Working Paper Series

Universidade de Vigo

http://webs.uvigo.es/rgea

______________________________________________________________________

______________________________________________________________________

Facultade de Ciencias Económicas e Empresariais, Campus As Lagoas-Marcosende,

RGEA

Working Paper Series

Information and Sigma-algebras

By Carlos Hervés-Beloso and Paulo K. Monteiro

6-12

Information and σ–algebras ^∗

Carlos Herv´ es-Beloso

RGEA. Universidad de Vigo, e-mail: [email protected]

Paulo K. Monteiro

EPGE/FGV, Rio de Janeiro, Brazil, e-mail: [email protected]

July 18, 2012

Abstract

In this work we clarify the relationship between the information that an agent receives from a signal, an experiment or from his own ability to determine the true state of nature that occurs and the infor- mation that an agent receives from a σ–algebra. We show for count- ably generated σ–algebras that the larger it is, larger is the informa- tion. The same is true for general σ–algebras after the removal of a negligible set of states.

Keywords: Signals, Partitions, σ–algebras, Measurability, Information mo- dels.

JEL Classification: C60, C70.

1 Introduction

In economic models that consider information, individual or collective, asym- metric or not, an informative signal, or equivalently, an information partition of the set of states of nature, is a primitive concept. From this a most natural

∗We thank the comments from the participants at Naples II Workshop on Equilibrium Analysis Under Ambiguity, 2011, SAET-2011 and Exeter workshop in honor of Cuong Le Van, 2011. In particular we gratefully acknowledge the comments of A. Citanna, B. Cor- net, J. Dubra, F. Echenique, M. Grandmont, M. Greinecker, F. Maccheroni, J.P. Torres- Mart´ınez and N. Yannelis. Carlos thanks the partial support of Research Grants ECO2009- 14457-C04-01 (Ministerio de Ciencia e Innovaci´on) and 10PXIB300141PR, RGEA (Xunta de Galicia and FEDER). Paulo acknowledges the financial support of CNPq, Brazil.

second step is to consider the σ–algebra generated by the partition as the in- formation σ–algebra. However, P. Billingsley [1], J. Dubra and F. Echenique [3] argued that the use of σ–algebras as the informational content of a sig- nal or a partition is frequently inadequate. Billingsley’s concern is based on the fact that sometimes the σ–algebra generated by the partition does not correspond to the heuristic equating of information. In J. Dubra and F.

Echenique’s paper “Information is not about measurability”, [3], it is shown by an example that the use of the σ–algebra generated by a partition, as a model of information, leads to a paradoxical conclusion: the decision-maker could prefer less information than more. This comes from the fact that finer partitions may not generate finer σ–algebras.

Consequently, the concern set either by Billingsley or by Dubra and Echenique regarding the use of σ–algebras and measure theory when working with information motivates us to elaborate on the meaning of information.

In this note we give a precise definition of the informational content of a signal (or a partition). We analyze the immediate consequences of the definition and elaborate on the concerns stated by Billingsley and Dubra and Echenique.

Our definition, which is a very natural interpretation of the informational content of a signal, is based on the primitive concept of the informed set. The family of informed sets is itself a σ–algebra, and therefore, one should not question whether to use σ–algebras as a tool in terms of whether they are more or less adequate, since the character of σ–algebra is inherent to the collection of the informed sets.

Next, we study the properties of the σ–algebra of the informed sets, show- ing that it is closed under arbitrary unions and, that it coincides with the σ–algebra generated by the partition, if and only if, the partition is a finite or countable set. We show that finer partitions generate finer σ–algebras of informed sets, and conversely, finer σ–algebras of informed sets come from finer partitions.

As a consequence of our definition, the concerns raised by Billingsley and by Dubra and Echenique have an adequate conceptual explanation and the pervasive intuition that larger σ–algebras have more informational content is true for the σ–algebra of the informed sets. For completeness we recast Theorem A in J. Dubra and F. Echenique (2004). Now, in their Blackwell type theorem the equivalence of (1) with (5) and (6) can be written in a natural way regarding the σ–algebra of the informed sets.

We next elaborate on what can be said in more general models, where the primitive are σ–algebras instead of partitions. We introduce the concept of a strongly Blackwell σ–algebra and note that it is a quite general property. We then show that for countably generated sub–σ–algebras of a strongly Black-

well σ–algebra, that the correspondence between finer σ–algebras and finer partitions is perfect. Finally if G is an arbitrary sub–σ–algebra of a strongly Blackwell probability space the partition of G is defined as the partition of any countably generated σ–algebra equivalent to G (modulo null sets.) In this way larger σ–algebras correspond to larger partitions.

The paper is structured as follows: In section 2 we present the definition of partitions and signals. In section 3 we review the examples by Billingsley and Dubra-Echenique, and in section 4 we set our main definition and elaborate on its consequences. In subsection 4.5 we restate Dubra and Echenique’s Blackwell-type theorem. In section 5 we analyze the case when information is given by a σ–algebra instead of partitions and in section 6 we set our conclusions.

2 Definitions

Let Ω 6= ∅ be the set of states of nature. The set of subsets of Ω is P(Ω) :=

{A : A ⊂ Ω}.

Definition 1 (partitions) A partition of Ω is a family of non-empty sub- sets of Ω, τ , τ ⊂ P(Ω) \ {∅}, such that

1. S{X : X ∈ τ } = Ω;

2. If X, Y ∈ τ and X 6= Y then X ∩ Y = ∅.

Thus if ω ∈ Ω there is one and only one element of τ that contains ω. We denote this element τ (ω) and say that it is the atom of τ that contains ω. If τ and τ⁰ are partitions of Ω, we denote by τ⁰ ≥ τ the property that τ⁰ is finer than τ . That is, every element of τ is a union of elements of τ⁰. Formally, τ⁰ ≥ τ if for all X ∈ τ , and for all z ∈ X there exists Y ∈ τ⁰, Y ⊂ X with z ∈ Y . If τ⁰ ≥ τ we also say that τ is coarser than τ⁰. The finest partition is {{ω} : ω ∈ Ω} and the coarsest partition is {Ω}.

Definition 2 (join and meet) If τ and η are partitions of Ω the joint of τ and η, denoted τ ∨ η, is the coarsest partition that is finer than τ and η.

And the meet of τ and η, denoted τ ∧ η, is the finest partition that is coarser that τ and η.

It is easy to see that τ ∨ η = {C ∩ D : C ∈ τ, D ∈ η, C ∩ D 6= ∅}. More- over if we have a family of partitions, τi, i ∈ I then

∨_i∈Iτ_i = {∩_i∈IC_i : C ∈ Π_i∈Iτ_i, ∩_i∈IC_i 6= ∅} .

Thus if ω ∈ Ω the atom of ∨_i∈Iτ_i that contains ω is ∩_i∈Iτ_i(ω). The following characterization of ∧_iτ_i is from Rohlin¹ (page 32):

Lemma 1 Two points x⁰ and x⁰⁰ belongs to the same atom of ∧_i∈Iτ_i if and only if there is a sequence

x⁰ = x₀, x₁, x₂, . . . , x_n, x_n+1 = x⁰⁰

such that for every 0 ≤ m ≤ n, xm and xm+1 belongs to the same atom of τi

for some i ∈ I.

Definition 3 (signal) A mapping f : Ω → S is a signal on Ω with values on S.

The signal f induces a partition of Ω, defined by τ_f = {f⁻¹(s) : s ∈ S}.

Conversely, any partition τ can be seen as the partition induced by a signal.

To see this, define the equivalence relation

ω ∼ ω⁰ if and only if τ (ω) = τ (ω⁰),

where τ (z) denotes the atom of τ containing z. Let Ω/∼ be the quotient set;

that is τ = Ω/∼ and define

f : Ω → τ = Ω/∼

as the natural projection f (ω) = τ (ω). It is clear that the partition induced by the signal f is, precisely, τ_f = τ .

We now consider partitions induced by σ–algebras. Given a set G ⊂ Ω, let us denote by 1_G the characteristic function of G, defined by 1_G(z) = 1 if z ∈ G and 1_G(z) = 0 otherwise.

Definition 4 Let G be a σ–algebra on Ω. We define an equivalence relation on Ω as follows

ω⁰ ≡ ω⁰⁰(mod G) ⇐⇒ 1_G(ω⁰) = 1_G(ω⁰⁰) , ∀G ∈ G.

The equivalence classes of this equivalence relation are the G atoms. It is easy to check that

atom (ω⁰, G) := {ω⁰⁰ : ω⁰⁰ ≡ ω⁰(mod G)} = ∩ {G ∈ G : ω⁰ ∈ G} . The partition determined by the mod G equivalence relation is denoted

τ (G) := {atom (ω, G) : ω ∈ Ω} .

1V. A. Rohlin appears in A.M.S. Translation (71). Later it is written V. A. Rohklin.

In Russian it is writtenV. A. Rohlin.

Definition 5 (countably generated σ–algebra) A σ–algebra G on Ω is countably generated if there is a countable subset U = {G_n: n ≥ 1} ⊂ G such that the sigma algebra generated by U , σ (U ) = G.

Definition 6 (separable σ–algebra) The σ–algebra G on Ω is separable if it is countably generated and contains the singletons: {ω} ∈ G for every ω ∈ Ω.

That is, a countably generated σ–algebra is generated by a countable set of events. The following lemma shows that countably generated σ–algebras are quite common. For a proof we refer the reader to Hoffmann–Jørgensen (1994) page 66, exerc. 1.14 and page 91, exerc. 1.80, 1.81. Alternatively see lemma 3.2.3 of [11].

Lemma 2 a) The Borelean σ–algebra of a separable metric space is separa- ble.

b) A countably generated σ–algebra can be written as the inverse image f⁻¹(C) of a measurable function² f : Ω → C where C is a separa- ble metric space and C its Boreleans.

c) Reciprocally if f : Ω → C is a function and C a separable metric space then if C is the Boreleans σ–algebra of C then the σ–algebra f⁻¹(C) is countably generated.

The following lemma shows that atoms of countably generated σ–algebras are measurable. The proof in Parthasarathy ([7], theorem 2.1 page 132) is elegant. We give another proof for the convenience of the reader.

Lemma 3 (i) If G is a countably generated σ–algebra then τ (G) ⊂ G. (ii) Moreover if {G_n : n ≥ 1} generates G then defining for a given ω ∈ Ω, H_n= G_n if ω ∈ G_n and H_n= G^c_n if ω /∈ G_n,

atom (ω, G) = \

n≥1

H_n. (*)

Proof. It suffices to prove that atom (ω, G) is G measurable for every ω ∈ Ω.

Thus, let us fix ω ∈ Ω. Then, since H_n ∈ G for every n it follows that atom (ω, G) ⊂T

n≥1H_n. Let ω⁰ ∈T

n≥1H_n. We now consider the set W = {G ∈ G : ω ∈ G ⇐⇒ ω⁰ ∈ G} .

This set is a σ–algebra that contains {G_n : n ≥ 1} and therefore W = G.

Therefore ω⁰ ∈ atom (ω, G).

2Without loss of generality we may take C = R.

3 Key examples

Let the state of the world be a real number between 0 and 1, so the set of possible states is Ω = [0, 1] and suppose that the state of the world is uni- formly distributed. Thus, we consider the probability space (Ω, B, λ) where B = B([0, 1]) are the Boreleans and λ is the Lebesgue measure on [0, 1].

Example 1 (Billingsley) Let τ = {{ω}, ω ∈ Ω} be the finest possible par- tition. The σ–algebra generated by τ is

G := σ(τ ) = {A ⊂ Ω : A or A^c is countable}.

Billingsley³ ([1]) notes the following

(i) G contains no information about H ∈ B–in the sense that B and G are independent;

(ii) G contains all the information about H–in the sense that it tells the observer exactly which ω was drawn.

Since (i) and (ii) are in intuitive opposition, Billingsley comments:

The source of the difficulty or apparent paradox here lies in the unnatural structure of the σ-field G rather than in any deficiency in the notion of independence. The heuristic equating of σ-fields and information is helpful even though it sometimes breaks down, and of course proofs are indifferent to whatever illusions and va- garies brought them into existence.

Let us now focus on Dubra and Echenique’s example.

Example 2 (Dubra and Echenique) Suppose that a decision-maker can choose either to be perfectly informed (i.e. he uses partition τ ), or only be told if the true state ω is smaller or not smaller than 1/2 . In the second case, the information is represented by the partition

τ⁰ =n [0,1

2); [1 2, 1]o

. The σ–algebra generated by τ⁰ is

σ(τ⁰) = n

∅, [0,1 2), [1

2, 1], Ωo .

3On example 4.9, page 58

Observe that τ is finer than τ⁰, but σ(τ ) and σ(τ⁰) are not comparable. Con- sider⁴ now a decision-maker who is a risk-neutral expected utility maximizer.

He has a choice to either buy a bond with a fixed return of ³₈ or a stock with returns S(ω) = ω. If he uses σ(τ ) his expected utility is ¹₂. However, if he uses σ(τ⁰), his expected utility is ₁₆⁹ > ¹₂.

Consequently, Dubra and Echenique write:

We do not argue that using σ–algebras as the informational con- tent of signals (partitions) is always inappropriate. We only want to emphasize that one should be careful when using σ–algebras as the informational content of signals (or partitions).

To exemplify the perils that Dubra and Echenique want alert us to, we first mention the following setup from Herv´es-Beloso, Martins-da-Rocha and Monteiro.

Example 3 In ([4]) we begin our model description with the assumption:

Each agent i = 1, . . . , I knows at the initial period t = 0 that at the next period t = 1 he will have an incomplete and private information in the sense that he will only observe the outcome of random variables measurable with respect to a sub-σ–algebra F_i of the given σ–algebra F .

Thus, if agent i could use another σ–algebra eF_i ⊃ F_i, would it be a good idea to use eFi?

The work by Yannelis ([9]) is mentioned by Dubra and Echenique to confirm the relevance of their warning. Item (3) on page 186 of [9]–the definition of the differential information exchange economy–states

(3) Fi is a (measurable) partition^† of (Ω, F) denoting the private information of agent i.

Footnote (†) states “In the sequel by an abuse of notation, we will still denote by an F_i, the σ–algebra that the partition F_i generates”. Yannelis is (implicitly) considering only countable partitions. However, and more importantly, considering the correct informational content of the partition, our results show that these concerns are irrelevant, because the results apply for countable and uncountable partitions.

4The reader will excuse us for not repeating the calculations done on page 178 of [3].

4 Main definition

Consider a signal f : Ω → S or the corresponding partition τ_f of Ω.

Definition 7 (informed set) A set A ⊂ Ω is an event or an informed set with respect to the information given by τ_f if

∀X ∈ τ_f, X ⊂ A or X ⊂ A^c.

Equivalently, A is an informed set with respect to the signal f , if for every s ∈ S,

f⁻¹(s) ⊂ A or f⁻¹(s) ⊂ A^c.

From now on, let us denote by I(τ_f) the family of sets informed by the partition τ_f, or equivalently the family of sets informed by the signal f.

Interpretation and examples

An informed set has a very natural meaning. An informed set is a set A ⊂ Ω such that, for every X ∈ τ_f if X occurs then necessarily A or necessarily A^c occurs.

On the other hand, A is not an informed set, if and only if, there exist ω ∈ A and ω⁰ ∈ A^c such that f (ω) = f (ω⁰).

If ω occurs, the decision-maker receives the image of the signal f (ω) = f (ω⁰), which corresponds to ω and also to ω⁰ and he does not know if A occurs or not.

Example 4 Let Ω = [0, 1]. As in example 2, if τ = {{ω}, ω ∈ Ω}, then the family of the informed sets

I(τ ) = P(Ω)(which is different from σ(τ )) On the other hand, if τ⁰ = {[0,¹₂); [¹₂, 1]},

I(τ⁰) = {∅; Ω; [0, 1/2); [1/2, 1]}(which is equal to σ(τ⁰)).

4.1 The σ–algebra of informed sets

The following theorem emphasizes that I(τf) (the informational content of a signal f , or equivalently, of the partition τ_f) is itself a σ–algebra.

Theorem 1 The family of events I(τ_f) is a σ–algebra that contains τ_f.

Proof: First, note that since τ_f is a partition, τ_f ⊂ I(τ_f). The definition of I(τ_f) is symmetric in (A, A^c). Thus, if A ∈ I(τ_f) then A^c ∈ I(τ_f). Now suppose that A_i ∈ I(τ_f) for every i in the index set I. Let A = ∪iA_i. Suppose X ∈ τ_f. If for some i, X ⊂ A_i then X ⊂ A. If for every i it is not true that X ⊂ A_i then X ⊂ A^c_i for every i. Thus, X ⊂ ∩_iA^c_i = A^c. Hence, A ∈ I(τ_f).

In the following proposition we show that to have a finer partition is equivalent to have more information.

Proposition 1 τ ≥ τ⁰ if and only if I(τ ) ⊃ I(τ⁰).

Proof. We first prove that τ ≥ τ⁰ implies I(τ⁰) ⊆ I(τ ). Let A ∈ I(τ⁰) and Y ∈ τ. There exists X ∈ τ⁰ such that Y ⊆ X. Then either X ⊆ A and thus Y ⊆ A or X ⊆ A^cand thus Y ⊆ A^c. We now prove that I(τ⁰) ⊆ I(τ ) implies τ ≥ τ⁰. Let X ∈ τ⁰ and z ∈ X. Let us consider the unique element Y ∈ τ such that z ∈ Y. As X ∈ I(τ ) then Y ⊆ X (since Y ⊂ X^c is impossible).

Remark 1 It is easy to see that a set is informed, if and only if, it is formed by a possibly uncountable union of elements of the partition. Consequently, the σ–algebra of informed sets coincides with the σ–algebra generated by ar- bitrary unions of the elements of the partition. Thus prop. 1 is contained in theorem A of Dubra and Echenique.⁵

In the following proposition, σ^u(∪_i∈II(τ_i)) denotes the smallest σ–algebra closed for arbitrary unions that contains ∪_i∈II(τ_i).

Proposition 2 If τ_i, i ∈ I is a family of partitions of Ω then:

1. ∩_i∈I I(τ_i) = I(∧_i∈Iτ_i) 2. σ^u(∪_i∈II(τ_i)) = I(∨_i∈Iτ_i).

Proof. (1) It is clear from the previous proposition that ∩i∈II(τi) ⊃ I(∧i∈Iτi).

Suppose X ∈ ∩_i∈II(τ_i). If ω ∈ X then for every i ∈ I, τ_i(ω) ⊂ X. Let D be the atom of ∧_i∈Iτ_i that contains ω. If d⁰⁰ ∈ D there exists a sequence x0 = ω, x1, . . . , xn, xn+1 = d⁰⁰ such that every two successive terms belong to the same atom of τ_i for some i. Thus x₁ ∈ τ_i0(ω) and therefore x₁ ∈ X.

Analogously from x₂ ∈ τ_i1(x₁) we get x₂ ∈ X. Proceeding inductively we get d⁰⁰∈ X. Finally this implies that X ∈ I(∧i∈Iτi).

(2) From the previous proposition we have that ∪_i∈II(τ_i) ⊂ I(∨_i∈Iτ_i).

Thus σ^u(∪_i∈II(τ_i)) ⊂ I(∨_i∈Iτ_i). The contains inclusion follows from σ^u(∪_i∈II(τ_i)) being closed for arbitrary intersections.

5We owe this observation to M. Greinecker and J. Dubra.

4.2 Measurable signals

Consider that the set of states is a measurable space (Ω, F ). A signal f on Ω with values on (S, B) is measurable if f⁻¹(B) ∈ F for every B ∈ B.

The σ–algebra generated by f , denoted by σ(f, B), is the smallest σ–

algebra on Ω for which f is measurable.

We omit the (easy) proof of the following:

Proposition 3 I(τ_f) = σ(f, P(S)).

4.3 Finite or countable partitions

The following result shows that, in the case of an uncountable partition, the σ-algebra generated by the partition does not contain all the information given by the associated signal.

Proposition 4 The σ–algebra of informed sets is equal to σ (τ ) if and only if the partition τ is a countable set.

Proof: Let τ = {X_j; j ∈ N} be a countable partition of Ω. It is immediate that I(τ ) ⊃ σ(τ ). For any A ∈ I(τ ) let J = {j ∈ N; Xj ⊂ A}. Thus,

∪_j∈JX_j ⊂ A. Since A ∈ I(τ ) for every j 6∈ J, X_j ⊂ A^c and therefore

∪_j∈JX_j = A. Since J is countable A ∈ σ(τ ). Suppose now that τ is an uncountable set. There is a subset Y of τ such that Y and Y^care uncountable.

Note that A ∈ σ (τ ) if and only if there is a countable set X ⊂ τ such that A = ∪ {x : x ∈ X} or that A^c= {x : x ∈ X}. Thus, ∪ {y : y ∈ Y } /∈ σ (τ ).

Remark 2 The previous proposition shows that the informed sets σ–algebra is larger than the σ–algebra generated from the partition if the partition is uncountable. For example if τ = {{ω} : ω ∈ Ω} the collection of informed sets is P(Ω). We need the following theorem of Ulam (1930):

Theorem 2 (Ulam) A finite measure defined for all subsets of a set X of power ℵ₁ vanishes identically if it is equal to zero for every singleton of X.

For a proof see Oxtoby (1980) page 25. Under the continuum hypothesis (i.e. ℵ₁ = |R|) the only finite measures defined on P(R) are the measures that vanishes on the complement of a countable set.

4.4 A Blackwell theorem

We finish this section restating, for completeness, Theorem A of Dubra and Echenique ([3]) in the informed sets context. See also Zhang (2008).

Let C denote the set of consequences. An act is a function a : Ω → C. The set of possible acts is A = C^Ω. A decision-maker is a complete, transitive, preference relation % on A. The decision-maker receives the value y of a given signal f : Ω → Y_f and then chooses a consequence.

An act a : Ω → C is f -compatible if a(ω) = a(ω⁰) whenever f (ω) = f (ω⁰).

The decision-maker % prefers signal f to g if and only if, for any g-compatible act a, there exists an f -compatible act ˆa such that ˆa % a.

Proposition 5 Let f : Ω → Y_f and g : Ω → Y_g be two signals. The following statements are equivalent:

a) The decision-maker prefers the signal f to g.

b) I(τ_g) ⊆ I(τ_f).

c) There exists h : Y_f → Y_g such that g = h ◦ f .

5 Sigma-algebras as information

In scholarly practice we do not have a partition ready for use. The best we can get is a family of random variables (the signals) and the values which are assumed as results of, let us say an experiment. From a signal a partition is defined as before and it is, in general, an uncountable partition.

In this section we consider the scenario where the starting points are σ–

algebras, instead of signals or partitions. We examine what can be said in general terms when the signal corresponds to the partition of a σ–algebra (Definition 3). We argue that for most practical purposes we may consider σ–algebras as information.

5.1 Partitions from countably generated σ–algebras

First, we examine the case of a countably generated σ–algebra. For this we need a few definitions.

Definition 8 A Polish space is a complete separable metric space.

Definition 9 (analytic set) If (S, d) is a metric space the subset A ⊂ S is analytic if there is a Polish space X and a continuous function f : X → S such that A = f (X).

Definition 10 (Blackwell σ–algebra) If (Ω, G) is a measurable space, G is a Blackwell σ–algebra if it is separable and there is no other separable sub–

σ–algebra. G is a strongly Blackwell σ–algebra if it is separable and every two countably generated sub–σ–algebras with the same atoms coincide.

This terminology⁶ is from Rao and Rao (1981). The following theorem was first proved by Blackwell (1956) for Lusin⁷ spaces:

Theorem 3 (Blackwell) If A is a strongly Blackwell σ–algebra on Ω, E ⊂ A is a countably generated σ–algebra and A ∈ A is a union of atoms of E, then A ∈ E .

Blackwell σ–algebras are quite common. The reader can see a long list of examples on page 445 of [5]. For us it suffices to mention that the Boreleans σ–algebra of a Polish space is a strongly Blackwell σ–algebra.

Theorem 4 Let (Ω, A) be a measure space. If A is a strongly Blackwell σ–algebra and G is a countably generated sub-σ-algebra. Then

I(τ (G)) ∩ A = G.

Proof. That G ⊂ I(τ (G)) ∩ A is obvious. Suppose now that B ∈ I(τ (G)) ∩ A. Thus since B is an informed set it is a union of atoms of G. Therefore by Blackwell’s theorems above, B ∈ G.

Remark 3 Thus, although σ–algebras are not closed for uncountable unions, countably generated sub–σ-algebras of a strongly Blackwell σ–algebra are closed for uncountable unions as long as the union is measurable in the larger σ–

algebra.

We now prove an analog of proposition 5. Let (Ω, A), (Y, B), (Z, C) be measurable spaces. We suppose A strongly Blackwell, B and C separable.

Proposition 6 Let f : Ω → Y and g : Ω → Z be measurable signals, f (Ω) ∈ B. The following statements are equivalent:

a) Every decision maker prefers the signal f to signal g.

b) σ(g, C) ⊂ σ(f, B).

c) There exists h : Y → Z measurable such that g = h ◦ f

6We thank the referee for this reference.

7A Lusin space is a pair (Ω, B) where Ω is analytic and B is the Boreleans σ–algebra.

Proof. From (a) we get τ_g ≤ τ_f. Suppose now that X ∈ σ(g, C). Then there exists C ∈ C such that X = g⁻¹(C). Thus X being a τ_g informative set is a τ_f informative set as well. Since X ∈ A and σ(f, B) is countably generated, X ∈ σ(f, B). This proves (b). From (b) we have that g is σ(f, B) measurable and by the Doob–Dynkin theorem (see 6.4.1, page 443 of Hoffmann–Jørgensen 1994) there exists h as desired. That (c) implies (a) is obvious since g⁻¹(z) =

(h ◦ f )⁻¹(z) = f⁻¹(h⁻¹(z)). QED

5.2 Partitions from general σ–algebras

From now on, let us consider a fixed probability space (Ω, A, P ). We suppose that A is a strongly Blackwell σ–algebra. We define the symmetric difference of two sets A, B by A∆B := (A \ B) ∪ (B \ A).

Definition 11 (equivalence) If B and C are sub–σ–algebras of A, then B and C are equivalent if

1. for every B ∈ B there is a C ∈ C such that P (B∆C) = 0;

2. for every C ∈ C there is a B ∈ B such that P (B∆C) = 0.

The following lemma is proved in [11], lemma 3.2.2.

Lemma 4 If B is a sub–σ–algebra of A, then it is equivalent to a countably generated sub–σ–algebra of A.

Remark 4 This lemma is at first mysterious. But note that d (A, B) = P (A∆B) is a pseudo-metric on the set of events. Since A is separable, this pseudo-metric space is separable as a pseudo-metric space. Thus, any subspace is separable as well.

Definition 12 The partition of the σ–algebra B ⊂ A (with respect to P ) is the partition of any countably generated sub–σ–algebra C ⊂ A that is equiva- lent to B.

Naturally, there are several countably generated σ–algebras equivalent to B. The following lemma shows that the partitions are essentially the same.

Lemma 5 Suppose C and D are equivalent countably generated sub–σ–algebras of A, then there is a null set N ∈ A such that

τ C_Ω\N
= τ D_Ω\N
.

Here C_Ω\N = {C ∩ (Ω \ N ) ; C ∈ C} and analogously for D_Ω\N. For a proof see lemma 3.2.7 in [11].

Our last theorem states, in an informal sense, that information and mea- surability are equivalent as long as the information is suitably defined through equivalent countably generated σ–algebras.

Theorem 5 Suppose B and C are σ–algebras contained in A. Then, except for the removal of a null subset of Ω, B ⊂ C if and only if the partition of C is finer than the partition of B.

Proof. Suppose B ⊂ C. Let the family n ˜Bn, n ≥ 1 o

generate the σ–algebra B equivalent to B. Let ˜˜ C be separable equivalent to C. There is, for every integer n ≥ 1 a null set N_n such that ˜B_n∆N_n ∈ ˜C. Define N := ∪_nN_n. Thus, ˜Bn belong to the trace of ˜C with Ω \ N . Therefore, the atom of ˜C containing ω ∈ Ω \ N is a subset of the atom of ˜B containing ω. Reciprocally, if τ ˜C_Ω\N

is finer than τ  ˜B_Ω\N

then, ˜B_Ω\N ⊂ ˜C_Ω\N and therefore, B ⊂ C up to equivalence.

Example 5 The σ–algebra in Billingsley and in Dubra and Echenique’s ex- ample,

G = {A ⊂ Ω : A or A^c is countable}

the information (partition) corresponding the sigma-algebra G is τ = {Ω}

(no information) and is not {{ω}, ω ∈ Ω} (the full information). It is not countably generated. But it is equivalent to {∅, Ω} with partition τ (G) = {Ω}.

6 Conclusions

This paper deals with the problem of clarifying the meaning of the infor- mation that an agent may receive from a signal, an experiment, or from his own ability to precise the actual state of nature that occurs. A signal is a partition of the set of the different states of nature, and it is common to read in the literature that the σ–algebra generated by the partition (the signal) is considered as the informational content of the signal due to technical rea- sons. On the other hand, Billingsley (1995) and Dubra and Echenique (2004) showed concerns on the use of σ–algebras generated by partitions when using them as the informational content of a signal.

We have given a precise and natural definition of the informational con- tent of a signal, analyzed its immediate consequences, and showed that the collection of the informed sets is itself a σ–algebra. Thus, the fact of consid- ering σ–algebras to model the informational content of a signal is not due to technical reasons; the family of informed sets is itself a σ-algebra.

We studied the properties of the σ–algebra of informed sets, showing that it is closed under arbitrary unions and that, for finite or countable partitions, it coincides with the σ–algebra generated by the partition. This result validates the use of the σ-algebra generated by the partition as the informational content of a signal, as it is the case of several articles, and in particular of the papers quoted, [4] and [9]. We showed that finer partitions generate finer σ–algebras of informed sets, and conversely, finer σ-algebras of informed sets come from finer partitions. This result provides a formal and solid basis to the heuristics and intuitions related to the informational content of a signal. Also, as a consequence of the definition of informed set, the concerns set by Billingsley and by Dubra and Echenique have a conceptually satisfactory explanation.

Finally, we elaborate in the more general case in which information is given directly by σ–algebras. We established the informational content of a σ–algebra and extended our results to the case in which information is defined by countably generated sub–σ–algebras of a strongly Blackwell σ–

algebra. For general σ–algebras that might not be countably generated we conclude that information and measurability are equivalent as long as the information is suitably defined through an equivalent countably generated σ–algebra.

References

[1] Billingsley, P.: Probability and Measure, third edition, John Wiley (1995)

[2] Blackwell, D.: 1956, On a Class of Probability Spaces, Proceedings of the Third Berkeley Symp. on Math. Stat. and Prob., 1-6 (1956)

[3] Dubra, J. and Echenique, F.: Information is not about measurability, Math. Soc. Sciences 47, 177–185 (2004)

[4] Herv´es–Beloso, C., Martins-da-Rocha, V. F. and Monteiro, P. K.: Equi- librium theory with asymmetric information and infinitely many states, Econ. Theory 38, 295-320 (2009)

[5] Hoffmann–Jørgensen, J.: Probability with a view toward statistics, Chapmann & Hall (1994)

[6] Oxtoby, J. C., Measure and Category, second edition, Graduate Texts in Mathematics 2, Springer–Verlag (1980)

[7] Parthasarathy, K. R.: Probability Measures on Metric Spaces, AMS Chelsea (2005)

[8] Rohlin, V. A.: On the fundamental ideas of measure theory, Amer.

Math. Soc. Transl. 71 (1952)

[9] Yannelis, N.C.: The core of an economy with differential information, Econ. Theory 1 (2), 183–197 (1991)

[10] Rao, K. P.S. Bhaskara and Rao, B. V.: Borel spaces, Dissertationes Math. v. 190 (1981)

[11] Stinchcombe, M. B.: Bayesian Information Topologies, J. Math. Econ.

19, 233–253 (1990)

[12] Zhang, Z.: Comparison of Information Structures with Infinite States of Nature, Thesis, Johns Hopkins University (2008)