral with respect to μ.

Definition 4.9

We call coproduct presentable any tribe 𝒯 that can be obtained as a coproduct of a sequence of algebras in RMVσω. If, in addition, each algebra of the sequence carries a σ-state, we call coproduct state the state obtained via the product measure, as described above.

With the same notations of the previous lemmas, for any finite set of indexes i₁,…,i_k ∈ I, define the map

where the product · is the usual ring-product of real-valued functions. We note that β is a k-additive function and it is also well defined, since each a_ij can be embedded in IRL(X)|_V and the product of 𝓑𝒜(V)-measurable functions is 𝓑𝒜(V)-measurable, thus it does belong to IRL(X)|V.

The next theorem shows how, in the finite case, the summands of the coproduct are independent with respect to the coproduct state.

Theorem 4.10

With the notation fixed above, if |I|ω, the σ-complete Riesz MV-algebras {(IRL(X_i)}_Vi,s_i)}_{i ∈ I}are (IRL(X)|V,s)-independent.

Proof

Take β as defined in equation (1). Take any subset of {(IRL(X_i)|V_i,s_i)}_{i ∈ I} of cardinality k ∈ ℕ. Let h ∈ ℕ be another integer such that |I| =k+h and assume that I = {i₁,…,i_k} ∪ {j₁,…,j_h}. Furthermore, let 1V_j1,…,1V_jh denote the functions that are identically equal to 1 on V_j1,…,V_jh. We remark that, by definition of s and β, the measure associated to s is the product measure of the μ_i’s, and each μ_i is a finite measure. Thus, we can apply (iteratively) the well known Fubini-Tonelli theorem (see, e.g. [14Section 36]Halmos): for any a_i1 ∈ IRL(X_i1)|V_i1,…,a_ik ∈ IRL(X_ik)|V_ik,

settling the claim.

Definition 4.11

Let A∈RMVσω. Take the coproduct ⨁ωA of A with itself countably many times. A σ-state s:⨁ωA→[0,1] is called weakly exchangeable if the associate measure, on the product of countable copies of (𝕍(I),S(𝕍(I)), is exchangeable in the classical sense. Similarly, the σ-state s is called weakly presentable if the associated measure is presentable in the classical sense.

As a straightforward remark, the coproduct state is weakly exchangeable.

Lemma 4.12

For any n ≤ ωand anyA∈RMVσω, endowed with aσ-state s: A → [0,1], the coproduct state on⨁i=1nAi, whereAi ≃ Afor anyi, is weakly exchangeable.

Proof

It is a consequence of the fact that a product measure on a countable power Vⁿ is exchangeable, see [15].

The next theorem is our weak version of de Finetti’s theorem for states, which depends upon the classical result.

Theorem 4.13

(Weak de Finetti’s exchangeability). Let X be a countable set. A state on IRL(X) is weakly exchangeable if, and only if, it is weakly presentable.

Proof

Since IRL(X) is presented by the trivial ideal, for any state of ⨁ωIRL(X), the associated measure is defined on countable copies of 𝓑𝒜([0,1]^X). Then, since [0,1]^X is a compact and Hausdorff space, we apply Theorem 2.4.

We close this section with some remarks on these notions. While it is somewhat natural to define independence directly on both stochastic processes and sequences of tribes, and it is also natural to define exchangeability of processes as done in Definition 3.8, it is still unclear how to define “strongly” exchangeable and “strongly” presentable states is in this framework, where the adverb strongly is to be intended as “without reference to the associated measure”. Since Definition 4.11 is completely dependent on the associated measure, it will be interesting to understand if and how we can adjust the definition in this sense.

To give more strength to our results, one possibility is to explore states in a categorial setting. With this aim in mind, in the next section we extend the Butnariu-Klement and Kroupa-Panti representations for σ-states and states, into categorical dualities.

Proposition 5.1

Let f(X,μ_X) → (Y,μ_Y) be morphism inBDKH, that is, a continuous and cozero closed function. Letf̃(C(Y),sY) → (C(X),sX) be its corresponding homomorphism inRMVσ. Thenμ_Y = μ_X ⚬ f^–1if, and only if, s_Y = s_X ⚬ f̃.

Proof

By hypothesis f(X,μ_X) → (Y,μ_Y) is continuous, cozero-closed and measure preserving and μ_Y is the pushforward measure of μ_X with respect to f. Hence, by the property of change of variables of pushforward measures,

which settles one direction of the claim.

On the other hand, assume f̃ is a measure preserving σ-homomorphism. That is, for any g ∈ C(Y), s_X(f̃(g))=s_Y(g). Than, as before,

By the Kroupa-Panti theorem, the regular measure associated to s_Y is unique. Thus, if we prove that μ_X ⚬ f^–1 is a regular measure, we can infer that μ_Y = μ_X Π f^–1.

Following [13], we recall that (since Y is a compact space) it is enough to prove that μ_X ⚬ f^–1 is inner regular, that is, for any E Borel subset of Y,

Let 𝒦={K ⊆ Y|K⊆E,Kcompact}. One inequality of equation 2 follows from the fact that for any K ∈ 𝒦, (μ_X ⚬ f^–1)(K) ≤ (μ_X ⚬ f^–1)(E), whence (μ_X ⚬ f^–1)(E) is greater or equal of their supremum, that is,

Let us prove the other inequality.

To do so, let

and

For any C ∈ 𝓗, f(C)⊆E is a compact set. Hence f^–1(f(C))∈ 𝒟. Moreover, C ⊆ f−1(f(C)) implies that μ_X(C) ≤ μ_X(f^–1(f(C))). Consequently,

Corollary 5.2

The functor

yields a duality between pBDKH and pRMVV _σ .

Proof. The restriction of Kakutani's duality proved in [7] and the Kroupa-Panti theorem imply that the functor is well defined and essentially surjective. It is full and faithful by the same duality of [7] and Theorem 5.1 which implies that measure-preserving morphisms in BDKH correspond to state-preserving morphisms in RMV_σ.

Remark 5.3

Despite RMV_σ being closed under coproducts, we notice that BDKH, the category dual to the whole RMV_σ is not closed under the usual product of topological spaces. Indeed 2 = {0,1} with the discrete topology is an object there, while 2^ω is not, since it is the Stone space of the Lindenbaum-Tarski algebra of the classical propositional calculus, which is not σ-complete. Consequently, products in BDKH do not coincide with products in Top, the category of topological spaces and continuous maps.

The duality obtained in Corollary 5.2 is built by combining two results: one is based on the classical Kakutani's duality, the other being the Kroupa-Panti integral representation. Both of these references work using the topological space of maximal MV-ideals of a σ-complete Riesz MV-algebra. Following the same ideas, the second duality is built on the integral representation of σ-states given in 1,3] , that is, Theorem 2.3 Consider the following categories:

1. p R M V σ ω is the category whose objects are triplets (IRL(X), I, sx ), where IRL(X)/I ≃ IRL(X)|V(I)∈RMVσω and s a σ-state on IRL(X)|_𝕍(I), and whose morphisms are statepreserving homomorphisms of RMVσω.

   That is, h: (IRL(X),J, s_X) → (IRL(Y),K , s_F) is induced by a unique h^*: IRL(X) → IRL(Y) such that h^*(J) ⊆ K and s_X = s_Y o h.

2. pBaire ^ω is the category whose objects are pairs (V, μ), where V ∈ Baire^ω and μ is a probability measure on 𝓑𝓐(V), the σ-algebra of Baire subsets of V. Morphisms in pBaire^ω are tuples of IRL-maps that are also measure-preserving, that is, f: (V, μ_V) → (W, μ_W) such that μ_V = μ_V o f⁻¹.

Consider now the functors Vp:pRMVσω→pBaireω and Ip:pBaireω→pRMVσω defined upon the functors and 𝒱 of 𝒥 as follows:

1. 𝒱^p(IRL(X), I, s) = (𝕍(I), μ_s), where μ_s is given by Theorem 2.3

2. for V ⊆ [0,1]^X, 𝒥^p(V, μ) = (IRL(X), 𝕀(V), s_μ), with s_μ: IRL(X)|_V → [0,1] defined by integration with respect to μ. Notice that 𝕍(𝕀(V)) = V.

3. on arrows, with the obvious notations, 𝒱^p(h) = 𝒱(h) and 𝒥^p(η) = 𝒱(η). We remark that 𝒱(η) is defined by precomposition with η.

THEOREM 5.4

The functors 𝒱^p and 𝒥^p give a duality.

Proof. Take (IRL(X),J,s)∈pRMVσω. By Lemma 4.6, IRL(X)|_𝕍(J) is the algebra of all 𝓑𝓐(𝕍(J))-measurable functions. Thus, the measure μ_s given in Theorem 2.3 is a measure on 𝓑𝓐(𝕍(J)) and consequently 𝒱^p(IRL(X), j, s) ∈ pBaire^ω. Similarly, for any (V, μ) ∈ bBaire^ω, with V ⊆ [0,1]^X, the triplet (IRL(X), 𝕀(V), s_μ) = J^p(V, μ) is an object in pRMVσω.

Furthermore, by [7: Corollary 4.17] and Theorem 2.3

Whence, in order to prove that we have a duality, it is enough to show that a state- preserving morphism in RMVσω is mapped to a measure-preserving map of Baire^ω, and viceversa.

Let h: (IRL(X), j, s_X) → (IRL(X), K, s_Y) be a morphism in RMVσω induced by some h^*. Then s_X = s_Y o h with s_X: IRL(X)|_𝕍(J) → [0,1] and s_Y: IRL(Y)|_𝕍(K) → [0,1].

Take A ∈ 𝓑𝓐(𝕍(J)). Then μ_sX(A) = s_X(𝒳_A) = s_X(h(𝒳_A)). For brevity, let η denote 𝒱^p(h). Note that it follows from [7] that the inverse functor of acts as the pre-composition with η. More precisely, for any p ∈ IRL(X)|_V, h(p) = p o η. Then, by definition of η, it follows that ?(3_A) = 3y−1(A). Indeed, ?(3_A) is again a Boolean member of IRL(Y)) |_𝕍(K) and

Then, μ_sX (A) = s_Y(𝒳_η−1(A)) = μ_sY (η⁻¹(A)), settling the claim.

Finally, we see how h(p): = p o η is state-preserving for any morphism η: (V ⊆ [0,1]^X, μ_X) → (W ⊆ [0; 1]^Y , μY ) in Baire^ω that is measure-preserving. By definition of arrows in pBaire^ω, let η : = (η_y)_y∈Y and each η_y ∈ IRL(X). We first remark that η is measurable between (V, 𝓑𝓐(V)) and (W, 𝓑𝓐(W)). Indeed, for any generator E of 𝓑𝓐(W), E=W∩∏y∈YAy with A_y ∈ 𝓑𝓐([0,1]), see [16 Lemma 1.2]. Then

Since each η_y belongs to IRL(X), it follows that ⋂y∈Yηy−1(Ay)∈BA([0,1]X) and V∩⋂y∈Yηy−1(Ay)∈

𝓑𝓐(V), making η a measurable function.

Now, using the property of change of variables of the pushforward measure, which is μ_Y by hypothesis, we have

which settles the claim.