Rich and Saturated Adapted Spaces

H. Jerome Keisler

1. Introduction

2. Law Mappings

3. Basic Sections and Neocompact Sets

4. Probability Spaces

5. Adapted Spaces with Finite Time Sets

6. Atomless Adapted Spaces

7. Richness and Saturation

8. Adapted Spaces with Infinite Time Sets

9. Right Continuous Adapted Spaces

1 Introduction

In the paper [HK] the notions of an adapted distribution and of a saturated adapted probability space were introduced. The adapted distribution of a random variable on an adapted space (with values in a complete separable metric space) is the natural analogue of the distribution of a random variable on a probability space. An adapted space Ω is saturated if for any random variable x on Ω and pair of random variables x̄ and ȳ on another adapted space Γ such that x and x̄ have the same adapted distribution, there is a random variable y on Ω such that (x, y) and (x̄, ȳ) have the same adapted distribution. For stochastic differential equations and a wide variety of other existence problems, every existence theorem which holds on some adapted space holds on a saturated adapted space.

The paper [FK1] introduced a new method for proving existence theorems in probability theory, based on the notion of a neocompact set of random variables. A set of random variables on an adapted space is said to be basic if it is either compact or is the set of all random variables which are measurable at time t and whose law belongs to a compact set C of measures, for some t and C. The family of neocompact sets is the closure of the family of basic sets under finite unions and Cartesian products, countable intersections, existential projections, and “universal projections with respect to a nonempty basic set”. An adapted space is said to be rich if the family of neocompact sets is countably compact. On a rich adapted space, the neocompact sets play a role analogous to the compact sets. They were used in the papers [FK1] and [CK] to prove a variety of optimization and existence theorems. The existence of rich adapted spaces for any linearly ordered set of times was proved in [FK2].

The purpose of this paper is to find the relationship between richness and saturation. Our main theorem is that richness and saturation are equivalent for adapted spaces with a countable set of times. For example, the two notions are equivalent for probability spaces, for discrete time adapted spaces, and for adapted spaces with dyadic rational times. We also show that for any rich adapted space with dyadic rational times, the associated right continuous adapted space with real times is saturated but cannot be rich.

Our proofs will use a “quantifier elimination” theorem from the paper [K5] which shows that in a rich or saturated space with a countable time set, the neocompact sets can be represented in a simple form. The paper [K5], which was aimed primarily at model theorists, introduced a very general notion called a law structure, which is an abstraction of the distribution and the adapted distribution in probability theory. This paper is aimed at probabilists, and applies the results of [K5] to probability spaces and adapted spaces.

In Section 2 we introduce the notion of a law mapping on a probability space Ω, which is a special case of the notion of a law structure from [K5]. We shall also state the results we need from [K5]. In Section 3 we prove some general results about law mappings. The rest of the paper deals with the particular law mappings which correspond to the distribution of a random variable and adapted distribution of a stochastic process. In Sections 4 through 8 we prove our main results showing that saturation is equivalent to richness for probability spaces, adapted spaces with finite time sets, and adapted spaces with infinite time sets. Finally, in Section 9 we prove that every rich adapted space with rational times induces a saturated right continuous adapted space with real times.

I wish to thank Sergio Fajardo and Siu-Ah Ng for helpful suggestions on this article. This research was supported in part by the National Science Foundation and the Vilas Trust Fund.

2 Law Mappings

In this section we introduce the notion of a law mapping on a probability space Ω, and state the theorems we shall need from the paper [K5].

Throughout this paper we let K = (K, τ), M = (M, ρ), and N = (N, σ) be complete separable metric spaces, and let Ω = (Ω, P, G) be a probability space. For each complete separable M, we use the corresponding script letter M to denote the metric space M = (L^{0}(Ω, M), ρ_{0}) of all equivalence classes of P -measurable functions from Ω into M. Here two functions are equivalent if they are equal P - almost surely, and ρ_{0} is the metric of convergence in probability on L^{0}(Ω, M),

ρ_{0}(x, y) = inf{ε : P [ρ(x(ω), y(ω)) ≤ ε] ≥ 1 − ε}.

We shall let M_{Ω} be the family of all the metric spaces L^{0}(Ω, M) where M is a complete separable metric space, so that M, N , K are arbitrary elements of M_{Ω}.

A Cartesian product M × N with the metric ρ × σ defined by

(ρ × σ)((x, y), (x̄, ȳ)) = max(ρ(x, x̄), σ(y, ȳ))

is again a complete separable metric space. The metrics ρ_{0} × σ_{0} and (ρ × σ)_{0} on M × N = L^{0}(Ω, M × N) are different, but determine the same topology. A similar remark holds for countable Cartesian products ∏_{n} K_{n} with the metric τ = ∏_{n} τ_{n} defined by

τ(x, y) = ∑_{n} min(1, τ_{n}(x_{n}, y_{n}))/2^{n}.

A subset of a topological space Λ is relatively compact if it is contained in a compact subset of Λ. Recall that a topological space is first countable if every point has a countable neighborhood base. For example, every metrizable space is first countable. In a first countable space, a set is closed if and only if it contains the limit of any convergent sequence of points in the set.

For each continuous function f : M → N, we let f̂: M → N be the function defined by (f̂(x))(ω) = f(x(ω)). f̂ is continuous from M to N.

The space of Borel probability measures on M with the Prohorov metric

d(μ, ν) = inf{ε : μ(C) ≤ ν(C^{ε}) + ε for all closed C ⊆ M}

is denoted by Meas(M). It is again a complete separable metric space, and convergence in Meas(M) is the same as weak convergence. Each random variable x ∈ M induces the measure law(x) ∈ Meas(M) such that (law(x))(S) = P [x^{−1}(S)] for each Borel S ⊆ M, and the function law : M → Meas(M) is continuous.

Definition 2.1 A law mapping on Ω is a pair (λ, Λ) which assigns to each M ∈ M_{Ω} a first countable Hausdorff space Λ(M) and a continuous function λ : M → Λ(M) such that:

1. For each x, x_{1}, x_{2}, … in M, if λ(x_{n}) → λ(x) in Λ(M), then law(x_{n}) → law(x) in Meas(M).

2. If A ⊆ M, B ⊆ N, and the images λ(A) and λ(B) are relatively compact in Λ(M) and Λ(N), then the image λ(A×B) is relatively compact in Λ(M×N).

3. For each continuous function f : M → N there is a continuous function f̄: λ(M) → λ(N) such that the following diagram is commutative:

[figure: M; λ; λ(M); f̄; λ(N); f; λ; N]

Moreover, if f is a metric isometry of M onto N, then f̄ is a homeomorphism from λ(M) to λ(N).

Since λ is continuous, convergence in probability implies convergence in λ. Condition (2.1.1) says that convergence in λ in turn implies convergence in law. One consequence of condition (2.1.1) is that λ(x) = λ(y) implies law(x) = law(y). Another consequence is that λ(x, y) = λ(z, z) implies x = y.

In condition (2.1.3), λ(M) denotes the image of M under the function λ, which is a possibly proper subspace of the space Λ(M). (2.1.3) says that for each continuous f : M → N, the function λ(x) ↦ λ(f(x)) is well-defined and continuous, and is denoted by f̄.

Condition (2.1.3) is often applied to projections. If f is the projection from M×N to M, then f̄ sends λ(x, y) to λ(x) and is called the projection function from λ(M× N) to λ(M). There is an analogous projection function from λ(M × N) to λ(N). Combining the two projections, it follows that the function λ(x, y) ↦ (λ(x), λ(y)) is a continuous surjection from λ(M × N) to the product space λ(M) × λ(N). In general, this function is many-one, and the space λ(M × N) is more complicated than the product space λ(M) × λ(N).

The canonical example of a law mapping on Ω, developed in Section 4, is the pair (λ, Λ) where Λ(M) is the space Meas(M) and λ(x) = law(x). In this case, λ(x, y) is the law of the joint random variable (x, y), and λ(x) and λ(y) are the laws of the marginals x and y.

We shall usually suppress the target space Λ and speak of a law mapping λ. However, it should be kept in mind that a law mapping can be changed in an essential way by extending or restricting the space Λ(M).

In the paper [K5] the notion of a law structure (M, λ, Λ) was introduced in the more general setting where the family M_{Ω} is replaced by a family M of sets closed under finite Cartesian products. In that setting, the sets X ∈ M do not have metrics, the function λ is not required to be continuous, condition (2.1.1) is replaced by the weaker condition that λ(x, y) = λ(z, z) implies x = y, and condition (2.1.3) is required only for the case that f is a projection from one finite Cartesian product to another.

Thus whenever (λ, Λ) is a law mapping on a probability space Ω, the triple (M_{Ω}, λ, Λ) is a law structure in the sense of [K5].

We next introduce some properties of law mappings which were studied in [K5]. For each x ∈ M and each C ⊆ N, let

λ(x, C) = {λ(x, y) : y ∈ C}, λ(C, x) = {λ(y, x) : y ∈ C}.

Definition 2.2 Let λ be a law mapping on Ω.

λ has the back and forth property if for all x, x̄ ∈ M such that λ(x) = λ(x̄), we have λ(x, N) = λ(x̄, N) for all N. That is, if λ(x) = λ(x̄) then for every y ∈ N there exists ȳ ∈ N such that λ(x, y) = λ(x̄, ȳ).

λ is said to be dense if whenever x, x̄ ∈ M and λ(x) = λ(x̄), λ(x, N) and λ(x̄, N) have the same closure in Λ(M × N).

λ is said to be closed if λ(M) is closed in Λ(M) for all M ∈ M_{Ω}.

λ has the Skorokhod property if for every x ∈ M and sequence c_{n} which converges to λ(x) in λ(M), there exists a sequence x_{n} in M such that λ(x_{n}) = c_{n} for each n and x_{n} converges to x in M.

We shall see in Section 4 that the Skorokhod property is closely related to the Skorokhod representation theorem. The next proposition shows that the Skorokhod property for a law mapping is equivalent to a condition which does not mention the metric on M and was called the “strong open mapping property” in [K5].

Proposition 2.3 Let λ be a law mapping on Ω. Then λ has the Skorokhod property if and only if for each M and each y ∈ N, the projection from λ(M × N) to λ(M) restricted to λ(M, y) is open.

Proof: The second condition is equivalent to the following:

(1) For each x ∈ M, y ∈ N, and sequence c_{n} converging to λ(x) in λ(M), there is a sequence x_{n} in M such that λ(x_{n}) = c_{n} for all sufficiently large n, and λ(x_{n}, y) converges to λ(x, y) in λ(M × N).

The Skorokhod property implies (1) because if x_{n} → x in M then (x_{n}, y) → (x, y) in M × N, and by the continuity of λ, λ(x_{n}, y) → λ(x, y).

For the converse, assume (1). Let c_{n} → λ(x) in λ(M). By (1) there exist x_{n} in M such that λ(x_{n}) = c_{n} for all n, and λ(x_{n}, x) → λ(x, x). By (2.1.1) we have law(x_{n}, x) → law(x, x) in Meas(M × M). Therefore x_{n} → x in M, and the Skorokhod property is proved. □

We state a result from [K5].

Proposition 2.4 A law mapping λ on Ω has the back and forth property if and only if

(i) λ is and dense, and

(ii) Whenever λ(x, y_{n}) converges to λ(x̄, ȳ) in λ(M × N), there exists y ∈ N such that λ(x, y) = λ(x̄, ȳ). □

Condition (ii) in the above proposition is called completeness in [HK] and [K5]. We now review the notions of a basic set and a basic section from [K5]. Basic

sections play a central role in the study of neocompact sets.

Definition 2.5 A set B ⊆ M is basic for a law mapping λ on Ω if B is of the form

B = {x ∈ M : λ(x) ∈ B̂}

for some compact subset B̂ of Λ(M).

Let z ∈ K. A set C ⊆ M is called a basic section for λ with parameter z on Ω if C has the form

C = {x ∈ M : λ(x, z) ∈ Ĉ}

for some compact subset Ĉ of Λ(M × K).

We say that a family C of sets is countably compact if every decreasing chain C_{0} ⊃ C_{1} ⊃ ⋯ of nonempty sets in C has a nonempty intersection ⋂_{n} C_{n}.

Every basic section for λ is closed in M, because the function λ is continuous. The following proposition and theorem on basic sections were proved in [K5].

Proposition 2.6 Let λ be a law mapping on Ω.

(i) For every z ∈ K, every basic set for λ is a basic section for λ with parameter z.

(ii) Let y ∈ N and z ∈ K. Every basic section B ⊆ M for λ with parameter y is a basic section for λ with parameter (y, z).

(iii) If A ⊆ M and B ⊆ M are basic sections for λ, then A ∩ B and A ∪ B are basic sections for λ.

(iv) For each M ∈ M_{Ω}, every finite subset A = {x_{1}, …, x_{m}} of M is a basic section for λ with parameter z = (x_{1}, …, x_{m}) in the Cartesian power K = M^{m}.

(v) Suppose λ is closed and has the back and forth property. Let A ⊆ M be a basic section for λ with parameter z. Then for each y ∈ N, the set B = λ(A, y) is compact in Λ(M × N). □

Theorem 2.7 ([K1, Theorem 4.9 and Corollary 4.10]).

(i) A law mapping λ on Ω is closed if and only if the family of basic sets B ⊆ M for λ is countably compact.

(ii) A law mapping λ on Ω is closed and has the back and forth property if and only if it is dense and for each z ∈ K, the family of basic sections B ⊆ M for λ with parameter z is countably compact. □

We now introduce the notion of a neocompact set over a family of sets A, which corresponds to the notion of a neocompact formula over A in [K5]. We shall then state a quantifier elimination theorem from [K5], which shows that the neocompact sets can be represented in a simple form.

Definition 2.8 For each M ∈ M_{Ω}, let A(M) be a family of subsets of M. A neocompact set over A is a set which is built using the following rules.

(a) Every set in A(M) is neocompact over A.

(b) The union of two neocompact subsets of M over A is neocompact over A.

(c) If A ⊆ M and B ⊆ N are neocompact over A, then A × B is neocompact over A.

(d) If ⟨A_{n} : n ∈ N⟩ is a countable sequence of neocompact subsets of M over A, then the intersection ⋂_{n} A_{n} is a neocompact set over A.

(e) If A ⊆ M × N is neocompact over A, then the existential projection

{x ∈ M : (∃y ∈ N)(x, y) ∈ A}

is neocompact over A, and the analogous rule holds for each factor in a finite Cartesian product.

(f) If A ⊆ M × N is neocompact over A and ∅ ≠ C ∈ A(N), then the universal projection

{x ∈ M : (∀y ∈ C)(x, y) ∈ A}

is neocompact over A, and the analogous rule holds for each factor in a finite Cartesian product.

In [FK1] and [FK2], the family of neocompact sets over A is called the neocompact family generated by (M_{Ω}, A).

A function f : M → N is neocontinuous over A if the graph of f|C is neocompact over A for each neocompact set C ⊆ M over A.

Let z ∈ K. We say that a set C ⊆ M is a basic section over A with parameter z if C has the form

C = {x ∈ M : (x, z) ∈ D}  (1)

for some D ∈ A(M × K), and that C is a neocompact section over A with parameter z if C has the form (2) for some neocompact set D ⊆ M × K over A.

Thus C is a basic section for λ as previously defined if and only if C is a basic section over the family of basic sets for λ.

It is obvious that every neocompact set over A is a neocompact section over A. The following proposition is a converse.

Proposition 2.9 ([FK1, Proposition 3.6]). Suppose that for each M, every finite subset of M belongs to A(M). Then every neocompact section over A is a neocompact set over A. □

The next theorem was proved in [K5, Theorem 6.5 and Corollary 6.6].

Theorem 2.10 (Quantifier Elimination for Neocompact Formulas) Let λ be a closed law mapping on Ω, and let A(M) be the family of basic subsets of M for λ. The following are equivalent.

(i) λ has the back and forth and Skorokhod properties.

(ii) Each neocompact set over A is basic for λ.

(iii) Each neocompact section over A with parameter z is a basic section over A with parameter z. □

Corollary 2.11 Let λ be a closed law mapping with the back and forth and Skorokhod properties. Let B(M) be the family of subsets of M which are either finite or basic for λ. Then a set is neocompact over B if and only if it is a basic section for λ.

Proof: By Proposition 2.9, every basic section for λ is neocompact over B. By the Quantifier Elimination Theorem and Proposition 2.6, the family of basic sections for λ is closed under the rules (a)–(f) with respect to B(M). □

The existential quantifier step of the proof of the Quantifier Elimination Theorem 2.10 used the following result (Theorem 5.2 in [K5]), which will be useful in its own right in this paper.

Theorem 2.12 Let λ be a closed law mapping. The following are equivalent. (i) λ has the back and forth property. (ii) For every basic set A ⊆ M × N for λ, the set

B = {x ∈ M : (∃y ∈ N)(x, y) ∈ A}  (2)

is basic for λ.

The implication (ii) ⇒ (i) holds for all law mappings.

3 Basic Sections and Neocompact Sets

In this section we shall prove some additional results about basic sections for law mappings. Throughout this section we assume that λ is a law mapping on a probability space Ω.

Lemma 3.1 For each countable sequence ⟨C_{n}⟩ of basic sections for λ in M, there is a single space K ∈ M_{Ω} and z ∈ K such that each C_{n} is a basic section for λ with parameter z.

Proof: We have

C_{n} = {x ∈ M : λ(x, z_{n}) ∈ D_{n}}

where z_{n} ∈ K_{n} ∈ M_{Ω} and D_{n} is compact in Λ(M × K_{n}). Let K be the product space K = ∏_{n} K_{n}. Let z = ⟨z_{n}⟩ ∈ K. Fix n ∈ N. Let Ĉ_{n} be the closure of the set

{λ(u, z) : u ∈ M and λ(u, z_{n}) ∈ D_{n}}.

Ĉ_{n} is relatively compact and hence compact by (2.1.3) and (2.1.2). We claim that

C_{n} = {x ∈ M : λ(x, z) ∈ Ĉ_{n}}.

By the definition of Ĉ_{n}, C_{n} is contained in the right side. Suppose λ(x, z) ∈ Ĉ_{n}. Then

λ(x, z) = lim_{k→∞} λ(x_{k}, z)

for some sequence x_{k} in M with λ(x_{k}, z_{n}) ∈ D_{n}. By (2.1.3),

λ(x, z_{n}) = lim_{k→∞} λ(x_{k}, z_{n}),

so λ(x, z_{n}) ∈ D_{n} and x ∈ C_{n}. This proves the claim and shows that C_{n} is a basic section for λ with parameter z. □

Corollary 3.2 For each M the family of basic sections for λ in M is closed under countable intersections.

Proof: This follows from the preceding lemma and the equation

⋂_{n}{x ∈ M : λ(x, z) ∈ D_{n}} = {x ∈ M : λ(x, z) ∈ ⋂_{n} D_{n}} □

Corollary 3.3 If λ is closed and has the back and forth property, then for each M ∈ M_{Ω} the family of basic sections for λ in M is countably compact.

Proof: By Theorem 2.7 and Lemma 3.1. □

We shall call a set B ⊆ M basic/compact for λ if it is either a basic subset of M for λ or is a compact subset of M. We say that C is neocompact for λ if C is neocompact over the family of basic/compact sets for λ. By Proposition 2.9, every neocompact section over the basic/compact sets for λ is a neocompact set for λ.

Proposition 3.4 Each compact set C ⊆ M is a basic section for λ. Moreover, a set A ⊆ M is a basic section for λ if and only if A is a basic section over the basic/compact sets for λ.

Proof: Suppose C is nonempty, and choose a countable sequence z = ⟨z_{n}⟩ which is dense in C. Then z belongs to the countable Cartesian power K = M^{N}, and K ∈ M_{Ω}. Since λ is continuous on M × K, the set

D = {λ(x, z) : x ∈ C}

is compact in Λ(M × K). It suffices to show that

C = {x ∈ M : λ(x, z) ∈ D}.  (3)

Clearly C is contained in the right side. Let λ(x, z) ∈ D. Then λ(x, z) = λ(y, z) for some y ∈ C. Therefore some subsequence of z_{n} converges to y in M. To simplify notation suppose that lim_{n→∞} z_{n} = y. By (2.1.3), we have λ(x, z_{n}) = λ(y, z_{n}) for each n. Moreover,

lim_{n→∞} λ(x, z_{n}) = λ(x, y)

and

lim_{n→∞} λ(y, z_{n}) = λ(y, y).

Therefore λ(x, y) = λ(y, y). Thus by (2.1.1), we have x = y ∈ C. This proves (2). □

In many of the applications of neocompact sets in [FK1], the compact sets were included in the initial family A(M) which was used as the starting point in building the neocompact sets. Proposition 3.4 shows that every compact set is a basic section for λ. We now complete the picture by showing that the family of basic sections for λ is closed under universal projections with respect to a compact set.

Proposition 3.5 Let C ⊆ N be a nonempty compact set. If A is a basic section in M × N for λ then the set

B = {x ∈ M : (∀y ∈ C)(x, y) ∈ A}

is also a basic section for λ.

Proof: Let {y_{n} : n ∈ N} be a countable dense subset of C. The set A has the form

A = {(x, y) ∈ M × N : λ(x, y, z) ∈ Â}

for some z ∈ K ∈ M_{Ω} and some compact set Â. For each n, the set

B_{n} = {x ∈ M : λ(x, y_{n}, z) ∈ Â} = {x ∈ M : (x, y_{n}) ∈ A}

is a basic section for λ. By Corollary 3.2, the intersection ⋂_{n} B_{n} is a basic section for λ. We show that B = ⋂_{n} B_{n}. It is obvious that B ⊆ ⋂_{n} B_{n}. Suppose x ∈ ⋂_{n} B_{n}. Let y ∈ C. Then some sequence ⟨y_{k_{n}}⟩ of ⟨y_{n}⟩ converges to y. For each n we have λ(x, y_{k_{n}}, z) ∈ Â. Since Â is compact and λ is continuous, we have λ(x, y, z) ∈ Â, so (x, y) ∈ A and x ∈ B. □

Theorem 3.6 Suppose λ has the back and forth and Skorokhod properties. Then a set is neocompact for λ if and only if it is a basic section for λ. Moreover, for each M the family of neocompact subsets of M for λ is countably compact.

Proof: By the Quantifier Elimination Theorem 2.10 and Propositions 3.4 and 3.5 the family of basic sections for λ is closed under the rules (a)–(f) where A(M) is the family of basic/compact sets. Countable compactness follows from Corollary 3.3. □

For the remainder of this section we let λ_{k} be a sequence of law mappings on Ω. For each k ∈ N, we shall let λ⃗_{k} be the finite product

λ⃗_{k} = ⟨λ_{0}, …, λ_{k}⟩,

and let Λ⃗_{k}(M) be the topological product Λ_{0}(M) × ⋯ × Λ_{k}(M).

We also let λ be the countable product where λ(x) = ⟨λ_{k}(x) : k ∈ N⟩, and let Λ(M) be the countable topological product ∏_{k} Λ_{k}(M).

We state two more results which are proved in [K5].

Proposition 3.7 ([K1, Proposition 7.1 and Lemma 7.3]).

(i) For each k, (λ⃗_{k}, Λ⃗_{k}) is a law mapping on Ω, and (λ, Λ) is a law mapping on Ω.

(ii) Suppose that λ⃗_{k} is a dense law mapping for each k ∈ N. Then λ is a dense law mapping.

(iii) A set B is basic for λ if and only if B = ⋂_{k} B_{k} for some sequence of basic sets B_{k} for λ⃗_{k}. □

Theorem 3.8 ([K1, Theorem 7.7 and Corollary 7.8]). Suppose that λ is closed and has the back and forth property, and for each k, λ⃗_{k} is closed and has the back and forth and Skorokhod properties. Let A_{k}(M) be the family of all basic subsets of M for λ⃗_{k}, and let A(M) = ⋃_{k} A_{k}(M). Then every neocompact set over A is basic for λ, and every neocompact section over A is a basic section for λ. □

We now improve Theorem 3.8 by replacing the basic sets by the basic/compact sets.

Theorem 3.9 Suppose that λ is closed and has the back and forth property, and for each k, λ⃗_{k} is closed, and has the back and forth and Skorokhod properties. Let B_{k}(M) be the family of all basic/compact subsets of M for λ⃗_{k}, and let B(M) = ⋃_{k} B_{k}(M). Then a set is neocompact over B if and only if it is a basic section for λ. Moreover, for each M the family of neocompact subsets of M over B is countably compact.

Proof: By Proposition 2.9, every basic section for λ is neocompact over B. By Theorem 3.8, the family of basic sections for λ is closed under the operations (a)–(e) and under universal projections with respect to basic sets for λ⃗_{k}. By Proposition 3.5, for each k ∈ N the family of basic sections for λ⃗_{k} is closed under universal projections with respect to compact sets. By Proposition 3.7 (ii), every basic section for λ is the intersection of a descending chain of basic sections B_{k} for λ⃗_{k}, and it follows that the family of basic sections for λ is closed under universal projections with respect to compact sets. This shows that every neocompact set over B is a basic section for λ. Countable compactness follows from Corollary 3.3. □

4 Probability Spaces

In this section we study the law mapping (law, Meas) for an atomless probability space Ω. We shall see that this law mapping has the back and forth property if and only if Ω is rich. A measure space (Γ, G, Q) with 0 < Q(Γ) < ∞ is said to be atomless if for each set S ∈ G of measure Q(S) > 0 and each positive r < Q(S) there is a subset U ⊆ S in G such that Q(U) = r. Note that if Ω = (Ω, F, P) is an atomless probability space and P (Γ) > 0 then the restriction of Ω to Γ is an atomless measure space.

The following formula due to Strassen (see [EK, Theorem 1.2 on p. 96]) characterizes the Prohorov metric in terms of the metric of convergence in probability when Ω is an atomless probability space.

Lemma 4.1 Suppose Ω is atomless and M ∈ M_{Ω}. Then for all b, c ∈ Meas(M),

d(b, c) = inf{ρ_{0}(x, y) : x, y ∈ M, law(x) = b, law(y) = c}.

The next proposition rephrases some well known facts in our framework.

Proposition 4.2 Let Ω be an atomless probability space. Then law is a closed law mapping on Ω.

Proof: By Proposition 4.1, law : M → Meas(M) is uniformly continuous, and in fact, d(law(x), law(y)) ≤ ρ_{0}(x, y). Conditions (2.1.1) and (2.1.3) are easily checked, and condition (2.1.2) follows from the characterization of relative compactness given by Prohorov’s theorem. Thus law is a law mapping. It is well known that a probability space Ω is atomless if and only if law maps M onto Meas(M) for each M. Therefore (law, Meas) is closed on Ω. □

We now recall the notion of a saturated probability space from [HK].

Definition 4.3 We say that Ω = (Ω, P, G) is a saturated probability space if for every probability space Γ and all complete separable metric spaces M and N, if

x̄ ∈ L^{0}(Γ, M), ȳ ∈ L^{0}(Γ, N), x ∈ M,

and law(x) = law(x̄), then there exists y ∈ N such that law(x, y) = law(x̄, ȳ).

It is easily seen that every saturated probability space is atomless and has the back and forth property. It was shown in [HK] that uncountable powers of [0, 1] and atomless Loeb probability spaces are saturated, and thus that saturated probability spaces exist.

It is well known that the set of simple functions (functions with finite range) is dense in each M ∈ M_{Ω}. Every atomless probability space satisfies the special case of saturation where x is a simple function.

Proposition 4.4 Let Ω be an atomless probability space, and let Γ be another probability space. Then for every simple x ∈ M and every pair of random variables (x̄, ȳ) ∈ L^{0}(Γ, M × N) such that law(x̄) = law(x), there exists y ∈ N such that law(x, y) = law(x̄, ȳ).

Proof: Let Ω = (Ω, F, P) and Γ = (Γ, G, Q). Let {m_{1}, …, m_{k}} be the range of x, and let A_{j} = {ω : x(ω) = m_{j}} and B_{j} = {γ : x̄(γ) = m_{j}}. We may assume without loss of generality that P (A_{j}) > 0 for each j. Let ȳ_{j} be the restriction of ȳ to the set B_{j}. Since the restriction of Ω to A_{j} is atomless and Q(B_{j}) = P (A_{j}), there is a random variable y_{j} on A_{j} such that law(y_{j}) = law(ȳ_{j}). Now take y ∈ N such that y(ω) = y_{j}(ω) whenever ω ∈ A_{j} for j = 1, …, n. Then law(x, y) = law(x̄, ȳ). □

The next proposition applies the Skorokhod representation theorem in probability theory, and is the reason for our use of the name “Skorokhod property”.

Proposition 4.5 If Ω is an atomless probability space, then law has the Skorokhod property on Ω.

Proof: Let x ∈ M, and let c_{n} be a sequence converging to law(x) in Meas(M). We must find a sequence x_{n} converging to x in M such that law(x_{n}) = c_{n} for all n.

The Skorokhod representation theorem says that on some probability space Γ there are random variables z_{n}, z ∈ L^{0}(Γ, M) such that law(z_{n}) = c_{n} for all n, law(z) = law(x), and z_{n} → z almost surely (see [EK, p. 102]). Let Δ be a saturated probability space. Then there are random variables y_{n}, y in L^{0}(Δ, M) such that

law(y, ⟨y_{n}⟩) = law(z, ⟨z_{n}⟩).

It follows that y_{n} → y in L^{0}(Δ, M). Let u_{n}, n ∈ N be a sequence of simple random variables converging to x in M. Since Δ is saturated there is a sequence v_{n}, n ∈ N in L^{0}(Δ, M) such that

law(y, ⟨v_{n}⟩) = law(x, ⟨u_{n}⟩).

By the preceding proposition, for each n there exists x_{n} ∈ M such that law(u_{n}, x_{n}) = law(v_{n}, y_{n}). Then law(x_{n}) = c_{n} for all n. Since y_{n} and v_{n} both converge to y, we have ρ_{0}(y_{n}, v_{n}) → 0, and thus ρ_{0}(x_{n}, u_{n}) → 0. Therefore x_{n} → x in M. □

Proposition 4.6 Let Ω be an atomless probability space. Then law is dense on Ω.

Proof: Let x, x̄ ∈ M with law(x) = law(x̄), and let y ∈ N. Let ⟨x_{n}⟩ be a sequence of simple functions converging to x in M. Then law(x_{n}) → law(x̄). By the Skorokhod property there is a sequence ⟨x̄_{n}⟩ converging to x̄ in M such that law(x̄_{n}) = law(x_{n}) for each n. Then for each n, x̄_{n} is simple. By Proposition 4.4 there exists ȳ_{n} ∈ N such that law(x̄_{n}, ȳ_{n}) = law(x_{n}, y). We have (x_{n}, y) → (x, y) in M × N, and therefore law(x̄_{n}, ȳ_{n}) → law(x, y). Moreover, since x̄_{n} → x̄ in M, d(law(x̄_{n}, ȳ_{n}), law(x̄, ȳ_{n})) → 0. Therefore law(x̄, ȳ_{n}) → law(x, y), so law(x̄, N) is dense in the closure of law(x, N). □

We now review the notion of a rich probability space from [FK1]. We shall see that richness and saturation are equivalent.

Definition 4.7 Ω is said to be a rich probability space if Ω is atomless and for each M ∈ M_{Ω}, the family of neocompact subsets of M for law is countably compact.

From the previous sections, a set B is basic for law in M if it is of the form

{x ∈ M : law(x) ∈ C}

for some compact set C ⊆ Meas(M), and is a basic section for law in M if it is of the form

{x ∈ M : law(x, z) ∈ D}

for some compact set D ⊆ Meas(M × N) and some z ∈ N.

Theorem 4.8 Let Ω be an atomless probability space. The following are equivalent. (i) Ω is saturated. (ii) law has the back and forth property on Ω. (iii) Ω is rich. (iv) For each M ∈ M_{Ω} the family of basic sections for law in M is countably

compact.

(v) For each M, N ∈ M_{Ω} and basic relation C for law in M×N, the existential projection

{x : ∃y(x, y) ∈ C}

is basic for law.

Proof: By Propositions 4.2 and 4.6, law is closed and dense on Ω.

We first prove that (i) and (ii) are equivalent. It is easily seen that (i) implies (ii). To prove (ii) implies (i), assume (ii) and let x̄ ∈ L^{0}(Γ, M), ȳ ∈ L^{0}(Γ, N), x ∈ M, and law(x) = law(x̄). Since Ω is atomless there exists (x′, y′) ∈ M × N such that law(x′, y′) = law(x̄, ȳ). Then law(x′) = law(x), and by the back and forth property for law on Ω there exists y ∈ N such that law(x, y) = law(x̄, ȳ).

Next we assume (i) and prove (iii). By Proposition 4.5, law has the Skorokhod property on Ω. Then Ω is rich by Theorem 3.6, so (iii) holds.

Since every basic section for law is neocompact, (iii) implies (iv).

Assume (iv). Since law is dense on Ω, it has the back and forth property on Ω by Theorem 2.7. Thus (iv) implies (ii). (v) is equivalent to (ii) by Theorem 2.12. □

We conclude this section with some examples arising in probability spaces which are not rich. By an ordinary probability space we shall mean a probability space of the form (Γ, μ, G) where Γ is a complete separable metric space and μ is the completion of a Borel probability measure on the family of Borel sets G in Γ. These spaces are the ones most commonly used in the literature. It is shown in [FK1] shows that no ordinary probability space is rich.

Consider an atomless ordinary probability space (Γ, μ, G), and let B = {O_{n} : n ∈ N} be a countable open basis for Γ. We say that a measurable set A is independent of a family of sets S in (Γ, μ, G) if

μ(A ∩ B) = μ(A)μ(B) for all B ∈ S.

The same terminology is applied to families of characteristic functions of sets.

For each n, let x_{n} be the characteristic function of O_{n}, considered as a random variable on Γ with values in the two-element space {0, 1}. Every measurable set in Γ can be approximated by sets in the basis B, and therefore no set of measure one half in Γ can be independent of {x_{n} : n ∈ N}.

Example 4.9 For each n, let C_{n} be the set of all z on Γ such that z is the characteristic function of a set of measure 1/2 and z is independent of x_{1}, …, x_{n}. Then C_{n} is a decreasing chain of nonempty neocompact subsets of L^{0}(Γ, {0, 1}) for law, but ⋂_{n} C_{n} is empty. In fact, each C_{n} is a basic section for law. This shows that Γ is not rich.

Example 4.10 Now consider the product space Γ × Γ, and let x̄_{n}(γ_{1}, γ_{2}) = x_{n}(γ_{1}). Let u be the characteristic function of a set of measure one half in Γ, and let ȳ(γ_{1}, γ_{2}) = u(γ_{2}) on Γ × Γ. Then law(⟨x_{n}⟩) = law(⟨x̄_{n}⟩), but there is no y on Γ such that law(⟨x_{n}⟩, y) = law(⟨x̄_{n}⟩, ȳ). This shows directly that Γ is not saturated. The example can by modified by taking x̄_{n}, ȳ on Γ itself, giving a direct example of the failure of the back and forth property on Γ.

Example 4.11 Let C be the set of all pairs (x, y) such that x = ⟨x_{n}⟩ is a sequence of characteristic functions of sets, and y is the characteristic function of a set of measure 1/2 which is independent of the the family {x_{n} : n ∈ N}. Then C is a basic relation for law on Γ. However, the existential projection D = {x : ∃y(⟨x_{n}⟩, y) ∈ C} is not closed in L^{0}(Γ, {0, 1}) and therefore cannot even be a basic section for law. In fact, if x = ⟨x_{1}, x_{2}, …, x_{n}, …⟩ ∉ D and z_{k} = ⟨x_{1}, …, x_{k}, 0, 0, …⟩, then z_{k} ∈ D and z_{k} → x.

5 Adapted Spaces with Finite Time Sets

We now apply our results to adapted probability spaces with finite time sets. In this and the next two sections we shall introduce law mappings for these adapted spaces, and prove that in this setting saturation is again equivalent to richness.

For the next four sections of this paper (through Section 8), we shall take Λ(M) to be the space

Λ(M) = R^{N} × Meas(M)

with the product metric. Whenever we introduce a law mapping, it will be understood that the target space is this particular space Λ(M).

Let T be a finite set of nonnegative real numbers. By a T-adapted (probability) space we mean a structure

Ω_{T} = (Ω, P, G_{∞}, G_{t})_{t∈T}

where (Ω, P, G_{∞}) is a probability space, G_{t} is a σ-subalgebra of G_{∞} for each t ∈ T, and G_{s} ⊆ G_{t} whenever s ≤ t in T.

Throughout this section we let Ω_{T} be a T-adapted probability space and let M ∈ M_{Ω}. R is the metric space L^{0}(Ω, R).

We now recall the notion of an adapted function, which was introduced in [HK].

Definition 5.1 The class of T-adapted functions on M is the least class of functions from M into R such that:

(i) For each bounded continuous function ϕ : M → R, the function (ϕ̂(x))(ω) = ϕ(x(ω)) is a T-adapted function;

(ii) If f_{1}, …, f_{m} are T-adapted functions on M and g : R^{m} → R is continuous, then h(x) = g(f_{1}(x), …, f_{m}(x)) is a T-adapted function;

(iii) If f is a T-adapted function and t ∈ T, then g(x)(ω) = E[f(x)|G_{t}](ω) is a T-adapted function.

Observe that each T-adapted function on M is uniformly bounded, so the expected value E[f(x)] is defined and finite for every T-adapted function f and every x ∈ M. Two processes x, y ∈ M are said to have the same adapted distribution if E[f(x)] = E[f(y)] for all T-adapted functions f.

We shall now take advantage of the separability of M to choose a countable set of T-adapted functions on M which is dense in an appropriate sense.

A set Ψ of bounded functions f : M → R is said to be bounded pointwise dense if every bounded Borel function g : M → R belongs to the closure of Ψ under pointwise convergence of uniformly bounded sequences of functions.

Suppose Ψ is bounded pointwise dense. Then Ψ separates points in M, that is, if u ≠ v in M then ψ(u) ≠ ψ(v) for some ψ ∈ Ψ. Moreover, Ψ is separating in Meas(M), that is, if b ≠ c in Meas(M) then ∫ ψ db ≠ ∫ ψ dc for some ψ ∈ Ψ.

For each complete separable metric space M, there exists a countable set Φ(M) of bounded continuous functions ϕ : M → R which is bounded pointwise dense (see [EK, Proposition 4.2]). For each M, we shall choose such a set Φ(M) once and for all.

Definition 5.2 The class of T-adapted functions built from Φ(M) is the least class of T-adapted functions on M such that:

(i′) For each function ϕ ∈ Φ(M), the corresponding function ϕ̂ : M → R is a T-adapted function built from Φ(M);

(ii′) If f_{1}, …, f_{m} are T-adapted functions built from Φ(M) and p : R^{m} → R is a polynomial with rational coefficients, then h(x) = p(f_{1}(x), …, f_{m}(x)) is a T-adapted function built from Φ(M);

(iii′) If f is a T-adapted function built from Φ(M) and t ∈ T, then g(x)(ω) = E[f(x)|G_{t}](ω) is a T-adapted function built from Φ(M).

There are only countably many T-adapted functions built from Φ(M). Let us arrange them in a list ⟨β_{k} : k ∈ N⟩. We now define the T-adapted law function.

Definition 5.3 By the T-adapted law of a random variable x ∈ M we mean the pair

law_{T}(x) = (⟨E[β_{k}(x)] : k ∈ N⟩, law(x))

in the space R^{N} × Meas(M).

The reader may wonder why the second coordinate law(x) is needed. One reason is to insure that condition (2.1.1) holds. Another reason is to insure that law_{T} is closed on Ω. The image of the first term ⟨E[β_{k}(x)] : k ∈ N⟩ is almost never closed in R^{N}, but we shall see in Proposition 6.7 that for all “atomless” T-adapted spaces the image of law_{T} on M is a closed subset of R^{N} × Meas(M). The second coordinate of law_{T} will also be needed in Lemma 7.8, which is used in the proof that saturated adapted spaces are rich.

The T-adapted law function is defined for every T-adapted space Ω_{T}, and it will sometimes be useful to compare law_{T}(x) and law_{T}(y) where x and y are random variables on two different T-adapted spaces Ω_{T} and Γ_{T}.

Note that for any x ∈ M and sequence x_{n} in M, if S ⊆ T and

lim_{n→∞} law_{T}(x_{n}) = law_{T}(x),

then

lim_{n→∞} law_{S}(x_{n}) = law_{S}(x).

We now give a series of lemmas which we shall use to show that law_{T} is a law mapping on Ω. The next lemma shows that the expected value of each T-adapted function depends continuously on the T-adapted law.

Lemma 5.4 Let x, x_{n} ∈ M. The following are equivalent:

(i) lim_{n→∞} law_{T}(x_{n}) = law_{T}(x).

(ii) lim_{n→∞} law(x_{n}) = law(x) and lim_{n→∞} E[f(x_{n})] = E[f(x)] for every Tadapted function f on M.

Proof: This follows from [HK, Theorem 2.26]. □

Lemma 5.5 Let f be a T-adapted function on M. Then f is continuous from M into R, and uniformly continuous on law^{−1}(C) for each compact set C ⊆ Meas(M).

Proof: We argue by induction on the steps used in constructing f. We begin with the basis step. Let C be a compact subset of Meas(M) and let ϕ : M → R be continuous and bounded by β. Let ε > 0. By Prohorov’s theorem, there is a compact set D ⊆ M such that μ(D) ≥ 1 − ε/(2β) for each μ ∈ C. Since D is compact there exists δ > 0 such that |ϕ(b) − ϕ(c)| < ε whenever b ∈ D, c ∈ M, and ρ(b, c) < δ. Taking δ < ε/(2β), we see that whenever x, y ∈ law^{−1}(C) and ρ_{0}(x, y) < δ, ϕ̂(x) is within ε of ϕ̂(y) in the metric of convergence in probability. Thus ϕ̂ is uniformly continuous on law^{−1}(C). The steps of the induction from f(x) to E[f(x)|G_{t}], and from f_{1}(x), …, f_{m}(x) to g(f_{1}(x), …, f_{m}(x)), are routine. □

Corollary 5.6 The function law_{T} is continuous from M into R^{N} × Meas(M), and uniformly continuous on law^{−1}(C) for each compact set C ⊆ Meas(M). □

Lemma 5.7 For each set A ⊆ M, the set law_{T}(A) ⊆ R^{N} × Meas(M) is relatively compact if and only if the set law(A) ⊆ Meas(M) is relatively compact.

Proof: If law_{T}(A) ⊆ C where C is compact then law(A) is contained in the compact set π(C) where π is the projection map π : R^{N} × Meas(M) → Meas(M). Suppose law(A) ⊆ D where D is compact. For each adapted function f, E[f(x)] is bounded uniformly in x. Then by the Tychonoff product theorem, law_{T}(M) ⊆ B × Meas(M) for some compact set B ⊆ R^{N}. Therefore law_{T}(A) is contained in the compact set B × D. □

We now show that law_{T} is a law mapping.

Proposition 5.8 For any T-adapted space Ω_{T}, law_{T} is a law mapping.

Proof: The continuity of law_{T} was established in Corollary 5.6. Condition (2.1.1) holds because law_{T}(x) is a pair whose second coordinate is law(x).

Suppose A ⊆ M, B ⊆ N, and law_{T}(A) and law_{T}(B) are relatively compact. By Lemma 5.7, law(A) and law(B) are relatively compact. It follows, e.g. by Prohorov’s theorem, that law(A × B) is relatively compact. Then by Lemma 5.7, law_{T}(A × B) is relatively compact. This proves (2.1.2) for law_{T}.

To prove condition (2.1.3), let h : M → N be continuous. For any adapted function f on M, the function g(x) = f(ĥ(x)) is an adapted function on M. Suppose that x, y ∈ M and law_{T}(x) = law_{T}(y). By Lemma 5.4, we have E[f(ĥ(x))] = E[f(ĥ(y))]. Moreover, law(ĥ(x)) = law(ĥ(y)). Therefore law_{T}(ĥ(x)) = law_{T}(ĥ(y)). This shows that the function h̄(law_{T}(x)) = law_{T}(ĥ(x)) is well-defined. Another application of Lemma 5.4 shows that h̄ is continuous, so (2.1.3) holds. □

6 Atomless Adapted Spaces

In this section we introduce atomless T-adapted spaces, and show that such spaces have natural law mappings which are closed on Ω. In the next section we shall see, using the notion of a saturated T-adapted space, that these law mappings are also dense and have the Skorokhod property. The notion of an atomless T-adapted spaces is taken from [HK]. For notational convenience we let 0 be the least element of T.

Definition 6.1 ([HK]) Let E and F be σ-subalgebras of G_{∞} with E ⊆ F. F is said to be atomless over E if for every U ∈ F of positive probability, there is a set V ⊆ U in F such that

0 < P [V |E] < P [U|E]

on a set of positive probability.

A T-adapted space Ω_{T} is said to be atomless if G_{0} is atomless over the trivial σ-algebra and G_{t} is atomless over G_{s} whenever s < t in T ∪ {∞}.

We now introduce a T-adapted analogue of a simple function x such that P [x = r] is rational for each r ∈ M. This notion will be useful in analyzing the law mapping law_{T}.

A finite algebra E of subsets of Ω will be called uniform if each atom of E has the same measure.

Let T = {t_{1}, …, t_{k}}, and put t_{k} + 1 = ∞. By a T-partition of Ω we shall mean a sequence E = ⟨E_{t}, t ∈ T ∪ {∞}⟩ of finite algebras such that E_{t} ⊆ G_{t}, and for each j ≤ k, E_{t_{j+1}} is generated by E_{t_{j}} and a uniform finite algebra whose atoms are independent of G_{t_{j}}. (A set S ⊆ Ω is independent of G_{s} if the conditional probability P [x|G_{s}] is constant). We say that a random variable x ∈ M is E-measurable if it is E_{∞}-measurable, and T-simple if x is E-measurable for some T-partition E. Note that if (x, y) is T-simple, then both x and y are T-simple, but the converse does not hold in general. Also, if (x, y) and (x, z) are both T-simple, then (x, y, z) is T-simple.

Given a T-partition E of Ω, let Ω/E_{t} be the set of all E_{t}-atoms and let Ω/E be the set of all E_{∞}-atoms. Two T-partitions E of Ω and F of Γ are equivalent if there is a bijection h : Ω/E → Γ/F such that h(Ω/E_{t}) = Γ/F_{t} for each t ∈ T, and h is called an isomorphism from E to F. If x is E-measurable and h is an isomorphism from E to F, h(x) is the F-measurable function y such that y(h(ω)) = x(ω) for each ω ∈ Ω/E.

The following lemma can be proved by an inductive argument using the results in Maharam [M]. It is a strengthening of the fact that every random variable can be approximated by simple random variables.

Lemma 6.2 Let Ω_{T} be atomless.

(i) Let Γ_{T} be a T-adapted space. For each T-partition F of Γ there is an equivalent T-partition E of Ω.

(ii) Let f be a T-adapted function. If E is a T-partition of Ω and x ∈ M is E-measurable, then f(x) is E-measurable, and for each t ∈ T, E[f(x)|G_{t}] is E_{t}- measurable. If x is T-simple then f(x) is T-simple.

(iii) Let E be a T-partition of Ω and let x be E-measurable. Then law_{T}(x) = law_{T}(y) if and only if y = h(x) for some T-partition F and isomorphism h : E → F.

(iv) For every M, the set of T-simple random variables is dense in M. In fact, for each T-simple y ∈ N, the set of x ∈ M such that (x, y) is T-simple is dense in M. □

Corollary 6.3 Let Ω_{T} be atomless, and let x ∈ L^{0}(Γ, M) be T-simple on some other T-adapted space Γ_{T}. Then there is a T-simple y ∈ L^{0}(Ω, M) such that law_{T}(x) = law_{T}(y).

Proof: Let x be E-measurable where E is a T-partition of Γ. By Lemma 6.2 (i) there is an equivalent T-partition F of Ω and an isomorphism h from E to F. By Lemma 6.2 (iii), y = h(x) is T-simple and law_{T}(x) = law_{T}(y). □

Here is a back and forth property for T-simple random variables.

Corollary 6.4 Let Ω_{T} be atomless. For each T-simple (x, y) ∈ M × N and Tsimple x̄ ∈ M such that law_{T}(x̄) = law_{T}(x), there exists ȳ ∈ N such that (x̄, ȳ) is T-simple and law_{T}(x̄, ȳ) = law_{T}(x, y).

Proof: Let (x, y) be E-measurable where E is a T-partition. Since law_{T}(x̄) = law_{T}(x), there is a T-partition Ē of Ω and an isomorphism h : E → Ē such that x̄ = h(x). Then ȳ = h(y) has the required properties. □

The following lemma is a consequence of Lemma 4.1 for atomless probability spaces. It will be used here as the first step in an inductive argument for T-adapted spaces.

Lemma 6.5 Suppose Ω is an atomless probability space, M ⊆ R^{n} is compact, and ε > 0. There exists δ > 0 and a finite set of polynomials p_{1}, …, p_{m} in n variables with rational coefficients such that for every uniform finite algebra E ⊆ G, and all E-measurable x, y ∈ M such that

|E[p_{i}(x)] − E[p_{i}(y)]| < δ for i = 1, …, m,  (4)

there is a permutation h of the atoms of E such that ρ_{0}(x, h(y)) < ε. □

Proof: By the compactness of M, there exist δ > 0 and p_{1}, …, p_{m} such that whenever x, y ∈ M and equation (4) holds, we have

d(law(x), law(y)) < ε.

Suppose x, y ∈ M satisfy equation (4). By Lemma 4.1, there exist x̄, ȳ ∈ M such that

law(x̄) = law(x), law(ȳ) = law(y),

and

ρ_{0}(x̄, ȳ) < ε.

Now suppose that E ⊆ G is a uniform finite algebra with k atoms E_{1}, …, E_{k} and that x and y are E-measurable. Then x is simple, and by Lemma 4.3 we may take x̄ = x. To complete the proof it suffices to show that ȳ may also be taken to be E-measurable.

Consider the set A = {z ∈ M : law(z) = law(y)}. Each z ∈ A is determined by a uniform F ⊆ G with k atoms, and an ordering of its atoms F_{1}, …, F_{k}. The joint distribution law(x, z) is determined by the k × k matrix p(x, z) = (P [E_{i} ∩ F_{j}]). The set p(x, A) of all such matrices is a convex polyhedron of dimension k^{2}, whose vertices are permutation matrices corresponding to E-measurable processes of the form z = h(y). For each α > 0, the probability that ρ(x(ω), z(ω)) ≥ α depends linearly on the matrix p(x, z). Therefore this probability takes its minimum at a vertex of p(x, A). It follows that the set of distances {ρ_{0}(x, z) : z ∈ A} has a minimum at a point ρ_{0}(x, ȳ) where ȳ ∈ A is E-measurable. □

Lemma 6.6 Let Ω_{T} be atomless, let M ∈ M_{Ω}, and let D ⊆ Meas(M) be compact. For each ε > 0 there exists δ > 0 such that whenever x, y ∈ law^{−1}(D), (x, y) is Tsimple, and law_{T}(y) is within δ of law_{T}(x), there exists z such that (y, z) is T-simple, law_{T}(z) = law_{T}(y), and ρ_{0}(x, z) < ε.

Proof: Let ε >0 and D ⊆ Meas(M) be compact. By Prohorov’s theorem there is a compact set C ⊆ M such that P [x(ω) ∈ C] > 1 − ε/3 whenever law(x) ∈ D. Since Φ(M) separates points in M and each ϕ_{n} ∈ Φ(M) is bounded and continuous, there exists δ_{1} > 0 and a finite subset {ϕ_{1}, …, ϕ_{m}} ⊆ Φ(M) such that δ_{1} < ε/3 and whenever b, c ∈ C,

|ϕ_{i}(b) − ϕ_{i}(c)| < δ_{1} for i = 1, …, m implies ρ(b, c) < ε.  (5)

Using Lemma 6.5 inductively for each t ∈ T in increasing order, we can find δ > 0 and finitely many adapted functions f_{1}, …, f_{k} built from {ϕ_{1}, …, ϕ_{m}} such that for every T-partition E of Ω and E-measurable x, y ∈ M, if

|E[f_{i}(x)] − E[f_{i}(y)]| < δ for i = 1, …, k,  (6)

there exists an automorphism h of E such that

⟨ϕ_{1}(x), …, ϕ_{m}(x)⟩ is within δ_{1} of ⟨ϕ_{1}(h(y)), …, ϕ_{m}(h(y))⟩  (7)

in the metric of convergence in probability in R^{m}.

Suppose x, y ∈ law^{−1}(D), (x, y) is T-simple, and law_{T}(y) is within δ of law_{T}(x). Then (6) holds, and both x and y are E-measurable for some T-partition E of Ω. Thus (7) holds for some automorphism h of E. Let z = h(y). Then (x, z) is T-simple and law_{T}(y) = law_{T}(z). By (5) and (7), we have

x(ω) ∈ C, z(ω) ∈ C, and ρ(x(ω), z(ω)) < ε

with probability at least 1 − (ε/3 + ε/3 + ε/3). Therefore ρ_{0}(x, z) < ε. □

We are now ready to show that law_{T} is closed on Ω when Ω_{T} is atomless. The proof will take advantage of the fact that the ordinary law was tacked on as a second coordinate to the law_{T} function.

Proposition 6.7 For every atomless T-adapted space Ω_{T}, law_{T} is closed on Ω.

Proof: Let (b, c) be a point in the closure of law_{T}(M) in R^{N}×Meas(M). We must find an x ∈ M such that law_{T}(x) = (b, c). By Lemma 6.2 (iv) there is a sequence x_{n} ∈ M such that law_{T}(x_{n}) converges to (b, c) and (x_{1}, …, x_{n}) is T-simple for each n. Then law(x_{n}) converges to c, and the set D = {c} ∪ {law(x_{n}) : n ∈ N} is compact. Let δ_{n} be the δ corresponding to D and ε = 2^{−n} in Lemma 6.6. Then x_{n} has a subsequence y_{n} such that law_{T}(y_{n}) is within δ_{n}/2 of (b, c), and hence law_{T}(y_{n+1}) within δ_{n} of law_{T}(y_{n}), for each n. By Lemma 6.6 there is a sequence z_{n} ∈ M such that (y_{n}, z_{n}) is T-simple, law_{T}(z_{n}) = law_{T}(y_{n}), and ρ_{0}(z_{n}, z_{n+1}) ≤ 2^{−n} for each n. Then the limit z = lim_{n→∞} z_{n} exists in M and law_{T}(z) = (b, c). □

Proposition 6.8 A T-adapted space Ω_{T} is atomless if and only if it is universal, that is, for every random variable x on some other T-adapted space Γ_{T} there exists y ∈ M such that law_{T}(y) = law_{T}(x).

Proof: Atomless implies universal by Corollary 6.3 and Proposition 6.7. Universal implies atomless by [HK, Lemma 4.4 (iv)]. □

7 Richness and Saturation

In this section we shall introduce the notion of a saturated T-adapted space from [HK], and a notion of a rich T-adapted space which is analogous to the rich continuous time adapted spaces from [FK1].

Definition 7.1 A T-adapted space Ω_{T} is saturated if for every T-adapted space Γ_{T} and all complete separable metric spaces M and N, if

x̄ ∈ L^{0}(Γ, M), ȳ ∈ L^{0}(Γ, N), x ∈ M,

and law_{T}(x) = law_{T}(x̄), then there exists y ∈ N such that law_{T}(x, y) = law_{T}(x̄, ȳ).

It is shown in [HK] that saturated T-adapted spaces exist. It is obvious that law_{T} has the back and forth property for every saturated T-adapted space Ω_{T}. Let us now prove that such spaces also have the Skorokhod property.

Our next order of business is to prove that when Ω_{T} is atomless, law_{T} has the Skorokhod property and is dense on Ω. Our arguments will parallel the corresponding methods for ordinary probability spaces in Section 4. The next result is a weak saturation property which holds for all atomless T-adapted spaces.

Proposition 7.2 Let Ω_{T} be atomless, and let Γ_{T} be another T-adapted space. Then for every T-simple x ∈ M and every pair of random variables (x̄, ȳ) ∈ L^{0}(Γ, M ×N) such that law_{T}(x̄) = law_{T}(x), there exists y ∈ N such that law_{T}(x, y) = law_{T}(x̄, ȳ).

Proof: Let x be E-measurable for some T-partition E of Ω_{T}. Then x̄ is Fmeasurable for some T-partition F of Γ_{T} which is equivalent to E. By Lemma 6.2, we have x̄ = h(x) for some isomorphism h : E → F. For each A ∈ E_{∞}, form the T-adapted space Ω_{A,T} = (A, G_{A,t}, P_{A}) where P_{A}(U) = P (A ∩ U) and G_{A,t} = {A ∩ U : U ∈ G_{t}}. Define Γ_{A,T} similarly. Then Ω_{A,T} is an atomless adapted measure space, and the measures of A in Ω_{A,T} and of h(A) in Γ_{h(A),T} are finite and equal. By Proposition 6.8, Ω_{A,T} is universal, so there exists y_{A} on Ω_{A,T} with the same T-adapted law as the restriction of ȳ to h(A) in Γ_{h(A),T}. Let y ∈ N be the random variable whose restriction to each A ∈ E_{∞} is y_{A}. Then law_{T}(x, y) = law_{T}(x̄, ȳ). □

Hoover obtained a generalization of the Skorokhod representation theorem for Tadapted spaces in [H1, Corollary 10.2], which shows in our terminology that atomless T-adapted spaces with the Skorokhod property exist. We now improve that result by showing that all atomless T-adapted spaces have the Skorokhod property.

Proposition 7.3 Let Ω_{T} be an atomless T-adapted space. Then law_{T} has the Skorokhod property on Ω.

Proof: We first prove the result in the case that Ω_{T} is a saturated T-adapted space, and then prove the general case.

Suppose that x_{n} is a sequence in M, x ∈ M, and law_{T}(x_{n}) converges to law_{T}(x). law_{T} is closed by Proposition 6.7. By Lemma 6.2 (iv), there are sequences y_{k}, z_{k,n} in M such that the pair (y_{k}, z_{k,n}) is T-simple for each k, n, y_{k} → x in probability, and z_{k,n} → x_{n} in probability for each n. Let D be the compact set

{law(y_{k}), law(x), law(z_{k,n}), law(x_{n}) : k, n ∈ N}.

For each k and n, let C_{k,n} be the set

C_{k,n} = {u ∈ M : law_{T}(u) ∈ {law_{T}(z_{m,n}) : k ≤ m} ∪ {law_{T}(x_{n})}}.

This set is basic for law_{T}. Consider an ε > 0. For each k and n, the set

B_{k,n} = C_{k,n} ∩ {u ∈ M : ρ_{0}(u, x) ≤ ε}

is a basic section for law_{T} with parameter x. For each n, for all sufficiently large k we have

d(law_{T}(z_{k,n}), law_{T}(y_{k})) < 2d(law_{T}(x_{n}), law_{T}(x)).

Applying Lemma 6.6 with the above compact set D, we see that for all sufficiently large n, ρ_{0}(y_{n}, x) < ε/2 and for all sufficiently large k there exists u ∈ M such that

law_{T}(u) = law_{T}(z_{k,n}) and ρ_{0}(u, y_{k}) < ε/2.

Then ρ_{0}(u, x) ≤ ε, and thus u ∈ B_{k,n}. Therefore for all sufficiently large n the sets B_{k,n} form a decreasing chain of nonempty sets as k → ∞. We assume at this point that Ω_{T} is saturated, so that it has the back and forth property. By Corollary 3.3, the family of basic sections in M for law_{T} is countably compact. Therefore for all sufficiently large n there exists u_{n} ∈ ⋂_{k} B_{k,n}. Then law_{T}(u_{n}) = law_{T}(x_{n}) and ρ_{0}(u_{n}, x) ≤ ε. Letting ε → 0, we obtain a sequence v_{n} in M such that law_{T}(v_{n}) = law_{T}(x_{n}) for each n and v_{n} → x in M. This proves the result in the case that Ω_{T} is saturated.

We now prove the general case. Let x ∈ M, and let c_{n} be a sequence converging to law_{T}(x) in Meas(M). We must find a sequence x_{n} converging to x in M such that law_{T}(x_{n}) = c_{n} for all n. Let Γ_{T} be a saturated adapted space. By the preceding paragraph, there are random variables z_{n}, z ∈ L^{0}(Γ, M) such that law_{T}(z_{n}) = c_{n} for all n, law_{T}(z) = law_{T}(x), and z_{n} → z in L^{0}(Γ, M). Let u_{n}, n ∈ N be a sequence of T-simple random variables converging to x in M. Since Γ_{T} is saturated there is a sequence v_{n}, n ∈ N in L^{0}(Γ, M) such that

law_{T}(y, ⟨v_{n}⟩) = law_{T}(x, ⟨u_{n}⟩).

By the preceding proposition, for each n there exists x_{n} ∈ M such that law_{T}(u_{n}, x_{n}) = law_{T}(v_{n}, y_{n}). Then law_{T}(x_{n}) = c_{n} for all n. Since y_{n} and v_{n} both converge to y, we have ρ_{0}(x_{n}, u_{n}) → 0, and thus x_{n} → x in M. □

Proposition 7.4 For every atomless T-adapted space Ω_{T}, law_{T} is dense on Ω.

Proof: The argument is the same as the proof of Proposition 4.6, but using T-simple processes and Proposition 7.2 instead of Proposition 4.4. □

We now turn to the notion of a rich T-adapted space. We shall consider two different families of basic sets, the family B_{Ω_{T}} of basic/compact sets for law_{T}, and a simpler family A_{Ω_{T}} which is defined in one step from the ordinary law function law(x) and the notion of a G_{t}-measurable function. Using this simpler family A_{Ω_{T}}, we were able to define rich adapted spaces in [FK1] without introducing the complicated adapted law function law_{T}.

Definition 7.5 For each M ∈ M_{Ω}, let A_{Ω_{T}}(M) and B_{Ω_{T}}(M) be the following families of subsets of M.

A ∈ A_{Ω_{T}}(M) iff A is compact or

A = {x ∈ M : x is G_{t} − measurable and law(x) ∈ D}

for some compact set D ⊆ Meas(M) and some t ∈ T ∪ {∞}.

A ∈ B_{Ω_{T}}(M) iff A is basic/compact for law_{T}, that is, A is compact or

A = {x ∈ M : law_{T}(x) ∈ C}

for some compact set C ⊆ R^{N} × Meas(M).

Recall that by Proposition 3.6, the family of basic sections over B_{Ω_{T}} is the same as the family of basic sections for law_{T}.

Definition 7.6 A T-adapted space Ω_{T} is said to be rich if Ω_{T} is atomless and for each M the family of neocompact subsets of M over A_{Ω_{T}} is countably compact.

We need the following result from [FK1].

Lemma 7.7 Suppose Ω_{T} is rich. The function law(·) is neocontinuous over A_{Ω_{T}}, and each T-adapted function on M is neocontinuous over A_{Ω_{T}}.

Proof: This follows from [FK1, Proposition 5.12 and Theorem 7.6]. □

The next lemma is another key point where we need the ordinary law function as the second coordinate of the T-adapted law function.

Lemma 7.8 Let Ω_{T} be a T-adapted space, and let M ∈ M_{Ω}.

(i) A_{Ω_{T}}(M) ⊆ B_{Ω_{T}}(M).

(ii) If Ω_{T} is rich, then every basic section for law_{T} is neocompact over A_{Ω_{T}}.

Proof: (i) Let A ∈ A_{Ω_{T}}(M). If A is compact then A ∈ B_{Ω_{T}}(M) by definition. The other possibility is that A is a set of the form

A = {x ∈ M : x is G_{t} − measurable and law(x) ∈ D}

for some t ∈ T ∪ {∞} and some compact D ⊆ Meas(M). By Lemma 5.7 the set law_{T}(law^{−1}(D)) has a compact closure C ⊆ R^{N} × D. We have law_{T}(x) ∈ C if and only if law(x) ∈ D, so law_{T}^{−1}(C) = law^{−1}(D).

In the case t = ∞, we have A = law^{−1}(D), and hence A = law_{T}^{−1}(C) ∈ B_{Ω_{T}}(M). Now suppose t ∈ T. Then an x ∈ M is G_{t}-measurable if and only if

E[|ϕ̂(x) − E[ϕ̂(x)|G_{t}]|] = 0

for every ϕ ∈ Φ(M). Since

|ϕ̂(x) − E[ϕ̂(x)|G_{t}]|

is a T-adapted function for each ϕ, it follows that

A = law_{T}^{−1}(B) ∩ law^{−1}(D) = law_{T}^{−1}(B ∩ C)

for some closed set B ⊆ R^{N} × Meas(M), and again A ∈ B_{Ω_{T}}(M).

(ii) Suppose Ω_{T} is rich. Since countable intersections of neocompact sets are neocompact, it follows from Lemma 7.7 that the law_{T} function is neocontinuous over A_{Ω_{T}}, where we take law_{T}(x) to be a random variable with a constant value in R^{N} × Meas(M). Let A be a basic section in M for law_{T}. Then for some K, A is a section of a set in B_{Ω_{T}}(M × K). Suppose that A is not already compact. Then A has the form

A = {x ∈ M : law_{T}(x, z) ∈ Â}

for some z ∈ K and some compact set Â in R^{N} × Meas(M × K). Then by [FK1, Proposition 3.9], A ∩ B is neocompact over A_{Ω_{T}} for each neocompact set B ⊆ M over A_{Ω_{T}}. The projection D̂ of Â into Meas(M × K) is compact, so A is contained in the neocompact set

D = {x ∈ M : law(x, z) ∈ D̂}

over A_{Ω_{T}}. Therefore A is neocompact over A_{Ω_{T}}. □

Theorem 7.9 Let Ω_{T} be an atomless T-adapted space. The following are equivalent.

(i) Ω_{T} is saturated.

(ii) law_{T} has the back and forth property.

(iii) Ω_{T} is rich.

(iv) For each M the family of basic sections for law_{T} is countably compact.

(v) For each basic relation C in M × N for law_{T}, the existential projection

{x : ∃y(x, y) ∈ C}

is basic for law_{T}.

Proof: law_{T} is closed by Proposition 6.7, dense by Proposition 7.4, and has the Skorokhod property by Proposition 7.3.

The equivalence of (i) and (ii) is proved exactly as in Theorem 4.8, using the fact that Ω_{T} is universal by Proposition 6.8. (v) is equivalent to (ii) by Theorem 2.12.

We assume (i) and prove (iii). The family of neocompact sets over B_{Ω_{T}} is countably compact by Theorem 3.6. Since A_{Ω_{T}}(M) ⊆ B_{Ω_{T}}(M), every neocompact set overA_{Ω_{T}} is neocompact over B_{Ω_{T}}. Thus (iii) holds.

(iii) implies (iv) by Lemma 7.8 (ii), and (iv) implies (ii) by Theorem 2.7. □

Since saturated T-adapted spaces exist by [HK, Lemma 5.7], it follows that rich T-adapted spaces exist. The following corollary gives four characterizations of the neocompact sets for a rich adapted space. It follows from Lemma 7.8 and the proof of Theorem 7.9.

Corollary 7.10 Let Ω_{T} be a rich T-adapted space. The following four families of subsets of M are the same.

(i) The family of neocompact sets over A_{Ω_{T}}.

(ii) The family of neocompact sets over B_{Ω_{T}}.

(iii) The family of neocompact sets for law_{T}.

(iv) The family of sets which are intersections of a set of the form

{x ∈ M : law(x, z) ∈ C}

and countably many sets of the form

{x ∈ M : E[f_{n}(x, z)] ∈ D_{n}}

where each f_{n} is a T-adapted function, z ∈ N, C is compact in Meas(M × N), and each D_{n} is compact in R. □

This follows from Lemma 7.8 and the proof of Theorem 7.9. Condition (iv) gives a characterization of the neocompact sets directly in terms of adapted functions rather than in terms of the law_{T} function.

8 Adapted Spaces with Infinite Time Sets

The paper [FK1] introduced rich adapted spaces with times indexed by the dyadic rationals. Each adapted space with times indexed by the dyadic rationals has an associated right continuous adapted space with times indexed by the nonnegative reals. Neocompact sets were applied to prove several optimization and existence theorems for such spaces.

In this section we shall consider adapted spaces with times in an arbitrary linearly ordered set, and apply our results on law mappings to such spaces. This general approach will include the natural special cases of adapted spaces with times indexed by the natural numbers (discrete time), by the dyadic rationals, and by the nonnegative reals. It is known from [FK2] and [FK1] that rich adapted spaces exist for every linearly ordered time set, but rich adapted spaces with right continuous filtrations on the reals never exist.

There are two cases where an adapted space induces a law mapping in a natural way. The first case, where the set of times is countable, is treated in this section. In this case, saturation is equivalent to richness. The second case, where the times are nonnegative reals and the adapted space is right continuous, is treated in the next section. We shall see that the right continuous adapted space which is associated with a rich adapted space on the dyadic rationals is saturated and satisfies a weak form of richness.

Let ⟨L, ≤⟩ be a linearly ordered set. For convenience we assume that L contains a least element 0, and use the convention that t < ∞ for all t ∈ L. By an L-adapted space we mean a structure

Ω_{L} = (Ω, P, G_{∞}, G_{t})_{t∈L}

such that (Ω, P, G_{∞}) is a complete probability space, G_{t} is a σ-subalgebra of G_{∞} for each t ∈ L, and G_{s} ⊆ G_{t} whenever s < t in L. We shall write G = G_{∞}, so that (Ω, P, G) is the probability space associated with the adapted space Ω.

For each finite subset T ⊆ L, each L-adapted space Ω_{L} has a corresponding T-adapted space

Ω_{T} = (Ω, P, G_{∞}, G_{t})_{t∈T}.

Definition 8.1 Let Ω_{L} be an L-adapted space. We say that f is an adapted function on M for Ω_{L} if f is a T-adapted function for the corresponding T-adapted space Ω_{T} for some finite subset T ⊆ L.

An L-adapted space Ω_{L} is atomless if G_{0} is atomless over the trivial σ-algebra, and G_{t} is atomless over G_{s} whenever s < t ∈ L ∪ {∞}. Note that Ω_{L} is atomless if and only if Ω_{T} is atomless for each finite T ⊆ L.

We now define families of basic sets A_{Ω_{L}} and B_{Ω_{L}} which generalize the families A_{Ω_{T}} and B_{Ω_{T}} introduced in the preceding section, and review the notion of a rich L-adapted space from [FK1] and [FK2].

Definition 8.2 Let M ∈ M_{Ω}. We define

A_{Ω_{L}}(M) = ⋃{A_{Ω_{T}}(M) : T ⊆ L and T is finite}.

That is, A ∈ A_{Ω_{L}}(M) if A is either compact or of the form

C = {x ∈ M : x is G_{t} − measurable and law(x) ∈ D}

for some t ∈ L ∪ {∞} and some compact set D ⊆ Meas(M).

Ω_{L} is a rich L-adapted space if Ω_{L} is atomless and for each M ∈ M_{Ω}, the family of neocompact subsets of M over A_{Ω_{L}} is countably compact.

We let

B_{Ω_{L}}(M) = ⋃{B_{Ω_{T}}(M) : T ⊆ L and T is finite}.

Thus C ∈ B_{Ω_{L}}(M) if and only if C is basic/compact for law_{T} for some finite T ⊆ L.

We shall need the following existence result from [FK2].

Theorem 8.3 For every linearly ordered set L, rich L-adapted spaces exist. □

In fact, it is proved in [FK2, Theorem 5.15] that every atomless Loeb L-adapted space is rich. The Loeb adapted spaces are constructed using methods from nonstandard analysis, and have been used extensively in the literature to prove existence theorems in probability theory (e.g. see [AFHL] or [K5]).

Lemma 8.4 Let Ω_{L} be an L-adapted space. A set is neocompact over A_{Ω_{L}} if and only if it is neocompact over A_{Ω_{K}} for some finite or countable K ⊆ L. A similar result holds for B_{Ω_{L}}. Thus if Ω_{L} is rich then the corresponding K-adapted space Ω_{K} is rich for every K ⊆ L. Conversely, if Ω_{K} is rich for every countable K ⊆ L then Ω_{L} is rich.

Proof: The family of sets which are neocompact over A_{Ω_{K}} for some finite or countable K ⊆ L is closed under the operations (a)-(f), and hence is the same as the family of neocompact sets over A_{Ω_{L}}. Similarly for B_{Ω_{L}}. □

Using the preceding lemma, we have an analogue of Lemma 7.8.

Lemma 8.5 Let Ω_{L} be an L-adapted space, and let M ∈ M_{Ω}.

(i) A_{Ω_{L}}(M) ⊆ B_{Ω_{L}}(M).

(ii) If Ω_{L} is rich, then every basic section over B_{Ω_{L}} is neocompact over A_{Ω_{L}}. □

Now let B be the set of nonnegative dyadic rationals. Because of Lemma 8.4 and the fact that every countable linearly ordered set can be embedded in B, we shall concentrate on B-adapted spaces. For each k ∈ N, let B_{k} be the finite set of multiples of 2^{−k} in the interval [0, 2^{k}]. Then B = ⋃_{k} B_{k}. Each B-adapted probability space Ω_{B} has a corresponding B_{k}-adapted space

Ω_{k} = (Ω, P, G_{∞}, G_{t})_{t∈B_{k}}.

We have not defined a law function corresponding to an arbitrary L-adapted space Ω_{L}. We shall now take advantage of the countability of the set B of dyadic rationals to introduce a law function corresponding to a B-adapted space Ω_{B}. By the k^{th}-adapted law of a random variable x ∈ M we mean the function

law_{k}(x) = law_{B_{k}}(x),

where law_{B_{k}} is the B_{k}-adapted law function introduced in Section 5.

In order to fit these finite adapted law functions into an infinite product as in Section 2, for k ≥ 1 we let β_{k,n}, n ∈ N, be a list of all the adapted functions for Ω_{B_{k}} built from Φ which are not adapted functions for Ω_{B_{k−1}}. Let

λ_{0}(x) = law(x), λ_{k}(x) = ⟨E[β_{k,n}(x)] : n ∈ N⟩.

Then the finite product λ⃗_{k}(x) is the k^{th} adapted law

law_{k}(x) = ⟨λ_{1}(x), …, λ_{k}(x), law(x)⟩.

(We put λ_{0}(x) = law(x) last in the sequence to conform with our practice in the preceding sections). The adapted law of x is the infinite product

law_{B}(x) = ⟨λ_{1}(x), …, λ_{k}(x), …, law(x)⟩.

Both law_{k}(x) and law_{B}(x) take values in R^{N} × Meas(M).

All of the lemmas 5.4 through 5.8 hold for B in place of T. In each case, the result for B is an easy consequence of the result for T.

Proposition 8.6 (i) law_{B} is a law mapping.

(ii) If Ω_{B} is atomless then law_{B} is dense.

(iii) If Ω_{B} is atomless then for any other B-adapted space Γ_{B} and any x ∈ L^{0}(Γ, M), law_{B}(x) is in the closure of law_{B}(M).

Proof: (i) follows from Propositions 3.7 (i) and 5.9. (ii) follows from Propositions 3.7 (ii) and 7.4.

We now prove (iii). For each k, Ω_{k} is an atomless B_{k}-adapted space, and by Proposition 6.8 there exists y_{k} ∈ M such that law_{k}(y_{k}) = law_{k}(x). Then

d(law_{B}(y_{k}), law_{B}(x)) ≤ 2^{−k},

and thus law_{B}(y_{k}) → law_{B}(x) in Λ(M). □

Proposition 8.7 Suppose Ω_{B} be atomless.

(i) If law_{B} has the back and forth property on Ω then law_{k} has the back and forth property on Ω for each k ∈ N.

(ii) If law_{B} is closed on Ω then law_{k} is closed on Ω for each k ∈ N.

Proof: (i) Assume that law_{B} has the back and forth property on Ω. Suppose that law_{k}(x, y_{n}) converges to law_{k}(x̄, ȳ) as n → ∞. Let

A = {(x_{n}, y) : n ∈ N}.

Then law_{k}(A) is relatively compact. By Lemma 5.7, the sets law(A) and law_{B}(A) are relatively compact. Therefore there is a subsequence ⟨y_{m}⟩ of ⟨y_{n}⟩ such that law_{B}(x, y_{m}) converges to a point law_{B}(x′, y′) as m → ∞. By Proposition 2.4 there exists y such that law_{B}(x, y_{m}) converges to law_{B}(x, y). Then law_{k}(x, y_{m}) converges to law_{k}(x, y), so law_{k}(x, y) = law_{k}(x̄, ȳ). Thus by Proposition 2.4, law_{k} has the back and forth property.

(ii) Assume that law_{B} is closed. Suppose that

lim_{n→∞} law_{k}(x_{n}) = c.

By Lemma 5.7, law_{B}({x_{n} : n ∈ N}) is relatively compact. Since law_{B} is closed, there is a subsequence ⟨x_{m}⟩ of ⟨x_{n}⟩ such that law_{B}(x_{m}) converges to law_{B}(x) for some x ∈ M. Then law_{k}(x_{m}) converges to law_{k}(x), and hence law_{k}(x) = c. This shows that law_{k} is closed. □

It will be convenient to take B_{0} to be the empty set and to identify law_{0} with the ordinary law function on the probability space Ω.

Definition 8.8 Following our earlier pattern, a B-adapted space Ω_{B} is saturated if for every B-adapted space Γ_{B} and all complete separable metric spaces M and N, if

x̄ ∈ L^{0}(Γ, M), ȳ ∈ L^{0}(Γ, N), x ∈ M,

and law_{B}(x) = law_{B}(x̄), then there exists y ∈ N such that law_{B}(x, y) = law_{B}(x̄, ȳ).

A B-adapted space Ω_{B} is universal if for every B-adapted space Γ_{B} and complete separable metric space M, for each x̄ ∈ L^{0}(Γ, M) there exists x ∈ M such that law_{B}(x) = law_{B}(x̄).

It is clear that every saturated B-adapted space is universal and has the back and forth property. The converse also holds, as one can see from the next theorem, which characterizes universal B-adapted spaces.

Theorem 8.9 A B-adapted space Ω_{B} is universal if and only if it is atomless and law_{B} is closed on Ω.

Proof: Suppose Ω_{B} is atomless and law_{B} is closed on Ω. Let Γ_{B} be another B-adapted space and let x ∈ L^{0}(Γ, M). By Proposition 8.6 (iii), law_{B}(x) is in the closure of law_{B}(M). Since law_{B} is closed on Ω, there exists y ∈ M such that law_{B}(y) = law_{B}(x). Thus Ω_{B} is universal.

For the converse, suppose Ω_{B} is universal. Then for each k, Ω_{k} is universal, and hence is atomless by Proposition 6.8. It follows that Ω_{B} is atomless. Suppose x_{n} ∈ M and law_{B}(x_{n}) → c in Λ(M). By Theorem 8.3, there exists a rich B-adapted space Γ_{B}. Then Γ_{B} is atomless. By Proposition 8.6 (ii), each law_{B}(x_{n}) and hence c belongs to the closure of law_{B}(L^{0}(Γ, M)). Therefore there is a sequence ⟨y_{n}⟩ in L^{0}(Γ, M) such that law_{B}(y_{n}) → c. Then the sets

B_{k} = {law_{B}(y_{n}) : n ≥ k} ∪ {c}

form a decreasing chain of compact sets, and their inverse images

C_{k} = {z ∈ L^{0}(Γ, M) : law_{B}(z) ∈ B_{k}}

form a decreasing chain of nonempty basic sets for law_{B} on Γ. Since Γ_{B} is rich, each of these sets is neocompact and by Lemma 8.5, and their intersection is nonempty. Thus there exists y ∈ L^{0}(Γ, M)) such that law_{B}(y) = c. Since Ω_{B} is universal, there exists x ∈ M with law_{B}(x) = c. This shows that law_{B} is closed on Ω. □

We now prove our main theorem on B-adapted spaces.

Theorem 8.10 Let Ω_{B} be an atomless B-adapted space. The following are equivalent.

(i) Ω_{B} is saturated.

(ii) law_{B} is closed and has the back and forth property.

(iii) Ω_{B} is rich.

(iv) For each M ∈ M_{Ω}, the family of basic sections in M for law_{B} is countably compact.

(v) For each basic relation C in M × N for law_{B}, the existential projection

{x : ∃y(x, y) ∈ C}

is basic for law_{B}.

Proof: law_{B} is dense on Ω by Proposition 8.6.

Assume (i). Clearly, Ω_{B} is universal and law_{B} has the back and forth property on Ω. By Theorem 8.9, law_{B} is closed on Ω, so (ii) holds.

Using Theorem 8.9 and the argument in the proof of Theorem 4.8, we see that (ii) implies (i). As in our previous results, (v) is equivalent to (ii) by Theorem 2.12.

We now assume (ii) and prove (iii). Let k ∈ N. By Proposition 8.7, law_{k} has the back and forth property and is closed. By Theorem 7.9, the corresponding B_{k}- adapted space Ω_{k} is saturated. By Proposition 7.3, law_{k} has the Skorokhod property for each k ∈ N. By Theorem 3.9, the family of basic sections for law_{B} is countably compact for each M, and every neocompact set over B_{Ω_{B}} is a basic section for law_{B}. By Lemma 8.5 (i), every neocompact set over A_{Ω_{B}} is a basic section for law_{B}, and (iii) holds.

We now assume (iii), that Ω_{B} is rich, and prove (iv). By Lemma 8.4, Ω_{k} is rich for each k ∈ N. By Proposition 3.7 (iii), every basic section C for law_{B} is an intersection of a chain of basic sections C_{k} for law_{k}, and by Lemma 7.8 (ii), each C_{k} is neocompact over A_{Ω_{B}}. It follows that the family of basic sections for law_{B} is countably compact, so (iv) holds.

Finally, (iv) implies (ii) by Theorem 2.7. □

It is natural to ask whether the above theorem can be improved to show that the family of neocompact sets for law_{B} is countably compact when Ω_{B} is rich. The following negative result shows that this can never happen.

Proposition 8.11 Let Ω_{B} be a universal B-adapted space, and let M = {0, 1}. Then:

(i) The function law_{B} does not have the Skorokhod property over Ω.

(ii) The family of neocompact subsets of M for law_{B} on Ω is not countably compact.

(iii) There is a basic set C ⊆ M×M for law_{B} and a nonempty basic set C ⊆ M for law_{B} such that the set {x : (∀y ∈ M)(x, y) ∈ C} is not basic for law_{B}.

Proof: By Theorem 8.9, Ω_{B} is atomless and law_{B} is closed on Ω. For each n ∈ N let t_{n} = 1 − 2^{−n}. By a result of Maharam [M2, p. 146], for each n there is a set S_{n} ∈ G_{t_{n+1}} of measure 1/2 which is independent of G_{t_{n}}. Let x_{n} ∈ M be the characteristic function of S_{n}. If k ≤ n, s ∈ B_{k}, and ϕ : M → R, then E[ϕ̂(x_{n})|G_{s}] has the constant value (ϕ(0) + ϕ(1))/2 when s ≤ t_{n}, and E[ϕ̂(x_{n})|G_{s}] = ϕ̂(x_{n}) when s > t_{n}. It follows that for each k we have law_{k}(x_{m}) = law_{k}(x_{n}) whenever k ≤ m, k ≤ n. Therefore the sequence b_{n} = law_{B}(x_{n}) is a Cauchy sequence and hence converges to a limit b_{∞} ∈ Λ(M). Since law_{B} is closed on Ω, there exists x ∈ M such that law_{B}(x) = b_{∞}. We observe that whenever law_{B}(y) = b_{m}, law_{B}(z) = b_{n}, and m ≠ n in N ∪ {∞}, y is independent of z and hence ρ_{0}(y, z) = 1/2. Thus there cannot be a sequence y_{n} ∈ M such that y_{n} → x_{∞} but law_{B}(y_{n}) = b_{n} for each n. This proves (i).

Let B_{m} = {y : law_{B}(y) ∈ {b_{0}, …, b_{m}, b_{∞}}} and B = ⋃_{m} B_{m}. Then each of the sets B_{m} and B is basic for law_{B} and nonempty. Moreover, for each m ∈ N the set

C_{m} = {(y, z) ∈ B_{m} × B : ρ_{0}(y, z) ≥ 1/2}

is basic for law_{B}. Let

D_{m} = {z ∈ M : (∀y ∈ B_{m})(y, z) ∈ C_{m}}.

Then D_{m} is a decreasing chain of neocompact sets for law_{B} on Ω. We have x_{n} ∈ D_{m} whenever m < n ∈ N, so each D_{m} is nonempty. However, the intersection ⋂_{m} D_{m} is empty, because if z ∈ ⋂_{m} D_{m} then z ∈ B_{m} for some m, and we would have (z, z) ∈ C_{m} and ρ_{0}(z, z) ≥ 1/2. This proves (ii).

To prove (iii), we show that at least one of the sets D_{m} is not basic for law_{B}. Suppose to the contrary that each D_{m} is basic for law_{B}. None of the sets D_{m} can be compact, because then D_{n} would be compact for all n > m and the intersection could not be empty. Thus for each m, D_{m} = {y : law_{B}(y) ∈ E_{m}} for some compact set E_{m}. Since x_{n} ∈ D_{m}, we have b_{n} ∈ E_{m} whenever m < n. Recalling that b_{n} → law_{B}(x_{∞}), we see that law_{B}(x_{∞}) ∈ E_{m}, and hence x_{∞} ∈ D_{m}, for each m. This contradicts the fact that ⋂_{m} D_{m} is empty, and proves (iii). □

The preceding proof also works in the same way when we use a decreasing sequence of times instead of an increasing sequence of times.

The following question about the Skorokhod property remains open.

Question 8.12 Is there an atomless B-adapted space Ω_{B} such that law_{B} has the Skorokhod property on Ω?

By Proposition 8.11, law_{B} cannot be both closed and have the Skorokhod property on Ω.

We now return to an arbitrary linearly ordered time set L. If L is countable, we may represent L as the union of a countable chain of finite subsets L_{k}, and define law_{L} in the same way as law_{B}. Then all of the results of this section hold for Ω_{L} as well as for Ω_{B}. (Proposition 8.11 holds whenever L is infinite). The following corollary holds even for uncountable L, and is a generalization of Corollary 7.10.

Corollary 8.13 Let Ω_{L} be a rich L-adapted space. The following four families of subsets of M are the same.

(i) The family of neocompact sets over A_{Ω_{L}}.

(ii) The family of neocompact sets over B_{Ω_{L}}.

(iii) The family of countable intersections of basic sections over B_{Ω_{L}}.

(iv) The family of sets which are intersections of a set of the form

{x ∈ M : law(x, z) ∈ C}

and countably many sets of the form

{x ∈ M : E[f_{n}(x, z)] ∈ D_{n}}

where each f_{n} is an L-adapted function, z ∈ N, C is compact in Meas(M × N), and each D_{n} is compact in R.

(v) If L is countable, then the family of basic sections for law_{L} on Ω is equal to the families (i)-(iv). □

Proof: The proof of Theorem 8.10 gives the result in the case that L = B. The general case now follows by Lemma 8.4 and Corollary 7.10. □

9 Right Continuous Adapted Spaces

We now consider continuous time adapted spaces. As is usual in the literature, we restrict our attention to the case where the filtration is complete and right continuous. By a right continuous adapted space we mean a structure

Ω_{R} = (Ω, P, F_{∞}, F_{t})_{t∈R_{+}}

where R_{+} = [0, ∞), F_{t} is a F_{∞}-complete σ-algebra, and F_{t} = ⋂_{s>t} F_{s} for each t ∈ R_{+}. Similarly, a right continuous B-adapted space is a structure (Ω, P, F_{∞}, F_{t})_{t∈B} where F_{t} is F_{∞}-complete and F_{t} = ⋂_{s>t} F_{s} for each t ∈ B.

Each B-adapted space Ω_{B} has a corresponding right continuous adapted space Ω_{R} where F_{∞} = G_{∞} and F_{t} is the G_{∞}-completion of the σ-algebra ⋂_{s>t} G_{s}. The filtration G_{t} for Ω_{B} is not necessary right continuous, and thus is not uniquely determined by the filtration F_{t} for Ω_{R}.

Each right continuous adapted space Ω_{R} has a corresponding right continuous B-adapted space

Ω_{B}^{rt} = (Ω, P, F_{∞}, F_{t})_{t∈B}.

The law function for Ω_{B}^{rt} will be denoted by law_{B}^{rt}.

If we start with a given B-adapted space Ω_{B}, then Ω_{B}^{rt} will denote the right continuous B-adapted space obtained from the right continuous adapted space Ω_{R} corresponding to Ω_{B}. Throughout this section we shall always assume that Ω_{B}, Ω_{R}, and Ω_{B}^{rt} are related in this way. Note that Ω_{R} is atomless if and only if Ω_{B} is atomless, and also if and only if Ω_{B}^{rt} is atomless.

We begin with a negative result. This result is an improvement of [FK1, Example 7.7], which showed that a right continuous B-adapted space cannot be rich, and hence cannot be saturated in the sense of the preceding section.

Proposition 9.1 Let Ω_{B} be atomless.

(i) The law mapping law_{B}^{rt} is not closed.

(ii) Neither Ω_{B}^{rt} nor Ω_{R} is rich in the sense of Definition 8.2.

(iii) If law_{B} is closed, then F_{t} is atomless over G_{t} for each t ∈ B.

Proof: (i) There exist x_{n} ∈ L^{0}(Ω, {0, 1}) such that E[x_{n}] = 1/2 and x_{n} is F_{t+1/n}- measurable but independent of F_{t}. Then law_{B}^{rt}(x_{n}) converges in R^{N} × Meas({0, 1}) but there is no x ∈ L^{0}(Ω, {0, 1}) such that law_{B}^{rt}(x_{n}) converges to law_{B}^{rt}(x). Therefore law_{B}^{rt} is not closed.

(ii) By (i) and Theorem 8.9, Ω_{B}^{rt} is not rich. Since A_{Ω_{B}^{rt}} ⊆ A_{Ω_{R}}, Ω_{R} is also not rich.

(iii) Suppose law_{B} is closed, and let x_{n} be as in the proof of part (i). By taking a subsequence we may assume without loss of generality that law_{B}(x_{n}) converges to some c ∈ R^{N} × Meas({0, 1}). Then there is an x ∈ L^{0}(Ω, {0, 1}) such that law_{B}(x) = c. For each s > t, x_{n} is G_{s}-measurable for all sufficiently large n, and hence x is G_{s}-measurable. Therefore x is F_{t}-measurable. However, for each n we have E[x_{n}|G_{t}] = 1/2 almost surely, and therefore E[x|G_{t}] = 1/2 almost surely. It follows that F_{t} is atomless over G_{t}. □

Combining the above proposition with Theorems 8.8 and 8.9, we see that right continuous B-adapted spaces are never saturated or even universal in the sense of the preceding section. To get around this difficulty, we use the notions of saturation and universality from [HK], which compare a right continuous adapted space with other right continuous adapted spaces rather than with arbitrary adapted spaces.

Definition 9.2 Two random variables x, y ∈ M are adapted equivalent on R, in symbols x ≡ y, if E[f(x)] = E[f(y)] for every adapted function f on M for Ω_{R}. This notion can also be applied to random variables on two different right continuous adapted spaces. Ω_{R} is universal if for every other right continuous adapted space Γ_{R} and random variable x̄ ∈ L^{0}(Γ, M) there exists x ∈ M such that x ≡ x̄. Ω_{R} is saturated if for every other right continuous adapted space Γ_{R}, if x̄ ∈ L^{0}(Γ, M), ȳ ∈ L^{0}(Γ, N), x ∈ M, and x ≡ x̄, then there exists y ∈ N such that (x, y) ≡ (x̄, ȳ).

The following result is proved in [HK, Corollary 2.13].

Proposition 9.3 Let Ω_{R} be a right continuous adapted space, M ∈ M_{Ω}, and x, y ∈ M. Then x ≡ y if and only if law_{B}^{rt}(x) = law_{B}^{rt}(y). □

The proof of the above proposition also yields the following result.

Proposition 9.4 Let Ω_{B} be an atomless B-adapted space and x, y ∈ M. If law_{B}(x) = law_{B}(y) then x ≡ y in Ω_{R}. □

Example 9.5 The converse of the above proposition is false. If law_{B} is closed, there exist x, y ∈ L^{0}(Ω, {0, 1}) such that x ≡ y but law_{B}(x) ≠ law_{B}(y).

To see this, note first that by Proposition 9.1, F_{0} is atomless over G_{0}. Take x to be the characteristic function of a G_{0}-measurable set of measure 1/2, and take y to be the characteristic function of an F_{0}-measurable set of measure 1/2 which is independent of G_{0}. Then x ≡ y, but E[(E[x|G_{0}])^{2}] = 1/2 and E[(E[y|G_{0}])^{2}] = 1/4. This shows that law_{B}(x) ≠ law_{B}(y). □

We next wish to show that richness for Ω_{B} implies saturation for Ω_{R}. In order to do this we shall need a law mapping (law_{R}, Λ) such that:

x ≡ y if and only if law_{R}(x) = law_{R}(y)  (8)

and

For each M, the function law_{R} is neocontinuous over A_{Ω_{B}}.  (9)

By Proposition 9.3, the function law_{B}^{rt} obtained from a rich B-adapted space Ω_{B} has property (8). However, [FK1, Example 7.7] shows that law_{B}^{rt} cannot have property (9). To build a law mapping with both properties (8) and (9) we shall introduce the notion of a conditional process from [HK], which is an adapted function with variable times.

We let ν be the Borel probability measure on R_{+} with exponential density, so that ν([s, t]) = e^{−s} − e^{−t}. L^{0}(R_{+}, R) will denote the space of measurable functions y : R_{+} → R with the metric of convergence in probability with respect to ν. It is a complete separable metric space. An n-fold stochastic process on Ω is a random variable on Ω with values in the complete separable metric space L^{0,n} = L^{0}((R_{+})^{n}, R). We shall let L^{0,n} = L^{0}(Ω, L^{0,n}) be the space of all n-fold stochastic process on Ω with the metric of convergence in probability.

Definition 9.6 The class of conditional processes on M for a right continuous adapted space Ω_{R} is the least class of functions from M into L^{0,n} such that:

(i) For each bounded continuous function ϕ : M → R, the function (ϕ̂(x))(ω) = ϕ(x(ω)) is a 0-fold conditional process on M.

(ii) If f_{1}, …, f_{m} are n-fold conditional processes on M and g : R^{m} → R is continuous, then h(x) = g(f_{1}(x), …, f_{m}(x)) is an n-fold conditional process on M.

(iii) If f is an n-fold conditional process on M and t⃗ varies over (R_{+})^{n} then

(g(x)(ω))(t⃗, s) = E[(f(x))(t⃗)|F_{s}](ω)

is an (n + 1)-fold conditional process on M.

Each conditional process on M is uniformly bounded. For each x ∈ M and n-fold conditional process f on M, we define the expected path E[f(x)](t⃗) ∈ L^{0,n} by

E[f(x)](t⃗) = E[(f(x)(·))(t⃗)].

We shall now define the right continuous law function law_{R}. As in the case of adapted functions, we choose a countable set of conditional processes on M which is dense in an appropriate sense. Recall that Φ(M) is a countable set of bounded continuous functions f : M → R which is bounded pointwise dense. The class of conditional processes built from Φ(M) is defined in the natural way analogous to the class of adapted functions built from Φ(M), and is a countable set which we arrange in a list ⟨γ_{0}, γ_{1}, …⟩. Each γ_{n} is a j(n)-fold conditional process for some j(n) ∈ N. For each M, the target space will be the product

Λ(M) = ∏{L^{0,j(n)} : n ∈ N} × Meas(M).

(An alternative, would have been to use the space of right continuous functions with left limits from (R_{+})^{j(n)} into R with the Skorokhod topology in place of L^{0,j(n)}).

Definition 9.7 Let Ω_{R} be a right continuous adapted space and x ∈ M. The continuous time adapted law of x is the pair

law_{R}(x) = (⟨E[γ_{0}(x)](·), E[γ_{1}(x)](·), …⟩, law(x))

in the metric space Λ(M).

Proposition 9.8 x ≡ y on R if and only if law_{R}(x) = law_{R}(y).

Proof: This follows from [HK, Corollary 2.15]. □

The proof of the next proposition is analogous to the proof of the corresponding result for the discrete time law function law_{T}.

Proposition 9.9 (i) The law_{R} function is continuous on M, and uniformly continuous on law^{−1}(C) for each compact set C ⊆ Meas(M).

(ii) For each set A ⊆ M, law_{R}(A) is relatively compact if and only law(A) is relatively compact.

(iii) law_{R} is a law mapping.

(iv) If Ω_{B} is rich then the law_{R} function is neocontinuous over A_{Ω_{B}} for each M.

Proof: Part (iv) follows from [FK1, Theorem 9.7]. □

One way of approximating the adapted space Ω_{R} is to approximate the filtration F_{t} by a finite step function. Let F_{t,k} = F_{s} where s is the least element of B_{k} ∪ {∞} such that s ≥ t, let Ω_{R,k} be the right continuous adapted space (Ω, P, F_{∞}, F_{t,k})_{t∈R_{+}}, and let law_{R,k} be the right continuous adapted law for Ω_{R,k}. By Proposition 9.9, law_{R,k} is a law mapping. Thus F_{t,k} is a step function with steps in B_{k}. For each conditional process f for Ω_{R}, let f_{k} be the corresponding conditional process for Ω_{R,k}. The paths of each n-fold conditional process f_{k}(x) for Ω_{R,k} are n-fold step functions which are constant on the interior of each cube with vertices in (B_{k})^{n}. Moreover, the value of the conditional process f_{k}(x) at t⃗ is equal to the value of the adapted function f(x)(s⃗) at the greatest s⃗ ≤ t⃗ in (B_{k})^{n}.

Note that law_{B_{k}}^{rt}(x) = law_{B_{k}}^{rt}(y) if and only if law_{R,k}(x) = law_{R,k}(y). In fact, the mapping law_{B_{k}}^{rt}(x) ↦ law_{R,k}(x) is a topological homeomorphism from law_{B_{k}}^{rt}(M) to law_{R,k}(M). However, this map does not preserve the metrics.

We shall use the upcrossing inequality to prove that law_{R,k}(x) converges to law_{R}(x) uniformly in Ω_{R} and x, and then show that law_{R} is dense.

Lemma 9.10 For each complete separable M and each ε > 0 there exists m ∈ N such that for all k ≥ m, all right continuous adapted spaces Ω_{R}, and every x ∈ M, law_{R,k}(x) is within ε of law_{R}(x).

Proof: It suffices to prove that for each conditional process f and ε > 0 there exists m(ε, f) ∈ N such that for all k ≥ m(ε, f), all Ω_{R}, and all x ∈ M, f_{k}(x) is within ε of f(x) in L^{0,n}. We do this by induction on the formation of f. The main difficulty is in the conditional expectation step. Assume the result holds for f(x)(t⃗) and let

g(x)(t⃗, s) = E[f(x)(t⃗)|F_{s}].

Let b be a uniform bound for f. For each δ > 0, k ≥ m(δ, f), and x ∈ M, the set U_{k} of all t⃗ ∈ (R_{+})^{n} such that f_{k}(x)(t⃗) is within δ of f(x)(t⃗) in R has ν^{n}-measure at least 1 − δ. Then for each t⃗ ∈ U_{k},

E[|f_{k}(x)(t⃗) − f(x)(t⃗)|] ≤ δ(1 + b).

For each (t⃗, s) ∈ (B_{k})^{n+1},

g_{k}(x)(t⃗, s) = E[f_{k}(x)(t⃗)|F_{s}].

Let h_{k} = g_{k} − g. Then for each t⃗, ⟨h_{k}(x)(t⃗, s) : s ∈ B_{k}⟩ is a martingale for Ω_{R,k}. By the maximal inequality for martingales, for each t⃗ ∈ U_{k} the set

V_{k}(t⃗) = \{ω ∈ Ω : sup\{|h_{k}(x)(t⃗, s)| : s ∈ B_{k}\} ≤ \sqrt{δ}\}

has measure at least 1 − \sqrt{δ}(1 + b).

Let ε > 0 and take δ so that \sqrt{δ}(1 + b) ≤ ε/2. Divide [−b, b] into a finite set I of intervals of length ε/8 such that the center of one interval is the starting point of the next. The set I has cardinality 16b/ε. By the upcrossing inequality (see [B2, p. 489]), for each t⃗ and k, the expected number of upcrossings of an interval of length ε/8 by either g_{k}(x)(t⃗, u) or g(x)(t⃗, u) for u ∈ R_{+} is at most 16b/ε.

Suppose t⃗ ∈ U_{k} and ω ∈ V_{k}(t⃗). Then

|h_{k}(x)(t⃗, u)(ω)| ≤ ε/2

for all u ∈ B_{k}. Let

W_{k}(ω, t⃗) = {u ∈ R_{+} : |h_{k}(x)(t⃗, u)(ω)| ≥ ε}.

If s < s′ ∈ B_{k} and W_{k}(ω, t⃗) meets [s, s′], then either g_{k}(x)(t⃗, v)(ω) or g(x)(t⃗, v)(ω) must have an upcrossing of one of the intervals in I while v ∈ [s, s′]. Therefore there exists m(ε, g) ∈ N depending only on ε and g such that for all k ≥ m(ε, g) and t⃗ ∈ U_{k}, the set of ω ∈ V_{k}(t⃗) such that ν(W_{k}(ω, t⃗)) ≥ ε has measure ≤ ε/2. Then for each t⃗ ∈ U_{k} the set of ω ∈ Ω such that ν(W_{k}(ω, t⃗)) < ε has measure at least 1 − ε. Since U_{k} has measure at least 1 − δ, it follows that for all Ω_{R} and all x ∈ M, g_{k}(x) is within ε of g(x) in L^{0,n+1}. □

Proposition 9.11 If Ω_{R} is an atomless right continuous adapted space, then law_{R} is dense.

Proof: By Proposition 7.4, law_{B_{k}}^{rt} is dense for each k. It follows that law_{R,k} is dense for each k. Suppose x, x̄ ∈ M, ȳ ∈ N, and law_{R}(x) = law_{R}(x̄). By Propositions 9.3 and 9.8, law_{R,k}(x) = law_{R,k}(x̄) for each k ∈ N. Let ε > 0. For each k we may choose y_{k} ∈ N such that law_{R,k}(x, y_{k}) is within ε of law_{R,k}(x̄, ȳ). By Lemma 9.10, law_{R,k}(x, y_{k}) is within ε of law_{R}(x, y_{k}) and law_{R,k}(x̄, ȳ) is within ε of law_{R}(x̄, ȳ) for all sufficiently large k. Therefore law_{R}(x, y_{k}) is within 3ε of law_{R}(x̄, ȳ), so law_{R} is dense. □

Corollary 9.12 Let Ω_{R} be an atomless right continuous adapted space. The following are equivalent:

(i) law_{R} has the back and forth property;

(ii) law_{B}^{rt} has the back and forth property.

Moreover, Ω_{R} is saturated if and only if Ω_{R} is universal and one of the above conditions (i)-(ii) holds.

Proof: law_{R} is dense by Proposition 9.11, and law_{B}^{rt} is dense by Proposition 8.5.

(i) is equivalent to (ii) by Propositions 9.8 and 9.3. It follows from Proposition 9.8 that Ω_{R} is saturated if and only if Ω_{R} is universal and (i) holds. □

Another proof of Corollary 9.12 is given by the proof of [HK, Theorem 5.2]. As pointed out by Hoover [H2], the statement of [HK, Theorem 5.2] was incorrect. Corollary 9.12 gives a corrected formulation of the result.

For any countable set L ⊆ R_{+}, one can define Ω_{L}^{rt} and law_{L}^{rt} in the same way as we defined Ω_{B}^{rt} and law_{B}^{rt}. Then Corollary 9.12 also holds for any countable dense set L ⊆ R_{+}. Hoover [H3] proved that if Ω_{R} is saturated then law_{L}^{rt} has the back and forth property for every countable L ⊆ R_{+}.

Proposition 9.13 Let Ω_{R} be an atomless right continuous adapted space. Then for each other right continuous adapted space Γ_{R} and each M, law_{R}(M) is dense in law_{R}(L^{0}(Γ, M)).

Proof: Let x ∈ L^{0}(Γ, M). By Proposition 6.8, for each k ∈ N we may choose y_{k} ∈ M such that law_{R,k}(y_{k}) = law_{R,k}(x). By Lemma 9.10, law_{R}(y_{k}) converges to law_{R}(x). □

Theorem 9.14 A right continuous adapted space Ω_{R} is universal if and only if Ω_{R} is atomless and law_{R} is closed.

Proof: Suppose first that Ω_{R} is universal. Then for each k, Ω_{R,k} is universal. By Proposition 6.8, Ω_{R,k} is atomless, so Ω_{R} is atomless. Let x_{n} be a sequence in M such that law_{B}(x_{n}) converges to a point c ∈ Λ(M). Let Γ_{B} be a rich B-adapted space. Then Γ_{R} is atomless, so by Proposition 9.13 there is a sequence y_{n} in L^{0}(Γ, M) such that law_{R}(y_{n}) converges to c. Then for each n, the set

C_{n} = {c} ∪ {law_{R}(y_{m}) : n ≤ m}

is compact. Since law_{R} is neocontinuous over A_{Γ_{B}}, the sets law_{R}^{−1}(C_{n}), n ∈ N, form a decreasing chain of nonempty neocompact sets. By countable compactness, the intersection of this chain is nonempty, so there exists y ∈ L^{0}(Γ, M) such that law_{R}(y) = c. Since Ω_{R} is universal, there exists x ∈ M such that law_{R}(x) = c, so law_{R} is closed.

Now suppose Ω_{R} is atomless and law_{R} is closed. Let Γ_{R} be another right continuous adapted space and let y ∈ L^{0}(Γ, M). By Proposition 9.13 there is a sequence x_{n}inM such that law_{R}(x_{n}) converges to law_{R}(y). Since law_{R} is closed, there exists x ∈ M with law_{R}(x) = law_{R}(y), so Ω_{R} is universal. □

Theorem 9.15 If Ω_{B} is a rich B-adapted space, then Ω_{R} is saturated.

Proof: Since Ω_{B} is rich, Ω_{R} is atomless. By Proposition 9.9 (iv), law_{R} is neocontinuous over A_{Ω_{B}}. Then each basic section for law_{R} is neocompact over A_{Ω_{B}}, and thus the family of basic sections in M for law_{R} is countably compact. law_{R} is dense by Proposition 9.11. By Theorem 2.7, law_{R} is closed and has the back and forth property. By Theorem 9.14, Ω_{R} is universal, and hence by Corollary 9.12, Ω_{R} is saturated. □

Since rich B-adapted spaces can never have right continuous filtrations, the converse of the above theorem is false. That is, there are spaces Ω_{B} such that Ω_{R} is saturated but Ω_{B} is not rich. The following related question is open.

Question 9.16 If Ω_{R} is a saturated right continuous adapted space, does there exist a rich B-adapted space Γ_{B} such that Γ_{R} = Ω_{R}?

The following negative result can be proved by the same construction that was used in the proof of Proposition 8.11 in the preceding section.

Proposition 9.17 Let Ω_{R} be a universal right continuous R-adapted space, and let M = {0, 1}. Then:

(i) The function law_{R} does not have the Skorokhod property on Ω.

(ii) The family of neocompact subsets of M for law_{R} on Ω is not countably compact.

(iii) There is a basic set C ⊆ M×M for law_{R} and a nonempty basic set C ⊆ M for law_{R} such that the set {x : (∀y ∈ M)(x, y) ∈ C} is not basic for law_{R}. □

The following question is analogous to Question 8.12 for B-adapted spaces.

Question 9.18 Is there an atomless right continuous adapted space Ω_{R} such that law_{R} has the Skorokhod property on Ω?

We shall now give a characterization of saturated right continuous adapted spaces Ω_{R} by a weaker analogue of richness which does not depend on the Skorokhod property.

Let us call a set C ⊆ M existentially definable over A if C is built from sets in A(M) using only the rules (a)–(e), that is, without the universal projection rule (f). The following weak quantifier elimination theorem is a consequence of Theorem 2.12 and is proved in [K5].

Theorem 9.19 (Existential Quantifier Elimination) Let λ be a closed law mapping. Let A(M) be the family of basic subsets of M for λ. The following are equivalent.

(i) λ has the back and forth property.

(ii) Each existentially definable set over A is basic for λ. □

The next result shows that saturation is equivalent to the analogue of richness for existentially definable sets.

Theorem 9.20 Let Ω_{R} be an atomless right continuous adapted space. The following are equivalent.

(i) Ω_{R} is saturated.

(ii) law_{R} is closed and has the back and forth property.

(iii) For each M ∈ M_{Ω}, the family of basic sections in M for the law mapping law_{R} is countably compact.

(iv) For each M ∈ M_{Ω}, the family of subsets of M which are existentially definable over the basic/compact sets for law_{R} is countably compact.

(v) law_{R} is closed and for every basic relation C ⊆ M × N for law_{R} the set {x : ∃y(x, y) ∈ C} is basic for law_{R}.

Proof: We first prove that (i) is equivalent to (ii). Assume (i). Then Ω_{R} is universal, and law_{R} is closed by Theorem 9.14. law_{R} has the back and forth property by Proposition 9.9. This proves that (i) implies (ii). Now assume (ii). Ω_{R} is universal by Theorem 9.14. Therefore Ω_{R} is saturated, and thus (ii) implies (i).

We now prove that (ii) is equivalent to (iii). (ii) implies (iii) by Theorem 2.7. Assume (iii). By Proposition 9.11, law_{R} is dense. Then (ii) follows by Theorem 2.7. (ii) is equivalent to (v) by Theorem 9.19. Finally, we prove that (iii) is equivalent to (iv). Assume (iii). law_{R} is dense by Proposition 9.11, and has the back and forth property by Theorem 2.7. By Theorem 9.19, the family of basic sections is closed under the operations (b)–(e), and (iv) follows. The implication from (iv) to (iii) is trivial. □

The main advantage of using a rich adapted space Ω_{B} instead of a saturated right continuous adapted space Ω_{R} is that we can use a rich adapted space to prove existence theorems without introducing the adapted law function (as in the paper [FK1]). Most of the applications of rich adapted spaces in the paper [FK1] use neocompact sets which are constructed using the universal projection rule (f), and thus require a rich space Ω_{B} rather than merely a saturated right continuous space Ω_{R}. In particular, this applies to most applications involving conditional expectations or stochastic integrals. However, some of the applications, such as the result that every continuous process has a closest Brownian motion in the metric of convergence in probability ([FK1, Corollary 12.2]), do not depend on the universal projection rule (f) and thus hold for any saturated right continuous space Ω_{R}.

The results in this section can be applied to adapted Loeb spaces. Let Ω_{B} be an atomless Loeb B-adapted space. By [FK2, Theorem 5.15], Ω_{B} is rich. Then Ω_{R} is saturated by Theorem 9.15, and thus Ω_{R} satisfies conditions (i)-(iv) in Theorem 9.19. A similar result was proved in [HK, Theorem 4.2]. (In that proof, formula (4.2.8) was stated without adequate justification. This gap can be corrected using Lemma 9.10 of this paper). The adapted Loeb spaces have a particularly nice subcollection, the hyperfinite adapted spaces. It is shown in [K3] that for each hyperfinite adapted space Ω_{B}, the associated right continuous adapted space Ω_{R} is universal and homogeneous. That is, for every pair of random variables x, y ∈ M, x ≡ y if and only if there is a measure preserving bijection h : Ω → Ω such that y(ω) = h(x(ω)) almost surely, and h(F_{t}) = F_{t} for all t ∈ R_{+}. It is easily seen that every universal homogeneous space Ω_{R} is saturated

The result in [K3] shows that for each hyperfinite adapted space Ω_{B}, the associated right continuous adapted space Ω_{R} is saturated.

References

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University of Wisconsin, Madison WI. keisler@math.wisc.edu

Mathematical Subject Classification, 1991:

Primary: 03C10, 03C90

Secondary: 54E35, 60G07, 60B10, 60E05

Keywords: elimination, quantifier, adapted, metric space, neocompact, neometric