Blackwell Spaces and 𝜖-Approximations of Markov Chains

Abstract

On a weakly Blackwell space we show how to define a Markov chain approximating problem, for the target problem. The approximating problem is proved to converge to the optimal reduced problem under different pseudometrics.

1. Introduction

A target problem (TP) is defined as a homogeneous Markov chain stopped once it reaches a given absorbing class, the target. Our purpose is to only use the necessary information relevant with respect to the target and in consequence to reduce the available information. A new Markov chain, associated with a new equivalent but reduced matrix is defined. In the (large) finite case, the problem has been solved for TPs: in [1–3], it has been proved that any TP on a finite set of states has its “best target” equivalent Markov chain. Moreover, this chain is unique and there exists a polynomial time algorithm to reach this optimum.

The question is now to find, in generality, an 𝜖-approximation of the Markov problem when the state space is measurable. The idea is to merge into one group the points that 𝜖-behave the same with respect to the objective and, at the same time, to keep an almost equivalent Markov chain, with respect to the other “groups”. The construction of these groups is done through equivalence relations. Each group will correspond to a class of equivalence. In fact, there are many other mathematical fields where approximation problems are faced by equivalences. For instance, in integration theory, we use simple functions, in functional analysis, we use the density of countable generated subspaces, and, in numerical analysis, we use the finite elements method.

In this paper, the approximation is made by means of discrete equivalences, which will be defined in the following. We prove that the sequence of approximations tends to the optimal exact equivalence relation defined in [1–3], when we refine the groups. Finer equivalence will imply better approximation, and accordingly the limit will be defined as a countably generated equivalence.

Under a very general Blackwell-type hypothesis on the measurable space, we show that it is equivalent to speak on countably generated equivalence relationships or on measurable real functions on the measurable space of states. If we do not work under this framework of Blackwell spaces, we can be faced to paradoxes, as it is explained by [4], of enlarging 𝜎-algebras, while decreasing the information available to a decision-maker. The 𝜖-approximation of the Markov chain depends always upon the kind of objective. In [5], Jerrum deals with ergodic Markov chains. His objective is to approximate the stationary distribution by means of a discrete approximating Markov chain, whose limit distribution is close in a certain sense to the original one. However, unlike our following work, his purpose is not the explicit and unified construction of the approximating process. In this paper, we focus on the target problem. We solve extensively the TP, where the objective is connected with the conditional probability of reaching the target 𝑇, namely ℙ(𝑋_𝑛∈𝑇∣𝑋₀=𝑥), for any 𝑛,𝑥. This part extends the work in [1–3].

2. Main Results

Let (𝑋,𝒳) be a measurable space, equipped with a assumption (A0) that will be explained when necessary. Let 𝑃 be any transition probability on (𝑋,𝒳). A homogeneous Markov process (𝑋_𝑛)_𝑛≥0 is naturally associated to (𝑋,𝒳,𝑃). In the target problem, we are interested in the probabilities of reaching the target class 𝑇 within 𝑛 steps, namely inℙ𝑋_𝑛∈𝑇∣𝑋₀=𝑥forany𝑛and𝑥. (2.1) The set 𝑇 is a priori given, and does not change through the computations. 𝑇 is supposed to be an absorbing set lying in 𝒳.

Definition 2.1. Let (𝑋,𝒳) be a measurable space and let 𝑇∈𝒳. Let ℱ⊆𝒳 be a sub 𝜎-algebra of 𝒳 such that 𝑇∈ℱ. A function 𝑃∶𝑋×ℱ→[0,1] is a transition probability on (𝑋,ℱ) if (i)𝑃(𝑥,⋅) is a probability measure on ℱ, for any 𝑥∈𝑋, (ii)𝑃(⋅,𝐹) is ℱ-measurable, for any 𝐹∈ℱ. Given a transition probability 𝑃 on (𝑋,ℱ), we denote by 𝑃^𝑛 the transition probability on (𝑋,ℱ) given inductively by 𝑃¹=𝑃;𝑃^𝑛+1(𝑥,𝐹)=_𝑋𝑃𝑥,𝑑𝑦𝑃^𝑛𝑦,𝐹. (2.2) We denote by TrP(𝑋,ℱ) the set of the transition probabilities on (𝑋,ℱ). We denote by 𝕋ℙ_𝑋=∪_ℱ⊆𝒳TrP(𝑋,ℱ) the set of all transition probabilities on 𝑋 that we equip with a suitable pseudometric 𝑑: 𝑑𝑃₁,𝑃₂=sup_𝑥𝑛𝛽^𝑛𝑃^𝑛₁(𝑥,𝑇)−𝑃^𝑛₂(𝑥,𝑇),with𝛽∈(0,1). (2.3) It is such that 𝑑𝑃₁,𝑃₂=0⟺𝑃^𝑛₁(𝑥,𝑇)=𝑃^𝑛₂(𝑥,𝑇),forany𝑛and𝑥. (2.4) This pseudometric, which is compatible with 𝑇, allows to approximate 𝑃 by simpler kernels.

A target problem is defined through a transition probability 𝑃∈(𝕋ℙ_𝑋,𝑑). More precisely, we have the following definition.

Definition 2.2. A target problem is a quadruple (𝑋,ℱ,𝑇,𝑃), where 𝑃∈TrP(𝑋,ℱ) and 𝑇∈ℱ. A simple target problem is a target problem where ℱ is generated by an at most countable partition of 𝑋.

The main purpose of this paper is to approximate any target problem by a sequence of simple target problems in the spirit of the construction of the Lebesgue integral, where the integral of a function 𝑓 is approximated by the integral of simple functions 𝑓_𝑛=∑_𝑖∈𝐼𝑐_𝑖𝐼_𝐶𝑖. The strategies will play the role of the approximating subdivisions (𝐶_𝑖)_𝑖∈𝐼.

Definition 2.3. We call strategy Str a sequence of maps (Str_𝑛)_𝑛≥0 from the set of the target problems to the set of the simple target problems. A strategy is a target algorithm if it is built as in Section 3.

In the “Lebesgue example” given above, the strategy is related to the “objective” of the problem (the integral) and the pseudometric 𝑑(𝑓,𝑓_𝑛)=∫|𝑓−𝑓_𝑛|𝑑𝑥 is required to go to 0 as 𝑛 goes to infinity. Here also, a strategy is meaningful if 𝑑(𝑃,Str_𝑛(𝑃)) tends to 0 as 𝑛 goes to infinity. Moreover, for what concerns applications, given a target problem (𝑋,𝒳,𝑇,𝑃) a good strategy should not need the computation of 𝑃^𝑛, 𝑛>1. The first main result of this paper states that the target algorithms are always good strategies.

Theorem 2.4. For any target problem (𝑋,ℱ,𝑇,𝑃) and any target algorithm Str, lim_𝑛→∞𝑑𝑃,𝑃_𝑛=0, (2.5) where (𝑋,ℱ_𝑛,𝑇,𝑃_𝑛)=Str_𝑛(𝑋,ℱ,𝑇,𝑃).

Two questions immediately arise: does the sequence (Str_𝑛(𝑃))_𝑛≥0 have a limit (and in which sense)? Moreover, since 𝑑 is defined as a pseudometric, does this limit depend on the choice of Str?

The extension of the concept of compatible projection given in [1–3] to our framework will enable us to understand better the answer to these questions. A measurable set 𝐴≠∅ of a measurable space (𝑋,𝒳) is an 𝒳-atom if it has no nonempty measurable proper subset. No two distinct atoms intersect. If the 𝜎-field is countably generated, say by the sequence {𝐴_𝑛} then the atoms of 𝒳 are of the form ∩_𝑛𝐶_𝑛 where each 𝐶_𝑛 is either 𝐴_𝑛 or 𝑋⧵𝐴_𝑛.

Definition 2.5. An equivalence relationship 𝜋 on a measurable space (𝑋,𝒳) is measurable (discrete) if there exists a (discrete) random variable 𝑓∶(𝑋,𝒳)→(ℝ,ℬ_ℝ) (ℬ_ℝ denotes the Borel 𝜎-algebra), such that 𝑥𝜋𝑦⟺𝑓(𝑥)=𝑓𝑦, (2.6) and we denote it by 𝜋=𝜋_𝑓. Let (𝑋,ℱ,𝑇,𝑃) be a target problem. A compatible projection is a measurable equivalency 𝜋_𝑓 such that 𝑇∈𝜎(𝑓) and 𝑃(𝑥,𝐹)=𝑃𝑦,𝐹,∀𝑥𝜋_𝑓𝑦,∀𝐹∈𝜎𝑓. (2.7) We say that 𝜋⊇𝜋^′ if 𝜋 corresponds to partitions finer than 𝜋^′. Finally, a compatible projection 𝜋 is said to be optimal if 𝜋⊇𝜋^′, for any other compatible projection 𝜋^′.

Remark 2.6. This definition is well posed if 𝜋_𝑓=𝜋_𝑔⟺𝜎𝑓=𝜎𝑔. (2.8) Assumption (A0) ensures that the definition of measurable equivalency is indeed well posed. This assumption will be stated and discussed in Section 4.

Theorem 2.7. If 𝜋=𝜋_𝑓 is a compatible projection for the target problem (𝑋,ℱ,𝑇,𝑃), then there exists a target problem (𝑋,𝜎(𝑓),𝑇,𝑃_𝜋). such that 𝑃_𝜋(𝑥,𝐹)=𝑃(𝑥,𝐹) for any 𝐹∈𝜎(𝑓).

It is not said “a priori” that an optimal compatible projection must exist. If it is the case, then this equivalence is obviously unique.

Theorem 2.8. For any target problem (𝑋,ℱ,𝑇,𝑃), there exists a (unique) optimal compatible projection 𝜋.

To conclude the main results, let us first go back to the Lebesgue example. The simple functions 𝑓_𝑛=∑_𝑖∈𝐼𝑐_𝑖𝐼_𝐶𝑖 are chosen so that 𝜎(𝐶_𝑖,𝑖∈𝐼) increases to 𝜎(𝑓) and 𝑓_𝑛(𝑥)→𝑓(𝑥). The following theorem guarantees these two similar facts by showing the “convergence” of any strategy to the optimal problem.

Theorem 2.9. Let Str_𝑛(𝑋,ℱ,𝑇,𝑃)=(𝑋,ℱ_𝑛,𝑇,𝑃_𝑛), with Str target algorithm and let 𝜋 be the optimal compatible projection associated to the target problem (𝑋,ℱ,𝑇,𝑃). Then (i)ℱ_𝑛⊆ℱ_𝑛+1 for any 𝑛, and ∨_𝑛ℱ_𝑛≥0=ℱ_𝜋; (ii)lim_𝑛𝑃_𝑛(𝑥,𝐹)=𝑃_𝜋(𝑥,𝐹), for any (𝑥,𝐹)∈(𝑋×∪_𝑚ℱ_𝑚).

Remark 2.10 (The topology top). In Theorems 2.4–2.9, we have proved the convergence of (𝑃_𝑛)_𝑛 to 𝑃_𝜋 with respect to the pseudometric 𝑑. The pseudometric topology top is the topology induced by the open balls 𝐵_𝑟(𝑃)={𝑄∈𝕋ℙ_𝑋∶𝑑(𝑃,𝑄)<𝑟}, which form a basis for the topology. Accordingly, the previous theorems may be reread in terms of convergence of 𝑃_𝑛 to 𝑃 on the topological space (𝕋ℙ_𝑋,Top).

2.1. Connection with Weak Convergence

Given a strategy (𝑋,ℱ_𝑛,𝑇,𝑃_𝑛)_𝑛≥0, if we want to show a sort of weak convergence of 𝑃_𝑛(𝑥,⋅) to 𝑃(𝑥,⋅), for any 𝑥, we face the two following problems: (i)each 𝑃_𝑛(𝑥,⋅) is defined on a different domain (namely, on ℱ_𝑛), (ii)we did not have required a topology on 𝑋.

We overcome the first restriction by introducing a new definition of probability convergence. The idea is given in the following example.

Example 2.11. Let ℱ_𝑛=𝜎({(𝑖2^−𝑛,(𝑖+1)2^−𝑛], 𝑖=0,…,2^𝑛−1}) be the 𝜎-algebra on (0,1] generated by the dyadic subdivision. Suppose we know that 𝜈_𝑛∶ℱ_𝑛→[0,1] is the unique probability on ℱ_𝑛 s.t. for any 𝑖, 𝜈_𝑛((𝑖2^−𝑛,(𝑖+1)2^−𝑛])=2^−𝑛. Even if 𝜈_𝑛 is not defined on the Borel sets of (0,1], it is clear that in “some” sense, it must happen that 𝜈_𝑛→𝜈_∗, where 𝜈_∗ is the Lebesgue measure on the Borel sets of (0,1]. Note that the cumulative function of 𝜈_𝑛 is not defined, and therefore a standard weak convergence cannot be verified.
In fact, we know that𝜈_𝑛𝑖−∞,2^𝑛=𝜈_𝑛𝑖0,2^𝑛=𝑖2^𝑛, (2.9) that is, in this case, as 𝑛→∞, we can determine the cumulative function on a dense subset. This fact allows to hope that 𝜈_𝑛→𝜈_∗ in a particular sense.

Definition 2.12. Let (𝑋,𝒳,(𝒳_𝑛)_𝑛≥0) be a filtered space, and set 𝒳_∞=∨_𝑛≥0𝒳_𝑛. Let 𝜈_𝑛∶𝒳_𝑛→[0,1], 𝑛≥1 and 𝜈_∞∶𝒳_∞→[0,1] be probability measures. One says that 𝜈_𝑛converges totally to 𝜈_∞ on the topological space (𝑋,𝜏) as 𝑛 tends to infinity if 𝜈→_𝑛^𝑤_𝜏𝜈_∞ (converges in weak sense on (𝑋,𝜏)), for any 𝜈_𝑛∶𝒳_∞→[0,1], such that 𝜈_{𝑛|𝒳𝑛}=𝜈_𝑛. One writes 𝜈→_𝑛^tot_𝜏𝜈_∞.

Going back to the example, it is simple to check that 𝜈→_𝑛^tot_𝜏(0,1]𝜈_∗, where 𝜈_𝑛,𝜈_∗ are given in Example 2.11 and 𝜏(0,1] is the standard topology on (0,1]. In fact, let (𝜈_𝑛)_𝑛≥1 be any extension of (𝜈_𝑛)_𝑛≥1 to the Borel sets of (0,1]. For any 𝑡∈(0,1), we have by (2.9) that1𝑡−2^𝑛≤𝐹_𝜈𝑛(𝑡)1≤𝑡+2^𝑛, (2.10) where 𝐹_𝜈𝑛 is the cumulative function of 𝜈_𝑛, which implies the weak convergence of 𝜈_𝑛 to 𝜈_∗ and, therefore, 𝜈→_𝑛^tot_𝜏(0,1]𝜈_∗.

For what concerns the topology on 𝑋, we will define the topological space (𝑋,𝜚_𝑃) induced by the pseudometric 𝑑_𝑃 associated to the target problem (𝑋,ℱ,𝑇,𝑃), and the pseudometric 𝑑. In this way 𝜚_𝑃 is only defined with the data of the problem. One may ask: is this topology too poor? The answer is no, since it is defined by the pseudometric 𝑑_𝑃. In fact, 𝑑_𝑃(𝑥,𝑦)<𝜖 means that 𝑥 and 𝑦 play “almost the same role” with respect to 𝑇. A direct algorithm which takes 𝑑_𝑃 into account needs the computation of 𝑃^𝑛 at each step. In any case, even if 𝑑_𝑃 may not be computable, it defines a nontrivial interesting topology 𝜚_𝑃 on 𝑋. As expected, we have the following theorem.

Theorem 2.13. Let Str_𝑛(𝑋,ℱ,𝑇,𝑃)=(𝑋,ℱ_𝑛,𝑇,𝑃_𝑛), with Str target algorithm. Then, for any given 𝑥, 𝑃_𝑛(𝑥,⋅)⟶^tot_𝜚𝑃𝑃(𝑥,⋅). (2.11)

3. The Target Algorithm

In this section, we introduce the core of the approximating target problem, namely a set of strategies Str which solves the target problem.

Given a measurable space (𝑋,𝒳) and a target problem (𝑋,ℱ,𝑇,𝑃), the target algorithm is built in the spirit of the exact one given in [1, 2], which starts from the largest classes 𝑇 and 𝑋⧵𝑇 and then reaches the optimal classes according to a backward construction.

The target algorithm defines a strategy Str=(Str_𝑛)_𝑛≥0, whereStr_𝑛(𝑋,ℱ,𝑇,𝑃)=𝑋,ℱ_𝑛,𝑇,𝑃_𝑛, (3.1) and it consists of three steps: (1)the choice of a sequence (∼_𝜖𝑛)_𝑛≥1 of equivalences on the simplex defined on the unit ball of ℓ₁ with 𝜖_𝑛→0; (2)the definition of a filtration (ℱ_𝑛)_𝑛 based on (∼_𝜖𝑛)_𝑛≥1 where each ℱ_𝑛 is generated by a countable partition of 𝑋; (3)the choice of a suitable measure 𝜇 and the definition of (𝑃_𝑛)_𝑛≥0.

3.1. Preliminary Results on Measurability and Equivalency and the Choice of (∼_𝜖𝑛)_𝑛≥1

Associated to each countably generated sub 𝜎-algebra 𝒜⊆𝒳, we define the equivalence relationship 𝜋_𝒜 induced by the atoms of 𝒜:𝑥𝜋_𝒜𝑦⟺[𝑥]_𝒜∶=∩{𝐴∈𝒜∶𝑥∈𝐴}=∩𝐴∈𝒜∶𝑦∈𝐴=∶𝑦_𝒜. (3.2) Thus, if (𝒜_𝑛)_𝑛 is a sequence of countably generated 𝜎-algebras, then𝜋_{∨𝑛𝒜𝑛}=∩_𝑛𝜋_𝒜𝑛. (3.3)

Now, the atoms of the 𝜎-algebra ℱ of each simple target problem (𝑋,ℱ,𝑇,𝑄) are at most countable, by definition. Then 𝑄 may be represented as a transition matrix on the state set ℕ. Each row of 𝑄 is a distribution probability on ℕ (i.e., a sequence (𝑝_𝑛)_𝑛≥1 in the simplex 𝑆 of ℓ₁). The first step of the target algorithm is to equip 𝑆 with the ℓ₁-norm and then to define an 𝜖-equivalence on 𝑆.

We will alternatively use both the discrete equivalencies and the countable measurable partitions, as a consequence of the following result, whose proof is left to the appendices.

Lemma 3.1. Given a measurable space (𝑋,𝒳), there exists a natural bijection between the set of discrete equivalencies on 𝑋 and the set of the countable measurable partitions of it.

Let 𝑆_ℓ1 be the unit sphere in ℓ₁ and 𝑆={𝑥≥0}∩𝑆_ℓ1 be the simplex on ℓ₁. Let Ω_𝑛=[0,1], for any 𝑛, and 𝜏 be the standard topology on [0,1]. Denote by ℬ_[0,1] the Borel 𝜎-algebra on [0,1] generated by 𝜏. We look at 𝑆 as a subset of Π^∞_𝑛=1Ω_𝑛 so that the Borel 𝜎-algebra ℬ_𝑆 induced on 𝑆 is ⨂^∞_𝑛=1ℬ_[0,1]∩𝑆.

Definition 3.2. ∼_𝜖 is an 𝜖-equivalence on 𝑆 if it is the product of finite equivalences on each (Ω_𝑛,ℬ_[0,1]), and ‖𝑝−𝑞‖₁<𝜖 whenever 𝑝∼_𝜖𝑞.

Remark 3.3. The choice of the ℓ₁-norm on 𝑆 is linked to the total variation distance between probability measures. This distance between two probability measures 𝑃 and 𝑄 is defined by 𝑑_TV(𝑃,𝑄)=sup_𝐴∈Ω|𝑃(𝐴)−𝑄(𝐴)|. On the other hand, the total variation of a measure 𝜇 is ‖𝜇‖(Ω)=sup∑_𝑖|𝜇(𝐴_𝑖)|, where the supremum is taken over all the possible partitions of Ω. As (𝑃−𝑄)(Ω)=0, we have that 𝑑_TV(𝑃,𝑄)=(1/2)‖𝑃−𝑄‖; see [6]. To each 𝑝∈𝑆 corresponds the probability measure 𝑃 on ℕ with 𝑃(𝑖)=𝑝_𝑖 (and vice versa). In fact, 𝑝∈𝑆 implies 𝑝_𝑖≥0 and ∑_𝑖𝑝_𝑖=1. Therefore, since ‖𝑝−𝑞‖₁=‖𝑃−𝑄‖=2𝑑_TV(𝑃,𝑄), we have 𝑝∼_𝜖𝑞⟹𝑑_TV()<𝜖𝑃,𝑄2. (3.4)

Example 3.4. Define the 𝜖-cut as follows: 𝑝∼_𝜖𝑞⇔⌊𝑝_𝑛/𝜖2^−𝑛⌋=⌊𝑞_𝑛/𝜖2^−𝑛⌋, for all 𝑛, where ⌊𝑥⌋ denotes the entire part of 𝑥.  ∼_𝜖 is an 𝜖-equivalence on 𝑆. Indeed, (i)for each 𝑛, define 𝑝_𝑛∼_𝑛𝑞_𝑛⇔⌊𝑝_𝑛/𝜖2^−𝑛⌋=⌊𝑞_𝑛/𝜖2^−𝑛⌋. Then ∼_𝑛 is a finite equivalence on Ω_𝑛 and 𝑝∼_𝜖𝑞⇔(𝑝_𝑛∼_𝑛𝑞_𝑛), for all 𝑛, (ii)for any 𝑝∈𝑆𝑝=𝑞∈𝑆∶𝜋_∼𝜖𝑞=𝜋_∼𝜖𝑝=_𝑛2^𝑛𝑝_𝑛/𝜖𝜖2^𝑛,2^𝑛𝑝_𝑛/𝜖+1𝜖2^𝑛𝑆 (3.5) is measurable with respect to ℬ_𝑆,(iii)for all𝑝∼_𝜖𝑞, 𝑝−𝑞₁≤_𝑛𝜖2^−𝑛=𝜖. (3.6)

3.2. The Choice of (ℱ_𝑛)_𝑛≥0

The following algorithm is a good candidate to be a strategy for the approximating problem we are facing. Given a sequence (∼_𝜖𝑛)_𝑛≥1 of 𝜖-equivalences on 𝑆, we define (ℱ_𝑛)_𝑛≥0 inductively. Consider the equivalence classes given by ℱ_𝑛−1 and divide them again according to the next rule. Starting from any two points in the same class, we check whether the probabilities to attain any other ℱ_𝑛−1-classes are 𝜖-the same. Mathematically speaking: we have the following steps.

Step 3.2. ℱ₀=𝜎(𝑇)={∅,𝑇,𝑋⧵𝑇,𝑋}.

Step n. ℱ_𝑛 is based on the equivalence ℱ_𝑛−1 and on ∼_𝜖𝑛, inductively. ℱ_𝑛−1 is generated by a countable partition of 𝑋, say (𝐴_𝑖^(𝑛−1))_𝑖. We define, for any couple (𝑥,𝑦)∈𝑋², 𝑥𝜋_𝑛𝑦⟺𝑥𝜋_𝑛−1𝑦∧𝑃𝑥,𝐴_𝑖^(𝑛−1)_𝑖∼_𝜖𝑛𝑃𝑦,𝐴_𝑖^(𝑛−1)_𝑖. (3.7) Lemma 3.6 shows that 𝜋_𝑛 is a discrete equivalency on (𝑋,𝒳), and therefore it defines ℱ_𝑛=𝜎(𝑋/𝜋_𝑛) as generated by a countable partition of 𝑋.

Remark 3.5. The choice of the “optimal” sequence (∼_𝜖𝑛)_𝑛≥1 is not the scope of this work. We only note that the definition of ∼_𝜖 can be relaxed and the choice of the sequence (∼_𝜖𝑛)_𝑛≥1 may be done interactively, obtaining a fewer number of classes (𝐴_𝑖^(𝑛))_𝑖 at each step.

Lemma 3.6. Let (𝑋,ℱ,𝑇,𝑃) be a target problem. (ℱ_𝑛)_𝑛≥0 defined as above, is a filtration on (𝑋,ℱ) and for any 𝑛∈ℕ, 𝜋_𝑛 is a finite (and hence discrete) equivalency on (𝑋,𝒳).

Proof. The monotonicity of (ℱ_𝑛)_𝑛≥0 is a simple consequence of (3.7).
The statement is true for 𝑛=0, since 𝑇∈𝒳. For the induction step, let {𝐴₁^(𝑛−1),𝐴₂^(𝑛−1),…,𝐴_𝑘^(𝑛−1)_𝑛}∈𝒳 be the measurable partition of 𝑋 given by 𝑋/𝜋_𝑛−1. The map ℎ∶(𝑋,𝒳)→(𝑆,ℬ(𝑆)) given by 𝑥↦(𝑃(𝑥,𝐴_𝑖^(𝑛−1)))_𝑖 is therefore measurable. As ∼_𝜖𝑛 is a finite equivalency on ∏^𝑘𝑛₁(Ω_𝑛,ℬ_[0,1]), the map 𝜋_∼𝜖𝑛∘ℎ∶(𝑋,𝒳)→(𝑆/∼_𝜖𝑛,2^{𝑆/∼𝜖𝑛}) is also measurable, where 𝜋_∼𝜖𝑛 is the natural projection associated with ∼_𝜖𝑛. Thus, two points 𝑥,𝑦∈𝑋 are such that𝑃𝑥,𝐴_𝑖^{(𝑛−1)𝑘𝑛}_𝑖=1∼_𝜖𝑛𝑃𝑦,𝐴_𝑖^{(𝑛−1)𝑘𝑛}_𝑖=1 (3.8) if and only if their image by 𝜋_∼𝜖𝑛∘ℎ is the same point of 𝑆/∼_𝜖𝑛. It results that the new partition of 𝑋 built by 𝜋_𝑛 is obtained as an intersection of the sets 𝐴_𝑖^(𝑛−1),1≤𝑖≤𝑘_𝑛—which formed the 𝜋_𝑛−1-partition- with the counter-images of (∏^𝑘𝑛_𝑖=1Ω_𝑖)/∼_𝜖𝑛 by 𝜋_∼𝜖𝑛∘ℎ. Intersections between two measurable finite partitions of 𝑋 being a measurable finite partition of 𝑋, we are done.

3.3. The Choice of 𝜇 and the Definition of (𝑃_𝑛)_𝑛≥0

Before defining (𝑃_𝑛)_𝑛≥0, we need the following result, which will be proved in Section 5.

Theorem 3.7. Let (𝜋_𝑛)_𝑛≥0 be defined as in the previous section and let 𝜋_∞=∩_𝑛𝜋_𝑛. Then 𝜋_∞ is a compatible projection.

As a consequence of Theorems 2.7 and 3.7, a target problem (𝑋,∨_𝑛ℱ_𝑛,𝑇,𝑃_∞) is well defined. We intend to define 𝑃_𝑛 as the 𝜇-weighted mean average of 𝑃_∞ given the information carried by ℱ_𝑛.

More precisely, let 𝜇 be a probability measure on (𝑋,∨_𝑛ℱ_𝑛) such that 𝜇(𝐹)>0, for any 𝐹∈ℱ_𝑛,𝐹≠∅ (the existence of such a measure is shown in Example 3.8).

For any 𝐹∈ℱ_𝑛, let 𝑌^𝐹 be the ∨_𝑛ℱ_𝑛-random variable such that 𝑌^𝐹(𝜔)=𝑃_∞(𝜔,𝐹). Define𝑃_𝑛(𝑥,𝐹)=𝔼_𝜇𝑌^𝐹∣ℱ_𝑛(𝑥),∀𝑥∈𝑋,∀𝐹∈ℱ_𝑛. (3.9)𝑃_𝑛 is uniquely defined on (𝑋×ℱ_𝑛), the only 𝜇-null set of ℱ_𝑛 being the empty set. We claim that 𝑃_𝑛(𝑥,⋅) is a probability measure, for any 𝑥∈𝑋.

We give in the following an example of the measure 𝜇 that has been used in (3.9) which justifies its existence.

Example 3.8. Let (𝑌_𝑛)_𝑛≥0 be a sequence of independent and identically distributed geometric random variables, with ℙ_𝑌𝑖(𝑗)=1/2^𝑗, 𝑗∈ℕ. Let 𝒜_𝑛=𝜎(𝑌₀,…,𝑌_𝑛) and set 𝒜=∨𝒜_𝑛. There exists a probability measure ℙ on 𝒜 such that ℙ∩^𝑛_𝑖=0𝑌_𝑙𝑖=𝑦_𝑖=ℙ_𝑌𝑙1𝑦₁⊗⋯⊗ℙ_𝑌𝑙𝑛𝑦_𝑛=12^{∑𝑛𝑖=0𝑦𝑖}, (3.10) and thus, ℙ(𝐴)>0, for all 𝐴∈𝒜_𝑛,𝐴≠∅. Moreover, it follows that for any 𝑛, 𝐴₁∈𝒜_𝑛,𝐴₂∈𝜎𝑌_𝑛+1,𝐴₁≠∅,𝐴₂≠∅,⟹ℙ𝐴₁∩𝐴₂>0. (3.11) We check by induction that we can embed ℱ_𝑛 into 𝒜_𝑛, for any 𝑛≥0. The required measure 𝜇 will be the trace of ℙ on the embedded 𝜎-field ∨_𝑛ℱ_𝑛.
For 𝑛=0, define 𝑇↦{𝑌₀=1}, 𝑋⧵𝑇↦{𝑌₀≥2}. The embedding forms a nontrivial partition, and therefore the restriction of ℙ to the embedding of ℱ₀ defines a probability measure on ℱ₀ with 𝜇₀(𝐹)>0 if 𝐹≠∅.
For the induction step, suppose it is true for 𝑛. Given 𝐹_𝑖^(𝑛)∈ℱ_𝑛, we have 𝐹_𝑖^(𝑛)↦𝐴_𝑖^(𝑛), where (𝐴_𝑖^(𝑛))_𝑖 is a nontrivial partition in 𝒜_𝑛 and therefore the restriction of ℙ to the embedding of ℱ_𝑛 defines a probability measure 𝜇_𝑛 on ℱ_𝑛 with 𝜇_𝑛(𝐹)>0 if 𝐹≠∅.
Given 𝐹_𝑖^(𝑛), let 𝐻_𝑖^(𝑛+1)∶={𝐹_𝑗^(𝑛+1)∶𝐹_𝑗^(𝑛+1)⊆𝐹_𝑖^(𝑛)}. The monotonicity of 𝜋_𝑛 ensures that each 𝐹_𝑗^(𝑛+1) will belong to one and only one 𝐻_𝑖^(𝑛+1). Moreover, by definition of 𝐹_𝑗^(𝑛+1), we have that𝐹_𝑖^(𝑛)=∪𝐹_𝑗^(𝑛+1)∶𝐹_𝑗^(𝑛+1)∈𝐻_𝑖^(𝑛+1). (3.12) Since 𝑋/𝜋_𝑛+1 is at most countable, we may order 𝐻_𝑖^(𝑛+1) for any 𝑖. We have accordingly defined an injective map 𝑋/𝜋_𝑛+1→ℕ², where 𝐹_𝑗^(𝑛+1)⟼(𝑖,𝑘)⟺𝐹_𝑗^(𝑛+1)isthe𝑘thelementin𝐻_𝑖^(𝑛+1). (3.13)
According to the cardinality of 𝐻_𝑖^(𝑛+1), define the 𝑛+1-embedding𝐹_𝑗^(𝑛+1)⟼(𝑖,𝑘)⟼𝐴_𝑗^(𝑛+1)∶=𝐴_𝑖^(𝑛)∩𝑌_𝑛+1=𝑘if𝑘<#𝐻_𝑖^(𝑛+1),𝑌_𝑛+1≥𝑘if𝑘=#𝐻_𝑖^(𝑛+1). (3.14) By definition of 𝐴_𝑗^(𝑛+1) and (3.12), it follows that we have mapped ℱ_𝑛+1 into a partition in 𝒜_𝑛+1. Moreover, ℙ(𝐴_𝑗^(𝑛+1))>0 as a consequence of (3.11). The restriction of ℙ to the embedding of ℱ_𝑛+1 defines a probability measure on ℱ_𝑛+1 with 𝜇_𝑛+1(𝐹)>0 if 𝐹≠∅. Note that 𝜇_𝑛+1 is by construction an extension of 𝜇_𝑛 to ℱ_𝑛+1 since by (3.14), 𝜇_𝑛𝐹_𝑖^(𝑛)=_{𝐹𝑗(𝑛+1)∈𝐻𝑖(𝑛+1)}𝜇_𝑛+1𝐹_𝑗^(𝑛+1). (3.15)
Finally, the Carathéodory's extension theorem ensures the existence of the required 𝜇, as 𝜇(𝐹)=𝜇_𝑛(𝐹), for any 𝐹∈ℱ_𝑛. Note that 𝜇 is just mapped to the trace of ℙ on the embedded ℱ_∞.

4. Blackwell

The problem of approximation is mathematically different if we start from a Markov process with a countable set of states or with an uncountable one. Let us consider, for the moment, the countable case: 𝑋 is an at most countable set of the states and 𝒳=2^𝑋 is the power set. Each function on 𝑋 is measurable. If we take any equivalence relation on 𝑋, it is both measurable and identified by the 𝜎-algebra it induces (see Theorem 4.6). This is not in general the case when we deal with a measurable space (𝑋,𝒳), with 𝑋 uncountable. In this section, we want to connect the process of approximation with the upgrading information. More precisely, a measurable equivalence 𝜋=𝜋_𝑓 defines both the partition 𝑋/𝜋 and the sigma algebra 𝜎(𝑓). One wishes these two objects to be related, in the sense that ordering should be preserved. Example 4.5 shows a paradox concerning 𝜋_𝑓 and 𝜎(𝑓) when 𝑋 is uncountable. In fact, we have the following lemma.

Lemma 4.1. Let 𝒜₁⊆𝒜₂ be countably generated sub 𝜎-algebras of a measurable space (𝑋,𝒳). Then [𝑥]_𝒜1⊇[𝑥]_𝒜2.
In particular, let 𝑓,𝑔 be random variables. If 𝜎(𝑓)⊇𝜎(𝑔), then 𝜋_𝑓⊆𝜋_𝑔.

Proof. See Appendix A.

The problem is that even if a partition is more informative than another one, it is not true that it generates a finer 𝜎-algebra, that is, the following implication is not always true for any couple of random variables 𝑓 and 𝑔:𝜋_𝑓⊆𝜋_𝑔⟹𝜎𝑓⊇𝜎𝑔. (A0) Then Lemma 4.1 is not invertible, if we do not require the further assumption (A0) on the measurable space (𝑋,𝒳). This last fact connects the space (𝑋,𝒳) with the theory of Blackwell spaces (see Lemma 4.3). We will assume the sole assumption (A0).

Example 4.2 (𝜋_𝑓=𝜋_𝑔⇏𝜎(𝑓)=𝜎(𝑔)). We give here a counterexample to assumption (A0), where two random variables 𝑓,𝑔 generate two different sigma algebras 𝜎(𝑓)≠𝜎(𝑔) with the same set of atoms. Obviously, assumption (A0) does not hold. Let (𝑋,ℬ_𝑋) be a Polish space and suppose ℬ_𝑋⊊𝒳. Let 𝐴∈𝒳⧵ℬ_𝑋 and consider the sequence {𝐴_𝑛, 𝑛∈ℕ} that determines ℬ_𝑋, that is, ℬ_𝑋=𝜎(𝐴_𝑛, 𝑛∈ℕ). Let 𝒜=𝜎(𝐴,𝐴_𝑛,𝑛∈ℕ). ℬ_𝑋⊊𝒜. As a consequence of Lemma A.3, there exist two random variables 𝑓,𝑔 such that ℬ_𝑋=𝜎(𝑓) and 𝒜=𝜎(𝑔). The atoms of ℬ_𝑋 are the points of 𝑋, and then the atoms of 𝒜 are also the points of 𝑋, since ℬ_𝑋⊆𝒜.

We recall here the definition of Blackwell spaces. A measurable space (𝑋,𝒳) is said Blackwell if 𝒳 is a countably generated 𝜎-algebra of 𝑋 and 𝒜=𝒳 whenever 𝒜 is another countably generated 𝜎-algebra of 𝑋 such that 𝒜⊆𝒳, and 𝒜 has the same atoms as 𝒳. A metric space 𝑋 is Blackwell if, when endowed with its Borel 𝜎-algebra, it is Blackwell. The measurable space (𝑋,𝒳) is said to be a strongly Blackwell space if 𝒳 is a countably generated 𝜎-algebra of 𝑋 and (A1)𝒜₁=𝒜₂ if and only if the sets of their atoms coincide, where 𝒜₁ and 𝒜₂ are countably generated 𝜎-algebras with 𝒜_𝑖⊆𝒳, 𝑖=1,2.

For what concerns Blackwell spaces, the literature is quite extensive. Blackwell proved that every analytic subset of a Polish space is, with respect to its relative Borel 𝜎-field, a strongly Blackwell space (see [7]). Therefore, if (𝑋,ℬ_𝑋) is (an analytic subset of) a Polish space and ℬ_𝑋⊊𝒳, then (𝑋,𝒳) cannot be a weakly Blackwell space (see Example 4.2). Moreover, as any (at most) countable set equipped with any 𝜎-algebra may be seen as an analytic subset of a Polish space, then it is a strongly Blackwell space. More connections and examples involving Blackwell spaces, measurable sets and analytical sets in connection with continuum hypothesis (CH) may be found in [8–11]. Finally, note that assumption (A0) and assumption (A1) coincide, as the following lemma states.

Lemma 4.3. Let (𝑋,𝒳) be a measurable space. Then (A0) holds if and only if (A1) holds.

Proof. Lemma A.3 in Appendix A asserts that 𝒜⊆𝒳 is countably generated if and only if there exists a random variable 𝑓 such that 𝒜=𝜎(𝑓). In addition, as a consequence of Lemma 4.1, we have only to prove that (A1) implies (A0). By contradiction, assume (A1), 𝜋_𝑓⊆𝜋_𝑔, but 𝜎(𝑔)⊆𝜎̸(𝑓). We have 𝜎(𝑓,𝑔)≠𝜎(𝑓), and then 𝜋_{𝜎(𝑓,𝑔)}≠𝜋_𝑓 by (A1) and Lemma A.3. On the other hand, from (3.3), we have that 𝜋_{𝜎(𝑓,𝑔)}=𝜋_{𝜎(𝑓)∨𝜎(𝑔)}=𝜋_𝑓∩𝜋_𝑔=𝜋_𝑓.

We call weakly Blackwell space a measurable space (𝑋,𝒳) such that assumption (A0) holds. If (𝑋,𝒳) is a weakly Blackwell space, then (𝑋,ℱ) is a weakly Blackwell space, for any ℱ⊆𝒳. Moreover, every strong Blackwell space is both a Blackwell space and a weakly Blackwell space whilst the other inclusions are not true in general. In [12, 13], examples are provided of Blackwell spaces which may be shown not to be weakly Blackwell. The following example shows that a weakly Blackwell space need not be Blackwell.

Example 4.4 (weakly Blackwell ⇏ Blackwell). Let 𝑋 be an uncountable set and 𝒳 be the countable-cocountable 𝜎-algebra on 𝑋. 𝒳 is easily shown to be not countably generated, and therefore (𝑋,𝒳) is not a Blackwell space. Take any countably generated 𝜎-field 𝒜⊆𝒳, that is, 𝒜=𝜎({𝐴_𝑖, 𝑖∈ℕ}). (i)Since each set (or its complementary) of 𝒳 is countable, then, without loss of generality, we can assume the cardinality of 𝑋⧵𝐴_𝑖 to be countable. (ii)Each atom 𝐵 of 𝜎(𝐴_𝑖, 𝑖∈ℕ) is of the form 𝐵=_𝑖=1,2,…𝐶_𝑖,where𝐶_𝑖=𝐴_𝑖or𝐶_𝑖=𝑋⧵𝐴_𝑖,forany𝑖. (4.1)Note that the cardinality of the set 𝐴∶=∪_𝑖(𝑋⧵𝐴_𝑖) is countable, as it is a countable union of countable sets. As a consequence of (4.1), we face two types of atoms: (1)for any 𝑖, 𝐶_𝑖=𝐴_𝑖. This is the atom made by the intersections of all the uncountable generators. This is an uncountable atom, as it is equal to 𝑋⧵𝐴. (2)exists 𝑖 such that 𝐶_𝑖=𝑋⧵𝐴_𝑖. This implies that this atom is a subset of the countable set 𝐴. Therefore, all the atoms (except 𝑋⧵𝐴) are disjoint subsets of the countable set 𝐴 and hence they are countable. It follows that the number of atoms of 𝒜 is at most countable. Thus, (𝑋,𝒜) is a strongly Blackwell space, that is, (𝑋,𝒳) is a weakly Blackwell space.

Example 4.5 (Information and 𝜎-algebra (see [4])). Suppose 𝑋=[0,1], 𝒳=𝜎(𝒴,𝐴) where 𝒴 is the countable-cocountable 𝜎-algebra on 𝑋 and 𝐴=[0,1/2). Consider a decisionmaker who chooses action 1 if 𝑥<1/2 and action 2 if 𝑥≥1/2. Suppose now that the information is modeled either as the partition of all elements of 𝑋, 𝜏={𝑥, 𝑥∈𝑋} and in this case the decisionmaker is perfectly informed, or as the partition 𝜏^′={𝐴,𝑋⧵𝐴}. If we deal with 𝜎-algebras as a model of information then 𝜎(𝜏)=𝒴 and 𝜎(𝜏^′)=𝜎(𝐴). The partition 𝜏 is more informative than 𝜏^′, whereas 𝜎(𝜏) is not finer than 𝜎(𝜏^′). In fact 𝐴∉𝒴 and therefore if the decisionmaker uses 𝜎(𝜏) as its structure of information, believing it more detailed than 𝜎(𝜏^′), he will never know whether or not the event 𝐴 has occurred and can be led to take the wrong decision. In this case, 𝜎-algebras do not preserve information because they are not closed under arbitrary unions. However, if we deal with Blackwell spaces, any countably generated 𝜎-algebra is identified by its atoms and therefore will possess an informational content (see, e.g., [14]).

The following theorem, whose proof is in Appendix B, connects the measurability of any relation to the cardinality of the space and assumption (A0). It shows the main difference between the uncountable case and the countable one.

Theorem 4.6. Assume (CH). Let (𝑋,𝒳) be a measurable space. The following properties are equivalent: (1)any equivalence relation 𝜋 on 𝑋 is measurable and assumption (A0) holds, (2)(𝑋,2^𝑋) is a weakly Blackwell space, (3)𝑋 is countable and 𝒳=2^𝑋.

5. Proofs

The following theorem mathematically motivates our approximation problem: any limit of a monotone sequence of discrete equivalence relationships is a measurable equivalence.

Theorem 5.1. For all 𝑛∈ℕ, let 𝜋_𝑛 be a discrete equivalency. Then 𝜋_∞=∩_𝑛𝜋_𝑛 is a measurable equivalency. Conversely, for any measurable equivalency 𝜋, there exists a sequence (𝜋_𝑛)_𝑛≥0 of discrete equivalencies such that 𝜋_∞=∩_𝑛𝜋_𝑛.

Proof. See Appendix A.

Proof of Theorem 2.7. Let 𝜋=𝜋_𝑓 be a compatible projection. We define 𝑃_𝜋(𝑥,𝐹)∶=𝑃(𝑥,𝐹),∀𝑥∈𝑋,∀𝐹∈𝜎𝑓. (5.1) What remains to prove is that 𝑃_𝜋∈TrP(𝑋,𝒳,𝜎(𝑓)). More precisely, we have to show that 𝑃_𝜋(⋅,𝐹) is 𝜎(𝑓)-measurable, for all 𝐹∈𝜎(𝑓). By contradiction, there exists 𝐹∈𝜎(𝑓) such that the random variable 𝑌^𝐹(𝜔)=𝑃_𝜋(𝜔,𝐹) is not 𝜎(𝑓)-measurable. Then 𝜎(𝑌^𝐹)⊆𝜎̸(𝑓), and hence 𝜋_𝑌^𝐹⊇𝜋̸_𝑓 by assumption (A0), which contradicts (2.7).

Proof of Theorem 3.7. As a consequence of Theorem 5.1, 𝜋_∞=𝜋_𝑓, where 𝜎(𝑓)=∨_𝑛ℱ_𝑛. Define 𝑃_∞(𝑥,𝐹)∶=𝑃(𝑥,𝐹),∀𝑥∈𝑋,∀𝐹∈𝜎𝑓. (5.2) We will prove that, for any 𝐹∈𝜎(𝑓), 𝑃_∞(⋅,𝐹) is 𝜎(𝑓)-measurable and consequently 𝜋_∞ will be a compatible projection. This implies that there exists a measurable function ℎ_𝐹∶(ℝ,ℬ_𝑅)→(ℝ,ℬ_𝑅) so that 𝑃_∞(𝜔,𝐹)=ℎ_𝐹(𝑓(𝜔)). Therefore, if 𝑥𝜋_𝑓𝑦, then 𝑃_∞(𝑥,𝐹)=𝑃_∞(𝑦,𝐹), which is the thesis.
We need to show that for any 𝐹∈𝜎(𝑓) and 𝑡∈ℝ, we have𝐻∶={𝑥∶𝑃(𝑥,𝐹)≤𝑡}∈𝜎𝑓. (5.3) We first show that it is true when 𝐹∈ℱ_𝑛 by proving that 𝐻=_𝑚>𝑛𝜋_𝑚⁻¹𝜋_𝑚(𝐻), (5.4) which implies that 𝐻∈𝜎(𝑓). The inclusion 𝐻⊆∩_𝑚𝜋_𝑚⁻¹𝜋_𝑚(𝐻) is always true. For the other inclusion, let 𝑦∈∩_𝑚>𝑛𝜋_𝑚⁻¹𝜋_𝑚(𝐻). Let 𝑚>𝑛; there exists 𝑥_𝑚∈𝐻 such that 𝑦𝜋_𝑚𝑥_𝑚. Therefore, (3.7) and the definition of ∼_𝜖𝑛 imply 𝑃(𝑦,𝐹)≤𝑃(𝑥_𝑚,𝐹)+𝜖_𝑚≤𝑡+𝜖_𝑚, for any 𝑚>𝑛. As 𝜖_𝑚↘0, we obtain that 𝑦∈𝐻. Then (5.3) is true on the algebra Alg∶=∪_𝑛ℱ_𝑛.
Actually, let 𝐹_𝑛∈Alg such that 𝐹_𝑛↗𝐹. We prove that (5.3) holds for 𝐹 by showing that 𝐻={𝑥∶𝑃(𝑥,𝐹)≤𝑡}=_𝑛𝑥∶𝑃𝑥,𝐹_𝑛≤𝑡=∶_𝑛𝐻_𝑛. (5.5) Again, since 𝐹_𝑛⊆𝐹, then 𝑃(𝑥,𝐹_𝑛)≤𝑃(𝑥,𝐹) and therefore 𝐻⊆⋂_𝑛𝐻_𝑛. Conversely, the set ∩_𝑛𝐻_𝑛⧵𝐻 is empty since the sequence of 𝒳-measurable maps 𝑃(⋅,𝐹)−𝑃(⋅,𝐹_𝑛) converges to 0: 𝑃(⋅,𝐹)−𝑃⋅,𝐹_𝑛=𝑃⋅,𝐹⧵𝐹_𝑛⟶𝑃(⋅,∅)=0. (5.6) Then (5.3) is true on the monotone class generated by the algebra Alg=∪_𝑛ℱ_𝑛, that is, on 𝜎(𝑓).

Proof of Theorem 2.8. Given a target algorithm (𝑋,ℱ_𝑛,𝑇,𝑃_𝑛)_𝑛, let 𝜋_∞=𝜋_𝑓 be defined as in Theorem 3.7. We claim that 𝜋_∞ is optimal. Let 𝜓_𝑔 be another compatible projection and let (𝑋,𝜎(𝑔),𝑇,𝑃_𝑔) be the target problem given by Theorem 2.7. We are going to prove by induction on 𝑛 that ∀𝑛∈ℕ,ℱ_𝑛⊆𝜎𝑔. (5.7) In fact, for 𝑛=0 it is sufficient to note that ℱ₀=𝜎({𝑇})⊆𝜎(𝑔).
Equation (3.7) states that ℱ_𝑛=𝜎(ℱ_𝑛−1,ℎ_𝑛), where ℎ_𝑛 is the discrete random variable, given by Lemma 3.1, s.t.(5.8)
Let 𝑘_𝑖^(𝑛−1)∶𝑋→[0,1] be defined as 𝑘_𝑖^(𝑛−1)(𝑥)=𝑃(𝑥,𝐴_𝑖^(𝑛−1)). Then(5.9) Obviously, 𝜎(ℎ_𝑛)⊆𝜎(𝑘₁^(𝑛−1),𝑘₂^(𝑛−1),…). For the induction step, as 𝐴_𝑖^(𝑛−1)∈ℱ_𝑛−1⊆𝜎(𝑔), we have that 𝑃_𝑔(⋅,𝐴_𝑖^(𝑛−1)) is 𝜎(𝑔)-measurable, and therefore 𝜎(𝑘_𝑖^(𝑛−1))⊆𝜎(𝑔). Then ℱ_𝑛=𝜎(ℱ_𝑛−1,ℎ_𝑛)⊆𝜎(ℱ_𝑛−1,𝑘₁^(𝑛−1),𝑘₂^(𝑛−1),…)⊆𝜎(𝑔). Therefore 𝜎(𝑓)=∨_𝑛ℱ_𝑛⊆𝜎(𝑔), which implies 𝜋_∞⊇𝜓_𝑔 by Lemma 4.1, and hence 𝜋_∞ is optimal.

Corollary 5.2. 𝜋_∞ does not depend on the choice of Str.

Proof. 𝜋_∞=∩_𝑛𝜋_𝑛 is optimal, for all (𝜋_𝑛)_𝑛≥0=Str(𝑃). The optimal projection being unique, we are done.

Proof of Theorem 2.4. Let 𝜋_∞=𝜋_𝑓 be defined as in Theorem 3.7 and (𝑋,𝜎(𝑓),𝑇,𝑃_∞) be given by Theorem 2.7 so that 𝑃(𝑥,𝐹)=𝑃_∞(𝑥,𝐹) for any 𝐹∈𝜎(𝑓). Then each 𝑃_𝑛 of (3.9) can be rewritten as 𝑃_𝑛()=∫𝑥,𝐹^[𝑥]𝑛𝑃_∞(𝑥,𝐹)𝜇(𝑑𝑧)𝜇[𝑥]_𝑛,∀𝑥∈𝑋,∀𝐹∈ℱ_𝑛, (5.10) where [𝑥]_𝑛 is the 𝜋_𝑛-class of equivalence of 𝑥 and 𝜇([𝑥]_𝑛)>0 since [𝑥]_𝑛≠∅.
Note that 𝑑(𝑃,𝑃_𝑚)≤2∑_𝑛𝛽^𝑛. Then, for any 𝜖>0, there exists an 𝑁 so that ∑_𝑛>𝑁𝛽^𝑛≤𝜖/2. Therefore we are going to prove by induction on 𝑛 that sup_𝑥𝑃^𝑛_𝑚(𝑥,𝑇)−𝑃^𝑛(𝑥,𝑇)⟶0as𝑚tendstoinfinity, (5.11) which completes the proof. If 𝑛=1, then by definition of 𝜖_𝑚, since 𝑇∈ℱ_𝑚−1, we have that 𝑃_𝑚()(≤∫𝑥,𝑇−𝑃𝑥,𝑇)^[𝑥]𝑚𝑃_∞(𝑧,𝑇)−𝑃(𝑥,𝑇)𝜇(𝑑𝑧)𝜇[𝑥]_𝑚=∫_[𝑥]𝑚|𝑃(𝑧,𝑇)−𝑃(𝑥,𝑇)|𝜇(𝑑𝑧)𝜇[𝑥]_𝑚≤𝜖_𝑚∫_[𝑥]𝑚𝜇(𝑑𝑧)𝜇[𝑥]_𝑚=𝜖_𝑚. (5.12) For the induction step, we note that 𝑃_𝑚^𝑛+1(𝑥,𝑇)−𝑃^𝑛+1(𝑥,𝑇)≤_𝑖𝑃_𝑚𝑥,𝐴_𝑖^(𝑚)𝑃^𝑛_𝑚𝐴_𝑖^(𝑚),𝑇−_{𝐴𝑖(𝑚)}𝑃(𝑥,𝑑𝑧)𝑃^𝑛(𝑧,𝑇), (5.13) where (𝐴_𝑖^(𝑚))_𝑖 is the partition of 𝑋 given by 𝜋_𝑚. By induction hypothesis, for any 𝜖̃>0, 𝑃^𝑛_𝑚(𝑧,𝑇)−𝑃^𝑛(𝑧,𝑇)≤𝜖̃ (5.14) for 𝑚≥𝑚₀ large enough. Since [𝑧]_𝑚=𝐴_𝑖^(𝑚) if 𝑧∈𝐴_𝑖^(𝑚), it follows that _{𝐴𝑖(𝑚)}𝑃(𝑥,𝑑𝑧)𝑃^𝑛_𝑚𝐴_𝑖^(𝑚),𝑇−𝑃^𝑛(𝑧,𝑇)≤𝜖̃_{𝐴𝑖(𝑚)}𝑃(𝑥,𝑑𝑧). (5.15) Equation (5.13) becomes 𝑃_𝑚^𝑛+1(𝑥,𝑇)−𝑃^𝑛+1(𝑥,𝑇)≤𝜖̃+_𝑖𝑃^𝑛_𝑚𝐴_𝑖^(𝑚),𝑇𝑃_𝑚𝑥,𝐴_𝑖^(𝑚)−𝑃𝑥,𝐴_𝑖^(𝑚)≤𝜖̃+_𝑖𝑃_𝑚𝑥,𝐴_𝑖^(𝑚)−𝑃𝑥,𝐴_𝑖^(𝑚). (5.16) On the other hand, by (5.10), 𝑃_𝑚𝑥,𝐴_𝑖^(𝑚)−𝑃𝑥,𝐴_𝑖^(𝑚)=_[𝑥]𝑚𝑃_∞𝑧,𝐴_𝑖^(𝑚)−𝑃𝑥,𝐴_𝑖^(𝑚)𝜇[𝑥]_𝑚𝜇(𝑑𝑧). (5.17) The definition of ∼_𝜖𝑚+1 states that _𝑖𝑃_∞𝑧,𝐴_𝑖^(𝑚)−𝑃𝑥,𝐴_𝑖^(𝑚)≤𝜖_𝑚+1 (5.18) whenever 𝑧∈[𝑥]_𝑚 and therefore 𝑃_𝑚^𝑛+1(𝑥,𝑇)−𝑃^𝑛+1(𝑥,𝑇)≤𝜖̃+_{[𝑥]𝑚𝑖}𝑃_∞𝑧,𝐴_𝑖^(𝑚)−𝑃𝑥,𝐴_𝑖^(𝑚)𝜇(𝑑𝑧)𝜇[𝑥]_𝑚≤𝜖̃+𝜖_𝑚+1. (5.19) Since 𝜖_𝑚→0 as 𝑚 tends to infinity, we get the result.

Proof of Theorem 2.9. By (3.9) and Lemma 3.6, (𝑃_𝑛(⋅,𝐹))_𝑛≥𝑚 is a martingale with respect to the filtration (ℱ_𝑛)_𝑛≥𝑚, for any 𝐹∈ℱ_𝑚. Then, if 𝑌^𝐹(𝑥)=𝑃(𝑥,𝐹) as in (3.9), we have that 𝑃_𝑛(𝑥,𝐹)⟶_𝑛→∞𝔼_𝜇𝑌^𝐹∣∨_𝑛ℱ_𝑛(𝑥)=𝑌^𝐹(𝑥),for𝜇-a.e.𝑥∈𝑋,∀𝐹∈ℱ_𝑚. (5.20) Let 𝜋_∞=𝜋_𝑓 be defined as in Theorem 3.7 and (𝑋,𝜎(𝑓),𝑇,𝑃_∞) given by Theorem 2.7 so that 𝑃(𝑥,𝐹)=𝑃_∞(𝑥,𝐹) for any 𝐹∈𝜎(𝑓). Unfortunately, (5.20) is not enough to state that 𝑃_𝑛(𝑥,𝐹)⟶_𝑛→∞𝑃_∞(𝑥,𝐹),for𝑥∈𝑋,∀𝐹∈_𝑚ℱ_𝑚, (5.21) even if 𝜎(𝑓) is countably generated (see, e.g., [15], for counterexample). In fact, Polish assumption is assumed in [15] to guarantee (5.21).
Here, we will deal with the specific properties of 𝑃_𝑛 and 𝑃_∞ to deduce (5.21). Take 𝐹∈ℱ_𝑚 and 𝑛>𝑚. By (5.10) and the definition of 𝜖_𝑛, since 𝐹∈ℱ_𝑛−1, we have, for any 𝑥∈𝑋, that𝑃_𝑛()(≤∫𝑥,𝐹−𝑃𝑥,𝐹)^[𝑥]𝑛𝑃_∞(𝑧,𝐹)−𝑃(𝑥,𝐹)𝜇(𝑑𝑧)𝜇[𝑥]_𝑛=∫_[𝑥]𝑛|𝑃(𝑧,𝐹)−𝑃(𝑥,𝐹)|𝜇(𝑑𝑧)𝜇[𝑥]_𝑛≤𝜖_𝑛∫_[𝑥]𝑛𝜇(𝑑𝑧)𝜇[𝑥]_𝑛=𝜖_𝑛 (5.22) since the only 𝜇-null set in ℱ_𝑛 is the empty set. Then 𝑃_𝑛(𝑥,𝐹)⟶_𝑛→∞𝑃_∞(𝑥,𝐹) (5.23) for any 𝑥∈𝑋 and 𝐹∈∪_𝑚ℱ_𝑚.

5.1. Weak Convergence of Conditional Probabilities

Let the target problem (𝑋,ℱ,𝑇,𝑃) be given and let Str=(Str_𝑛)_𝑛, where Str_𝑛(𝑋,ℱ,𝑇,𝑃)=(𝑋,ℱ_𝑛,𝑇,𝑃_𝑛) be a target algorithm. In order to prove Theorem 2.13, which states the total convergence of the probability measure 𝑃_𝑛(𝑥,⋅) towards 𝑃(𝑥,⋅), we proceed as follows: (i)first, we define the topology 𝜚_𝑃 on 𝑋; (ii)then, we define a “natural” topology 𝜏_Str on 𝑋 associated to any target algorithm (Str_𝑛)_𝑛. We prove in Theorem 5.4 the total convergence of (𝑃_𝑛)_𝑛≥0 to 𝑃_∞, under this topology; (iii)then, we define the topology 𝜏_𝑃 on 𝑋 as the intersection of all the topologies 𝜏_Str; (iv)finally, we show Theorem 2.13 by proving that 𝜚_𝑃⊆𝜏_Str. The nontriviality of 𝜚_𝑃 will imply that of 𝜏_𝑃.

We introduce the pseudometric 𝑑_𝑃 on 𝑋 as follows:𝑑_𝑃𝑥,𝑦=_𝑛𝛽^𝑛𝑃^𝑛(𝑥,𝑇)−𝑃^𝑛𝑦,𝑇. (5.24)

Now, let 𝜏_Str be the topology generated by ∪_𝑛ℱ_𝑛. 𝐶 is a closed set if and only if 𝐶=∩_𝑛𝐶_𝑛, 𝐶_𝑛∈ℱ_𝑛. In fact, if 𝐶∈ℱ_𝑛, for a given 𝑛, then 𝐶∈ℱ_𝑛+𝑝, for any 𝑝 and therefore 𝐶 is closed. (𝑋,𝜏_Str) is a topological space.

Remark 5.3. Let us go back to Example 2.11. The topology defined by asking that the sets in each ℱ_𝑛 are closed is strictly finer than the standard topology. On the other hand, the same example may be explained with left closed-right opened dyadic subdivisions, which leads to a different topology that also contains the natural one. Any other “reasonable” choice of subdivision will lead to the same point: the topologies are different, and all contain the standard one. In the same manner, we are going to show that all the topologies 𝜏_Str contain the standard one, 𝜚_𝑃.

Theorem 5.4. Let the target problem (𝑋,ℱ,𝑇,𝑃) and the target algorithm (𝑋,ℱ_𝑛,𝑇,𝑃_𝑛)_𝑛 be given. For any target algorithm Str, 𝑃_𝑛(𝑥,⋅)⟶^tot_𝜏Str𝑃(𝑥,⋅),∀𝑥∈𝑋. (5.25)

Proof. Denote by 𝑃_𝑛 any extension of 𝑃_𝑛 to ∨_𝑛ℱ_𝑛. We have to check that limsup_𝑛𝑃_𝑛(𝑥,𝐶)≤𝑃(𝑥,𝐶), for any closed set 𝐶 of Str and 𝑥∈𝑋 (see, e.g., [6]).
Let {𝐶_𝑛∈ℱ_𝑛}, with 𝐶_𝑛⊇𝐶_𝑛+1 and 𝐶=∩_𝑛𝐶_𝑛 (take, e.g., 𝐶_𝑛 as the closure of 𝐶 in ℱ_𝑛).
Note that, since 𝐶∈∨_𝑛ℱ_𝑛, we have 𝑃(𝑥,𝐶)=𝑃_∞(𝑥,𝐶). But, 𝑃_𝑛(𝑥,𝐶)−𝑃_∞(𝑥,𝐶)≤𝑃_𝑛(𝑥,𝐶_𝑛−1)−𝑃_∞(𝑥,𝐶)=𝑃_𝑛(𝑥,𝐶_𝑛−1)−𝑃_∞(𝑥,𝐶). Actually, 𝑃_𝑛𝑥,𝐶_𝑛−1−𝑃_∞(𝑥,𝐶)=𝑃_𝑛𝑥,𝐶_𝑛−1−𝑃_∞𝑥,𝐶_𝑛−1𝐼+𝑃_∞𝑥,𝐶_𝑛−1−𝑃_∞(𝑥,𝐶_𝐼𝐼). (5.26)𝐼→0 as 𝑛 tends to infinity, from the target algorithm and 𝐼𝐼→0 as 𝑛 tends to infinity, from the continuity of the measure.

An example of a natural extension of 𝑃_𝑛 to 𝑃_𝑛 is given by𝑃_𝑛(𝑥,𝐹)=𝔼_𝜇𝑌^𝐹∣ℱ_𝑛(𝑥),∀𝑥∈𝑋,∀𝐹∈∨_𝑛ℱ_𝑛, (5.27) where, for any 𝐹∈∨_𝑛ℱ_𝑛, 𝑌^𝐹 is the ∨_𝑛ℱ_𝑛-random variable such that 𝑌^𝐹(𝜔)=𝑃_∞(𝜔,𝐹). As mentioned for 𝑃_𝑛, 𝑃_𝑛(𝑥,⋅) is a probability measure, for any 𝑥∈𝑋.

Corollary 5.5. For any fixed strategy Str(𝑃), let 𝑃_𝑛 be as in Theorem 2.4. We have 𝑃_𝑛(𝑥,⋅)⟶^tot_𝜏𝑃𝑃(𝑥,⋅), (5.28) for any given 𝑥.

In order to describe the topology 𝜏_𝑃, we will denote by [[𝐹]]_∗ the closure of a set 𝐹⊆𝑋 in a given topology ∗. Note that the monotonicity of 𝜋_𝑛 implies[[𝐹]]_𝜏Str=_𝑛[[𝐹]]_𝜏Str𝑛 (5.29) where 𝜏_Str𝑛 is the (discrete) topology on 𝑋 generated by ℱ_𝑛. Since 𝜏_𝑃 is the intersection of all the topologies 𝜏_Str, we have [[𝐹]]_𝜏𝑃⊇[[𝐹]]_𝜏Str=_𝑛[[𝐹]]_𝜏Str𝑛,∀𝐹∈2^𝑋,∀Str. (5.30)

Proof of Theorem 2.13. Let 𝐹 be the closed set in 𝜚_𝑃 so defined 𝐹∶=𝑦∈𝑋∶𝑑_𝑃𝑦,𝑥≥𝑟, (5.31) that is, 𝐹 is the complementary of an open ball in (𝑋,𝑑_𝑃) with center 𝑥 and radius 𝑟. If we show that 𝐹∈𝜏_𝑃, then we are done, as the arbitrary choice of 𝑥 and 𝑟 spans a base for the topology 𝜚_𝑃.
We are going to prove𝐹=[[𝐹]]_𝜏Str=_𝑚[[𝐹]]_𝜏Str𝑚,∀Str, (5.32) which implies [[𝐹]]_𝜏𝑃=𝐹. It is always true that 𝐹⊆[[𝐹]]_∗; we prove the nontrivial inclusion 𝐹⊇∩_𝑚[[𝐹]]_𝜏Str𝑚. Assume that 𝑦∈[[𝐹]]_𝜏Str. Now, 𝑦∈[[𝐹]]_𝜏Str𝑚, for any 𝑚, and then there exists a sequence (𝑦_𝑚)_𝑛≥0 with 𝑦_𝑚∈𝐹 such that 𝑦𝜋_𝑚𝑦_𝑚, for any 𝑚. Thus, 𝑦∈∩_𝑚[𝑦_𝑚]_𝑚, where [𝑥]_𝑚 is the 𝜋_𝑚-class of equivalence of 𝑥. Thus 𝑃^𝑛_𝑚𝑦_𝑚,𝑇=𝑃^𝑛_𝑚𝑦,𝑇,∀𝑚,𝑛 (5.33) since 𝑃_𝑚(⋅,𝑇) is ℱ_𝑚-measurable. By Theorem 2.4, for any 𝑛∈ℕ, 𝑃^𝑛𝑦,𝑇−𝑃^𝑛_𝑚𝑦,𝑇+𝑃^𝑛_𝑚𝑦_𝑚,𝑇−𝑃^𝑛𝑦_𝑚,𝑇⟶_𝑚→∞0. (5.34) Now, let 𝑁 be such that ∑^∞_𝑛=𝑁𝛽^𝑛≤𝜖/4 and take 𝑛₀ sufficiently large s.t. ^𝑁_𝑛=0𝑃^𝑛𝑦,𝑇−𝑃^𝑛_𝑛0𝑦,𝑇+𝑃^𝑛_𝑛0𝑦_𝑛0,𝑇−𝑃^𝑛𝑦_𝑛0≤𝜖,𝑇2. (5.35) We have 𝑑_𝑃𝑦_𝑛0,𝑦=_𝑛𝛽^𝑛𝑃^𝑛𝑦,𝑇−𝑃^𝑛𝑦_𝑛0≤,𝑇_𝑁𝑛=0𝑃^𝑛𝑦,𝑇−𝑃^𝑛𝑦_𝑛0,𝑇+2^∞_𝑛=𝑁𝛽^𝑛≤^𝑁_𝑛=0𝑃^𝑛𝑦,𝑇−𝑃^𝑛_𝑛0+𝑃𝑦,𝑇_𝑛𝑛0𝑦,𝑇−𝑃^𝑛_𝑛0𝑦_𝑛0,𝑇+𝑃^𝑛_𝑛0𝑦_𝑛0,𝑇−𝑃^𝑛𝑦_𝑛0𝜖,𝑇+24≤𝜖2+𝜖2=𝜖, (5.36) and therefore 𝑑_𝑃𝑥,𝑦≥𝑑_𝑃𝑥,𝑦_𝑛0−𝑑_𝑃𝑦_𝑛0,𝑦≥𝑟−𝜖. (5.37) The arbitrary choice of 𝜖 implies 𝑦∈𝐹, which is the thesis.

Appendices

A. Results on Equivalence Relations

In this appendix we give the proof of auxiliary results that connect equivalency with measurability.

Proof of Lemma 3.1. Let 𝜋=𝜋_𝑓 be a discrete equivalency on 𝑋. Then 𝑋/𝜋 defines a countable measurable partition of 𝑋. Conversely, let {𝐴₁,𝐴₂,…} be a countable measurable partition on 𝑋. Define 𝑓∶𝑋→ℕ s.t. 𝑓(𝑥)=𝑛⇔𝑥∈𝐴_𝑛. Therefore 𝑓 is measurable and 𝜋=𝜋_𝑓 is a discrete equivalency on 𝑋.

Lemma A.1. Let 𝑓,𝑔 be two random variables such that 𝑔(𝑥)<𝑔(𝑦)⇒𝑓(𝑥)<𝑓(𝑦). Then 𝜎(𝑔)⊆𝜎(𝑓).

Proof. Let 𝑡∈ℝ be fixed. We must prove that {𝑔≤𝑡}∈𝜎(𝑓). If {𝑔≤𝑡} or {𝑔>𝑡} are empty, then we are done. Assume then that {𝑔≤𝑡},{𝑔>𝑡}≠∅ and define 𝑡^∗=sup(𝑓({𝑔≤𝑡})). We have the following two cases.Case 1 (𝑡^∗∈𝑓({𝑔≤𝑡})∶∃𝑥^∗∈{𝑔≤𝑡} such that 𝑡^∗=𝑓(𝑥^∗)). By definition of 𝑡^∗, {𝑔≤𝑡}⊆{𝑓≤𝑡^∗}. Conversely, let 𝑦∈{𝑔>𝑡}. Since 𝑔(𝑥^∗)≤𝑡<𝑔(𝑦), then 𝑓(𝑥^∗)=𝑡^∗<𝑓(𝑦), that is, {𝑔>𝑡}⊆{𝑓>𝑡^∗}. Then {𝑔≤𝑡}={𝑓≤𝑡^∗}∈𝜎(𝑓).Case 2 (𝑡^∗∉𝑓({𝑔≤𝑡})∶∀𝑥∈{𝑔≤𝑡} we have that 𝑓(𝑥)<𝑡^∗). Then {𝑔≤𝑡}⊆{𝑓<𝑡^∗}. Conversely, let 𝑦∈{𝑔>𝑡}. Since ∀𝑥∈{𝑔≤𝑡}𝑔(𝑦)>𝑔(𝑥), then 𝑓(𝑦)>𝑓(𝑥), which implies 𝑓(𝑦)≥sup𝑓({𝑔≤𝑡})=𝑡^∗, that is, {𝑔>𝑡}⊆{𝑓≥𝑡^∗}. Then {𝑔≤𝑡}={𝑓<𝑡^∗}∈𝜎(𝑓).

The next lemma plays a central rôle. Its proof is common in set theory.

Lemma A.2. For all 𝑛∈ℕ, let 𝜋_𝑛 be a discrete measurable equivalency. Then there exists a random variable 𝑓 such that 𝜎(𝑓)=∨_𝑛𝜎(𝑋/𝜋_𝑛).

Proof. Before proving the core of the Lemma, we build a sequence (𝑔_𝑛)_𝑛∈ℕ of functions 𝑔_𝑛∶ℕ^𝑛→ℝ that will be used to define the function 𝑓.
Take ℎ∶ℕ∪{0}→[0,1) to be the increasing function ℎ(𝑚)=1−2^−𝑚 and let (𝑔_𝑛)_𝑛∈ℕ the sequence of function 𝑔_𝑛∶ℕ^𝑛→ℝ so defined: 𝑔₁𝑚₁=ℎ𝑚₁,𝑔−1₂𝑚₁,𝑚₂=𝑔₁𝑚₁+ℎ𝑚₂−1Δ𝑔₁𝑚₁,⋮𝑔_𝑛+1𝐦_𝑛,𝑚_𝑛+1=𝑔_𝑛𝐦_𝑛+ℎ𝑚_𝑛+1Δ𝑔_𝑛𝐦_𝑛,⋮ (A.1) where, for all 𝑛, 𝐦_𝑛=(𝑚₁,…,𝑚_𝑛) and Δ𝑔_𝑛𝐦_𝑛=𝑔_𝑛𝐦_𝑛−1,𝑚_𝑛+1−𝑔_𝑛𝐦_𝑛−1,𝑚_𝑛. (A.2) As a first consequence of the definition, note that for any choice of 𝑛 and 𝐦_𝑛+1, it holds that 𝑔_𝑛𝐦_𝑛−1,𝑚_𝑛≤𝑔_𝑛+1𝐦_𝑛+1<𝑔_𝑛𝐦_𝑛−1,𝑚_𝑛+1 (A.3) since ℎ∈[0,1). We now prove by induction on 𝑛₁+𝑛₂ that for any choice of 𝑛₁∈ℕ, 𝑛₂∈ℕ∪{0} and 𝐦_𝑛1+𝑛2, we have 𝑔_𝑛1𝐦_𝑛1−1,𝑚_𝑛1≤𝑔_𝑛1+𝑛2𝐦_𝑛1+𝑛2<𝑔_𝑛1𝐦_𝑛1−1,𝑚_𝑛1+1. (A.4) Equation (A.4) is clearly true for 𝑛₁+𝑛₂=1, since ℎ is strictly monotone. The same argument shows that (A.4) is always true for 𝑛₂=0 and therefore we check it only for 𝑛₂>0. We assume by induction that (A.4) is true for 𝑛₁+𝑛₂≤𝑛 and we prove it for 𝑛₁+𝑛₂=𝑛+1. By using twice the induction hypothesis, as 𝑛₂−1≥0, we obtain 𝑔_𝑛1𝐦_𝑛1−1,𝑚_𝑛1≤𝑔_{𝑛1+𝑛2−1}𝐦_{𝑛1+𝑛2−2},𝑚_{𝑛1+𝑛2−1}<𝑔_{𝑛1+𝑛2−1}𝐦_{𝑛1+𝑛2−2},𝑚_{𝑛1+𝑛2−1}+1≤𝑔_𝑛1𝐦_𝑛1−1,𝑚_𝑛1+1. (A.5) Equation (A.4) is now a consequence of (A.3).
Now, we come back to the proof of the lemma. First note that, without loss of generality, we can (and we do) require the sequence (𝜋_𝑛)_𝑛∈ℕ to be monotone, by taking the sequence 𝜋^′_𝑛=∩^𝑛_𝑖=1𝜋_𝑖 instead of 𝜋_𝑛. 𝜋^′_𝑛 is again a countable measurable equivalency on 𝑋. In fact, by Lemma 3.1 we can read this statement in trivial terms of partitions: an at most countable intersection of countable measurable partitions is still a countable measurable partition. Moreover, by definition, ∨_𝑖=11^𝑛𝜎(𝑋/𝜋_𝑖)=∨_𝑖=11^𝑛𝜎(𝑋/𝜋^′_𝑖).
Let 𝜏_𝑛=𝑋/𝜋_𝑛 be the increasing sequence of countable measurable dissections of 𝑋. We are going to give a consistent inductive method of numbering the set of atoms of 𝜏_𝑛 to build the functions 𝑓_𝑛. Let 𝜏₁={𝐴₁⁽¹⁾,𝐴₂⁽¹⁾,…} be any ordering of 𝜏₁. By induction, let {𝐴_𝐦^(𝑛+1)_𝑛,1,𝐴_𝐦^(𝑛+1)_𝑛,2,…} be the partition of the atom 𝐴_𝐦^(𝑛)_𝑛∈𝜏_𝑛 given by 𝜏_𝑛+1. Define, for any 𝑛∈ℕ,𝑓_𝑛(𝑥)=𝑔_𝑛𝐦_𝑛⟺𝑥∈𝐴_𝐦^(𝑛)_𝑛. (A.6) To complete the proof, we first show that 𝜎(𝑓_𝑛)=𝜎(𝑋/𝜋_𝑛), ∀𝑛, and then we prove 𝜎(𝑓)=𝜎(𝑓₁,𝑓₂,…) by proving that 𝑓_𝑛→𝑓 pointwise.
To prove that 𝜎(𝑓_𝑛)=𝜎(𝑋/𝜋_𝑛) we show that 𝑓_𝑛(𝑥)=𝑓_𝑛(𝑦)⇔∃𝐦_𝑛∶𝑥,𝑦∈𝐴_𝐦^(𝑛)_𝑛. One implication is a consequence of the fact that 𝑓_𝑛 is defined on the partition of 𝑋 given by 𝑋/𝜋_𝑛=𝜏_𝑛. For the converse, assume that 𝑥∈𝐴_𝐦^(𝑛)_𝑛≠𝐴_𝐦^(𝑛)_′𝑛∋𝑦 and consider 𝑛₁∶=min{𝑗≤𝑛∶𝑚_𝑗≠𝑚^′_𝑗}. Thus 𝐦_𝑛1−1=𝐦^′_𝑛1−1 and, without loss of generalities, 𝑚_𝑛1<𝑚^′_𝑛1. By (A.4), we have 𝑓_𝑛(𝑥)=𝑔_𝑛𝐦_𝑛<𝑔_𝑛1𝐦_𝑛1−1,𝑚_𝑛1+1≤𝑔_𝑛1𝐦^′_𝑛1−1,𝑚^′_𝑛1≤𝑔_𝑛𝐦^′_𝑛=𝑓_𝑛𝑦. (A.7) We are going to prove that 𝜎(𝑓)=𝜎(𝑓₁,𝑓₂,…).Proof of ⊆. The sequence (𝑓_𝑛)_𝑛 is monotone by definition and bounded by (A.4). Then ∃𝑓∶𝑓_𝑛↑𝑓 and thus 𝜎(𝑓)⊆𝜎(𝑓₁,𝑓₂,…).Proof of ⊇. Let 𝑛 be fixed, and take 𝑥,𝑦∈𝑋 with 𝑓_𝑛(𝑥)<𝑓_𝑛(𝑦). Then, for any ℎ≥0, 𝜏_𝑛⊆𝜏_𝑛+ℎ implies 𝑥∈𝐴_𝐦^(𝑛+ℎ)_𝑛+ℎ≠𝐴_𝐦^(𝑛+ℎ)_′𝑛+ℎ∋𝑦. As above, consider 𝑛₁∶=min{𝑗≤𝑛∶𝑚_𝑗≠𝑚^′_𝑗}. As 𝑓_𝑛(𝑥)<𝑓_𝑛(𝑦), we have 𝐦_𝑛1−1=𝐦^′_𝑛1−1 and 𝑚_𝑛1<𝑚^′_𝑛1. Again, by (A.4), for ℎ>𝑛₁+1−𝑛, 𝑓_𝑛+ℎ(𝑥)=𝑔_𝑛+ℎ𝐦_𝑛+ℎ<𝑔_𝑛1+1𝐦_𝑛1,𝑚_𝑛1+1+1=𝛼<𝑔_𝑛1𝐦_𝑛1−1,𝑚_𝑛1+1≤𝑔_𝑛𝐦^′_𝑛=𝑓_𝑛𝑦, (A.8) that is, ∀ℎ, 𝑓_𝑛+ℎ(𝑥)<𝛼<𝑓_𝑛(𝑦). As 𝑓_𝑙↑𝑓, 𝑓(𝑥)<𝑓(𝑦). Apply Lemma A.1 with 𝑔=𝑓_𝑛 to conclude that 𝜎(𝑓_𝑛)⊆𝜎(𝑓).

As a consequence of Lemma A.2, any countably generated sub-𝜎-algebra is generated by a measurable equivalence 𝜋, as the following lemma states.

Lemma A.3. 𝒜⊆𝒳 is countably generated if and only if there exists a random variable 𝑓 such that 𝒜=𝜎(𝑓).

Proof. ⇒ Let 𝒜=𝜎(𝐴₁,𝐴₂,…). Apply Lemma A.2 with 𝑋/𝜋_𝑛={𝐴_𝑛,𝑋⧵𝐴_𝑛}.
⇐ Take a countable base 𝐵₁,𝐵₂,… of ℬ_ℝ and simply note that 𝜎(𝑓)=𝜎({𝑓⁻¹(𝐵₁),𝑓⁻¹(𝐵₂),…}).

Proof of Lemma 4.1. Let 𝑥∈𝑋 be fixed. By hypothesis, 𝒜₁⊆𝒜₂. If 𝒜₁=𝜎(𝐴¹₁,𝐴¹₂,…) then 𝒜₂ will be of the form 𝒜₂=𝜎(𝐴¹₁,𝐴²₁,𝐴¹₂,𝐴²₂,…). Without loss of generality (if needed, by choosing 𝑋⧵𝐴^𝑗_𝑛 instead of 𝐴^𝑗_𝑛) we can require 𝑥∈𝐴^𝑗_𝑛, for any 𝑛∈ℕ and 𝑗=1,2. Then [𝑥]_𝒜2=∩_𝑛(𝐴¹_𝑛∩𝐴²_𝑛)⊆∩_𝑛𝐴¹_𝑛=[𝑥]_𝒜1.
The last part of the proof is a consequence of Lemma A.3 and of the first point, since𝑓⁻¹𝑓(𝑥)=[𝑥]_𝜋𝑓⊆[𝑥]_𝜋𝑔=𝑔⁻¹𝑔(𝑥), (A.9) or, equivalently, 𝑓(𝑥)=𝑓(𝑦)⇒𝑔(𝑥)=𝑔(𝑦) which is the thesis.

Proof of Theorem 5.1. Note that 𝑋/𝜋_∞⊆𝒳 is countable and generated by ∪_𝑛𝑋/𝜋_𝑛. Then 𝜋_∞ is a measurable equivalency by Lemma A.3.
Conversely, we can use the standard approximation technique: if 𝜋=𝜋_𝑓 is measurable, let 𝑓_𝑛=2^−𝑛⌊2^𝑛𝑓⌋ for any 𝑛. Since 𝑓_𝑛 are discrete random variables, 𝜋_𝑛 are defined through Lemma 3.1. By Lemma 4.1 and (3.3), the thesis 𝜋_𝑓=∩_𝑛𝜋_𝑛 will be a consequence of the fact that 𝜎(𝑓)=∨_𝑛𝜎(𝑓_𝑛).
𝜎(𝑓_𝑛)⊆𝜎(𝑓) by definition, which implies 𝜎(𝑓₁,𝑓₂,…)⊆𝜎(𝑓). Finally, as 𝑓_𝑛→𝑓, we have 𝜎(𝑓)⊆𝜎(𝑓₁,𝑓₂,…), which completes the proof.

B. Proof of Theorem 4.6

Before proving the theorem, we state the following lemma.

Lemma B.1. Let (𝑋,𝒳) be a measurable space. (1)If any equivalence relationship 𝜋 on 𝑋 is measurable, then 𝒳=2^𝑋 and card(𝑋)≤card(ℝ). (2)The converse is true under the axiom of choice.

Proof. (1)⇒(2). Let 𝜋_𝐼 be the identity relation: 𝑥𝜋_𝐼𝑦⇔𝑥=𝑦. By hypothesis, there exists 𝑓 such that 𝜋_𝐼=𝜋_𝑓, and thus 𝑓 is injective. Then card(𝑋)≤card(ℝ). Now, take 𝐴⊆𝑋 and let 𝜋_𝐴 be the relation so defined: 𝑥𝜋_𝐴𝑦⟺𝑥,𝑦⊆𝐴or𝑥,𝑦⊆𝑋⧵𝐴. (B.1) Since any equivalency is measurable, then there exists 𝑓∶(𝑋,𝒳)→(ℝ,ℬ_ℝ) such that 𝜋_𝐴=𝜋_𝑓. But 𝜎(𝑓)=𝜎(𝐴), which shows that 𝐴⊆𝑋⇒𝐴∈𝒳, that is, 𝒳=2^𝑋.
(2)⇒(1). Since card(𝑋)≤card(ℝ), there exists an injective function ℎ∶𝑋→ℝ. Let 𝜋 be an equivalence relationship on 𝑋, and define the following equivalence on ℝ:𝑟₁𝑅𝑟₂⟺𝑟₁,𝑟₂⊆ℎ(𝑋),ℎ⁻¹𝑟₁𝜋ℎ⁻¹𝑟₂or𝑟₁,𝑟₂⊆ℝ⧵ℎ(𝑋). (B.2) By definition of 𝑅, if we denote by 𝜋_𝑅 the canonical projection of ℝ on ℝ/𝑅, then 𝜋_𝑅∘ℎ∶𝑋→ℝ/𝑅 is such that 𝜋_𝑅∘ℎ(𝑥)=𝜋_𝑅∘ℎ𝑦⟺𝑥𝜋𝑦. (B.3) The axiom of choice ensures the existence of a injective map 𝑔∶ℝ/𝑅→ℝ. Then 𝑓∶=𝑔∘𝜋_𝑅∘ℎ∶𝑋→ℝ is such that 𝜋=𝜋_𝑓. 𝑓 is measurable since 𝒳=2^𝑋.

Proof of Theorem 4.6. (1)⇒(2). By Lemma B.1 and assumption (A0), (𝑋,2^𝑋) is weakly Blackwell.
(2)⇒(3). Assume 𝑋 is uncountable. By CH, exists 𝑌⊆𝑋 s.t. (i.e., 𝑌 is in bijection with ℝ via 𝑔₁). Take a bijection ⧵{0}. Then the map𝑔(𝑥)=𝑔₂𝑔₁(𝑥)if0𝑥∈𝑌,if𝑥∈𝑋⧵𝑌, (B.4) is a bijective map from {𝑌,{𝑋⧵𝑌}} to ℝ. Equip ℝ with the Borel 𝜎-algebra ℬ_ℝ and let 𝒜₁=𝑔⁻¹(ℬ_ℝ). 𝒜₁ is countably generated and its atoms are all the points in 𝑌 and the set 𝑋⧵𝑌. Now, take a nonBorel set 𝑁 of the real line. 𝒜₂=𝑔⁻¹(𝜎(ℬ_ℝ,𝑁)) is also countably generated, 𝒜₁⊊𝒜₂ and its atoms are all the points in 𝑌 and the set 𝑋⧵𝑌, too. Since 𝒜₁⊆2^𝑋 and 𝒜₂⊆2^𝑋, (𝑋,2^𝑋) is not a weakly Blackwell space by Lemma 4.3.
(3)⇒(1). Since 𝑋 is countable, then 𝑋/𝜋 is. Therefore, Lemma 3.1 ensures that any equivalence 𝜋 is measurable, since 𝒳=2^𝑋. Finally, just note that each countable set is strongly Blackwell. And thus Lemma 4.3 concludes the proof.

Acknowledgments

The authors are very grateful to the anonymous referees whose incisive comments improved the paper. This work was made while the first author was hosted in Jerusalem. He wishes to thank for the warm hospitality.