Embedding an analytic equivalence relation in the transitive closure of a Borel relation

The transitive closure of a reflexive, symmetric, analytic relation is an analytic equivalence relation. Does some smaller class contain the transitive closure of every reflexive, symmetric, closed relation? An essentially negative answer is provided here. Every analytic equivalence relation on an arbitrary Polish space is Borel embeddable in the transitive closure of the union of two smooth Borel equivalence relations on that space. In the case of the Baire space, the two smooth relations are closed and the embedding is homeomorphic.


Introduction
This note answers a question in descriptive set theory that arises in the context of the Bayesian theory of decisions and games. It concerns the notion of common knowledge, formalized by Robert Aumann [1976]. For an event A that is represented as a subset of a measurable space Ω, Aumann defines the event that an agent knows A to be the event A\ [Ω\ A] P , where P is the agent's information partition of Ω. 1 If P is the meet of individual agents' information partitions (in the lattice of partitions where P ′ ≤ P ′′ ⇐⇒ P ′′ refines P ′ ), then Aumann defines to be the event that A is common knowledge among the agents. 2 Aumann restricts attention to the case that Ω is countable (or that the Borel σ-algebra on Ω is generated by the elements of a countable partition), so that measurability issues do not arise. But, otherwise, the passage from information partitions to a common-knowledge partition is very badly behaved, as is the passage from an information partition P and an event A to the related event that A is known according to P. For example, let X be an arbitrary subset of (0, 1), and let Ω = [0, 2]. Consider two agents, whose information partitions are P 1 = {{ω, ω +1} | Date: 2012.03.07. 2010 Mathematics Subject Classification. 03E02. The author thanks the Human Capital Foundation, (http://www.hcfoundation.ru/en/) and particularly Andrey P. Vavilov, for research support through the Center for the Study of Auctions, Procurements, and Competition Policy (CAPCP, http://capcp.psu.edu/) at the Pennsylvania State University. Also he gratefully acknowledges the hospitality of CEMFI (http://www.cemfi.es/) where the research was conducted.
1 [A] P denotes {π | π ∈ P and π ∩ A = ∅}, the saturation of A with respect to P. If E is an equivalence relation, then [A] E denotes the saturation of A with respect to the partition induced by E. Aumann's definition corresponds to the truth condition for A in Kripke [1959]. 2 Aumann sketches an argument-reminiscent of a general principle in proof theory (cf. Pohlers [2009, Lemma 6.4.8, p. 89])-that this definition is equivalent to the intuitive, recursive definition of common knowledge: that A has occurred and that, for all n ∈ N, both agents know. . . that both agents know (n times) that A has occurred. ω ∈ X} ∪ {{ω} | ω / ∈ X} and P 2 = {{ω} | ω < 1} ∪ {[1, 2]}. Then X ∪ [1, 2] is the block of the common-knowledge partition that includes the block [1,2] in P 2 . From information partitions composed of the simplest events-singletons, pairs, and a closed interval-we have passed to a common-knowledge partition with a block that, depending on what is X, might even be outside the projective hierarchy. Knowledge of an event by individual agent is likewise problematic. In the present example, the event that agent 1 knows (0, 1) is (0, 1) \ X.
These measurability problems dictate that information partitions should be represented as equivalence relations. If E 1 and E 2 are Σ 1 1 (that is, analytic) equivalence relations, then the meet of the partitions that they induce is induced by the transitive closure of their union. This transitive closure is also a Σ 1 1 equivalence relation. 3 Moreover, if an information partition is represented by a Σ 1 1 relation and an event is Π 1 1 (that is, co-analytic), then knowledge of the event according to the information partition is also a Π 1 1 event. 4 This observation implies that knowledge of a Borel event is a universally measurable event-surely a threshold condition for incorporating the analysis of knowledge into Bayesian theory. However, it is easy to envision a noncooperative Bayesian game situation in which a player must condition on the event that some other event is common knowledge. 5 Then the game theorist needs to model the event that both players' receipt of their respective signals is common knowledge as being a ∆ 1 1 (that is, Borel) event, not just a Π 1 1 event. It might be thought possible to avoid this difficulty by making tighter modeling assumptions regarding both the agents' information structures and also the event to which the common-knowledge operator is to be applied.
In particular, in most applications to Bayesian decision theory and game theory, it is reasonable to specify each agent's information as a ∆ 1 1 equivalence relation, or even as a smooth or closed Borel relation rather than as an arbitrary Σ 1 1 equivalence 3 Composition is defined with a single existential quantifier, and thus takes a pair of Σ 1 1 relations to a Σ 1 1 relation. The countable union of Σ 1 1 relations is Σ 1 1 . Cf. Moschovakis [2009, Theorem 2B.2, p. 54].
4 This is equivalent, by (1.1), to the fact that the saturation of a Σ 1 1 set with respect to a Σ 1 1 equivalence relation is Σ 1 1 . This latter fact is true because the saturation is defined with a single extential quantifier.
5 In a game-theoretic analysis of the "coordinated-attack problem" (cf. Rubinstein [1989]), there is an equilibrium in which two players will take complementary actions iff it is common knowledge that each has received a signal. The two signals are privately observed by the respective players and are highly, but imperfectly, correlated. Thus the players must exchange infinitely many messages to one another (that is, 'I have received the signal', 'I have received your confirmation of receipt of the signal', 'I have received your confirmation of receipt of my confirmation of receipt of the signal',. . . ) in order to attain common knowledge. If the communications channel closes at any stage, then common knowledge is not reached, and the action will not be taken. Closure of the channel is irreversible. The game can be elaborated by supposing that the stochastic distribution of payoffs from coordination is dependent on whether common knowledge has been achieved. (For example, the communication channel may have to be used, and therefore must not be broken, in order to accomplish the task. The probability that the channel being open at that time is positive if common knowledge has been achieved, but is zero otherwise.) Thus, in order for the game theorist to prove that coordination conditional on attaining common knowledge is an equilibrium, the expected payoff from coordination conditional on common knowledge being attained must be well defined.
relation. 6 Thus it may be asked: if the graphs of E 1 and E 2 are in ∆ 1 1 or in some smaller class, then how is the graph of the transitive closure of E 1 ∪ E 2 restricted?
It will be shown here that no significant restriction of the common-knowledge partition is implied by such restriction of agents' information partitions. This finding is not surprising, since restricting the complexity of individuals' equivalence relations does not obviate the use of an existential quantifier to define the transitive closure of a relation. Nevertheless, the syntactic form of a specific description of a set does not determine the intrinsic complexity of the set, so it needs to be shown that common-knowledge equivalence relations derived from Borel equivalence relations are not lower in the projective hierarchy, as a class, than their definition would suggest. Moreover, proposition 2.1 will show that being the union of finitely many (in fact, of fewer that 2 ℵ0 ) Borel equivalence relations-that is, representability in the form, of which the transitive closure is an equivalence relation specifying common-knowledge-is a stronger property than being an arbitrary Borel, reflexive, symmetric relation.
To define the transitive closure of R ⊆ Ω × Ω, let R (1) = R and R (n+1) = RR (n) (that is, the composition of relations R and R (n) ). Denote the transitive closure of R by R + = n∈N+ R (n) . It will be proved here that, if Ω is a Polish space and E 0 ⊂ Ω × Ω is a Σ 1 1 equivalence relation, then there are smooth ∆ 1 1 equivalence relations E 1 and E 2 and a ∆ 1 If Ω is the Baire space, then E 1 and E 2 can be taken to be closed, Z can be taken to be open, and the Borel equivalence can be taken to be a homeomorphic equivalence.

The case of the Baire space
First take Ω to be the Baire space, N = N N . 8 Define subsets X and Y of N by X = {α|α 0 > 0} and Y = {α|α 0 = 0}. X and Y are both homeomorphic to N , and homeomorphisms f : Each of X and Y is both open and closed in N . It follows that, if Z is either X or Y , then A ⊆ Z is open (resp. closed, Borel, Σ 1 1 ) as a subset of A iff it is open (resp. closed, Borel, Σ 1 1 ) as a subset of Z. This invariance to the ambient space extends to product spaces. (For example a subset of X × Y is closed in X × Y iff it is closed in N × N .) In subsequent discussions, subsets of these subspaces will be characterized (for example, as being closed) without mentioning the subspace.
1 equivalence relation, then there are equivalence relations I and J on N × N , each of which has a closed graph, such that E = (I ∪ J) + ↾ X.
Before proceeding to the proof of this theorem, note that I ∪ J is a closed, reflexive, symmetric relation. Thus, theorem 2.1 has the following corollary.
6 Smoothness (also called tameness) and closedness are co-extensive for equivalence relations on standard Borel spaces. Cf. Harrington et al. [1990, proof of Theorem 1.1, p. 920]. Standard Borel spaces are defined below, in section 4.
. Let restriction take precedence over Boolean operations. For example, N is topologized as the product of discrete spaces. 9 Since Y is homeomorphic with N , g can be constructed from the function described by Moschovakis [2009, p. 31].
Corollary 2.2. If E ⊆ X × X is a Σ 1 1 equivalence relation, then there is a closed, reflexive, symmetric relation R on N × N , such that E = R + ↾ X.
Theorem 2.1 would follow from corollary 2.2 if every closed, reflexive, symmetric relation were the union of two closed equivalence relations, but that is not the case.
R = E; every E ∈ E is an equivalence relation; R is closed, reflexive, and symmetric; and 2 ℵ0 is the cardinality of E. There is no other set F of equivalence relations such that R = F . Thus, R is not a union of fewer that 2 ℵ0 equivalence relations.
Proof. The assertions regarding E are obvious from its construction. To obtain a contradiction from supposing that E were not unique, suppose that R were also the union of a set F = E of equivalence relations. Not F E. So, there must be some E ∈ F \ E. By symmetry, there must be three distinct points, α, β, γ such that

Proof of the theorem
Denote the diagonal (that is, identity) relation in N ×N by D = {(α, α)|α ∈ N }. D is closed.
Recall that a relation E ⊆ X × X is Σ 1 1 iff there is a set F such that Proof. Let F 0 satisfy (3.1). Let h be a homeomorphism from N to X, and define F 1 ⊆ X × X × X by (α, β, γ) ∈ F 0 ⇐⇒ (α, β, h(γ)) ∈ F 1 . F 1 also satisfies (3.1), then, and it is closed. By symmetry of E, F 1 is another closed set that satisfies (3.1). Consequently, F = F 1 ∪ F 1 is a t 12 -invariant closed set that satisfies (3.1).
Let i denote the identity function on N . If K, L, M, N are any sets, and p : K → L and q : M → N , then denote the product mapping by p × q : K × M → L × N .
10 A sub-sequence of subscripted alphas distinct from α i and α j having subscripts that are not increasing, which occurs if i = 1 or j = i + 1 or j = k, denotes the empty sequence.
The two closed equivalence relations that theorem 2.1 asserts to exist are defined from the homeomorphisms f and g introduced in section 2, and the closed, t 12invariant set F guaranteed to exist by lemma 3.1, as follows. Proof. D is closed because N is a metric space.
The function i × j is a homeomorphism from X × X × X × X to X × Y . Being a homeomorphism, it is an open mapping (which takes closed sets to closed sets).
The image under j × j of a closed subset of its domain is therefore closed in its range.
, which is closed. H, the image of this set under j × j, is therefore closed.
J, the union of two closed sets, is closed. Proof. All assertions except the one regarding closedness of GG and I are verified by straightforward calculations. That F is invariant under t 12 is used to show that H = H and that E ⊆ GH G. The proof that GG is closed is parallel to the proof that H is closed. According to the first part of this lemma, GG = [j ×j](t 24 (D ↾ X ×X ×X ×X ×X)∩(F ×F )).
I, the union of four closed sets, is closed.
Lemma 3.4. I and J are equivalence relations.
Proof. These relations are reflexive and symmetric, so their transitive closures are equivalence relations. Thus, the lemma is equivalent to the assertion that I = I + and J = J + . For any relation K, K (2) = K is sufficient for K = K + . In the following calculations of I (2) and J (2) , composition of relations is distributed over unions. Terms that evaluate by identities that were calculated in lemma 3.3 to a previous term or its sub-relation, are omitted from the expansion by terms in the pentultimate step of each calculation.
Proof of theorem 2.1. Lemmas 3.2-3.4 show that the each of the relations I and J on N × N , is an equivalence relation that has a closed graph. It remains to be shown that that E = (I ∪ J) + ∩ (X × X). Note that, since D ⊆ I ∪ J, The following calculation shows that (I ∪ J) (5) = (I ∪ J) (6) . The calculation is done recursively, according to the following recipe at each stage n > 1: (1) Begin with the equation (I ∪ J) (n+1) = (I ∪ J)(I ∪ J) (n) .
(2) Rewrite (I ∪ J) as D ∪ G ∪ G ∪ GG ∪ H according to (3.2), rewrite (I ∪ J) (n) according to the result of the previous step, and then distribute composition of relations over union in the resulting equation.
(3) For each identity stated in lemma 3.3, and for each identity that, for some K ∈ {G, G, H}, equates a composition KD or DK of K and D (or a restriction of D to a product set of which K is a subset) to K, do as follows: Going from left to right, apply the identity wherever possible. 11 Repeat this entire step (consisting of one pass per identity) until no further simplifications are possible. (4) Delete compositions of relations that include terms KL such that the range of K and the domain of L (viewed as correspondences) are disjoint, in which case the term denotes the empty relation. Delete D ↾ X (occurring as a term by itself), of which D is a superset. (6) Reorder terms lexicographically, in the order D < D ↾ Y < E < G < G < H. Delete repeated terms.
Thus (I ∪J) + = (I ∪J) (5) . Note that D ↾ Y , G, G, H and all relations of form or GQ or HQ or QG or QH (where variable Q ranges over compositions of G, G, H, and E), are disjoint from X ×X. Therefore, from the calculation in (3.4) of (I ∪J) (5) as a union of E with such relations, it follows that (I ∪ J) + ∩ (X × X) = E.

The general case of a standard Borel space
In this concluding section, theorem 2.1 is generalized in two ways to an arbitrary standard Borel space. A standard Borel space is a pair Ω 0 = (Ω 0 , B 0 ) such that, for some pair Ω = (Ω, B), B is the σ-algebra of Borel subsets of the set Ω under some Polish topology, Ω 0 ∈ B, and B 0 = {B 0 | ∃B [B ∈ B and B 0 = B ∩ Ω 0 ]}. A Borel isomorphism of standard Borel spaces Ω 0 and Ω is a ∆ 1 1 function k : Ω 0 → Ω such that k −1 : Ω → Ω 0 exists and is also ∆ 1 1 . A ∆ 1 1 subset of a standard Borel space is also a standard Borel space, and every two uncountable standard Borel spaces are isomorphic. 12 In both generalizations, the concept of smoothness of a Borel equivalence relation substitutes for the concept of closedness that appears in theorem 2.1. If E ⊆ Ω×Ω is a ∆ 1 1 equivalence relation, and if there is a set {Y n } n∈N of ∆ 1 1 sets such that (ω, ω ′ ) ∈ E ⇐⇒ ∀n [ω ∈ Y n ⇐⇒ ω ′ ∈ Y n ], then E is a smooth equivalence relation. By Harrington et al. [1990, proof of Theorem 1.1, p. 920], every equivalence relation with closed graph is smooth. If k : Ω 0 → Ω is ∆ 1 1 and E ⊆ Ω × Ω is a smooth ∆ 1 1 equivalence relation, then E 0 ⊆ Ω 0 ×Ω 0 defined by (ψ, ω) ∈ E 0 ⇐⇒ (k(ψ), k(ω)) ∈ E is also smooth, with E 0 -equivalence determined by {k −1 (Y n )} n∈N .
12 Mackey [1957, pp. 338-9]. Henceforth, B will be implicit and the structure Ω will be identified with the set Ω on which it is defined.
The second generalization of theorem 2.1 applies to a Σ 1 1 equivalence relation that, in a sense, does not occupy the entire product space Ω × Ω. Specifically, the set of points, the singletons of which are blocks of the partition induced by the relation, must have an uncountable ∆ 1 1 subset. 14 Corollary 4.2. Suppose Ω is a standard Borel space and that E ⊆ Ω × Ω is a Σ 1 1 equivalence relation such that, for some uncountable ∆ 1 1 set B ⊆ Ω, E ↾ B = D ↾ B. Define Ω 0 = Ω \ B. Then there are smooth ∆ 1 1 relations E 1 and E 2 , such that E \ D ⊆ E ↾ Ω 0 ∪ D ↾ B Finally, corollary 4.2 provides a negative answer to the question, raised in the introduction, of whether the saturations of Borel sets (or even of singletons) with respect to the transitive closures of unions of smooth Borel equivalence relations lie within any significantly restricted sub-class of Σ 1 1 . Corollary 4.3. Suppose Ω is a standard Borel space and that S ⊆ Ω is a Σ 1 1 set such that, for some ∆ 1 1 set Ω 0 , S ⊆ Ω 0 and Ω \ Ω 0 is uncountable. Then there are smooth ∆ 1 1 relations E 1 and E 2 , such that for every non-empty A ⊆ S, [A] (E1∪E2) + ∩ Ω 0 = S.