Equivalent or absolutely continuous probability measures with given marginals

Abstract Let (X,A) and (Y,B) be measurable spaces. Supposewe are given a probability α on A, a probability β on B and a probability μ on the product σ-field A ⊗ B. Is there a probability ν on A⊗B, with marginals α and β, such that ν ≪ μ or ν ~ μ ? Such a ν, provided it exists, may be useful with regard to equivalent martingale measures and mass transportation. Various conditions for the existence of ν are provided, distinguishing ν ≪ μ from ν ~ μ.


Introduction . The problem
Let (X, A) and (Y, B) be measurable spaces. This paper is concerned with the following questions: (a) Is there ν ∈ Γ(α, β) such that ν µ ? (b) Is there ν ∈ Γ(α, β) such that ν ∼ µ ? Problems (a)-(b) are motivated in Section 2. Here, we introduce some further notation and summarize the content of this paper. whenever X is a real P-integrable random variable. Given another p.m. Q on F, we write P Q to mean that P(A) = whenever A ∈ F and Q(A) = . Similarly, P ∼ Q stands for P Q and Q P. The notations P Q and P ∼ Q have the same meaning even if F is a eld (and not necessarily a σ-eld) and P, Q are nitely additive probabilities (and not necessarily p.m.'s). Further, P is perfect if, for each measurable function f : Ω → R, there is a real Borel set B such that B ⊂ f (Ω) and P(f ∈ B) = .
If Ω is separable metric and F the Borel σ-eld, then P is perfect if and only if it is tight. Thus, for P to be perfect, it su ces that Ω is a universally measurable subset (in particular, a Borel subset) of a Polish space and F the Borel σ-eld. We refer to Ramachandran [6] for more information on perfect p.m.'s.
In the sequel, the reference p.m. µ ∈ P is xed. Moreover, for each ν ∈ P, the marginals ν and ν of ν are meant as Finally, m denotes the Lebesgue measure on the Borel σ-eld of [ , ] and we adopt the following convention. If X or Y are topological spaces, then A or B are always taken to be the Borel σ-elds.

. Outline
This paper consists of four sections. Section 2 provides some motivations to problems (a)-(b) while Section 5 includes concluding remarks. The core of the paper are Sections 3 and 4 which are concerned with problems (a) and (b), respectively. The main results are various conditions for problems (a)-(b) to admit a solution. Among other things, nitely additive solutions are considered as well.
We also note that, in addition to their possible applied interest, problems (a)-(b) are quite natural from the foundational point of view. Nevertheless, to our knowledge, they have been neglected so far. Apart from a recent paper [2, Example 15] we are not aware of any explicit reference.
In particular, problems (a)-(b) are not covered by the well known results by Strassen [8]. More precisely, such results do not apply to problem (b), for Λ fails to be closed in any reasonable topology on P. Instead, Strassen's ideas can be adapted to problem (a), since Λ is sequentially closed if P is given the topology of setwise convergence. Some Strassen-type solutions to problem (a) are actually provided by Theorems 5 and 6.
We next present a few examples. To x ideas, we focus on some speci c issues, but the ensuing remarks essentially extend to all areas where Γ(α, β) is involved.

Example 1. (Mass transportation).
Let C be a non-negative measurable function on X × Y. Here, C(x, y) is regarded as the cost per unit mass for transporting a material from x ∈ X to y ∈ Y. Such units are distributed according to α, before transportation, and according to β after transportation. Therefore, each member of Γ(α, β) is called a transport plan. Given Λ ⊂ Γ(α, β), say that ν is an optimal transport plan for Λ if ν ∈ Λ and Eν(C) = min λ∈Λ E λ (C). In this framework, it could be reasonable to choose Λ such that Λ ⊂ Λ or Λ ⊂ Λ , provided of course Λ ≠ ∅ or Λ ≠ ∅. As to Λ ⊂ Λ , sometimes, it makes sense to focus only on those transport plans which have a density with respect to some reference measure µ. This happens for instance in Korman and McCann [5], with X = Y = R p and µ equivalent to Lebesgue measure, in order to take capacity constraints into account. A further (concrete) reason for taking Λ ⊂ Λ is the following. It may be that some H ∈ A ⊗ B is "forbidden", in the sense that (x, y) ∈ H does not make sense for the problem at hand. Situations of this type are usually modeled by letting C = ∞ on H. An alternative option could be obtained by letting Λ ⊂ Λ and taking µ such that µ(H) = . Finally, quite analogous considerations hold for Λ ⊂ Λ .
We hope to devote further work to more speci c applications in the near future. As an interesting hint provided by one of the referees, the results in this paper could be applicable to nd Monge solutions in those cases where the underlying optimal transport plan is not unique.

Example 2. (Equivalent martingale measures)
. Let X = {X t : ≤ t ≤ } and Y = {Y t : ≤ t ≤ } be real cadlag processes on the probability space (Ω, F, P). A p.m. Q on F is said to be an equivalent martingale measure (e.m.m.) if Q ∼ P and both X and Y are Q-martingales.
Let D be the set of real cadlag functions on [ , ], equipped with the Skorohod topology, and let S be the Borel σ-eld on D. We make two simplifying assumptions. Firstly, X and Y are taken to be canonical processes. Namely, we let where t ∈ [ , ] and ω = (ω , ω ) ∈ D × D. Secondly, and more importantly, X and Y are required to be Q-martingales with respect to their canonical ltrations only.
Under these assumptions, existence of an e.m.m. ts nicely into the framework of this paper. It su ces to let X = Y = D, µ = P, and to choose α and β such that where µ and µ are the marginals of µ. In fact, if such α and β do not exist, no e.m.m. is available. Otherwise, if α and β exist, an e.m.m. is exactly a solution to problem (b). And the condition µ × µ µ guarantees the existence of an e.m.m. by Theorem 11 below.
The situation is more complicated, even though more realistic, when X and Y are asked to be martingales with respect to a common ltration {F t : ≤ t ≤ } on Ω = D × D. In this case, existence of an e.m.m. can not be easily seen as a particular case of problem (b). In fact, to decide whether X and Y are martingales with respect to {F t : ≤ t ≤ }, one needs some further information beyond α and β; see also Section 5.

Example 3. (Contingency tables).
For de niteness, a contingency table is identi ed with a non-negative p × q matrix T = (t i,j ) such that i,j t i,j = . If S and T are contingency tables, write S T if t i,j = ⇒ s i,j = , and S ∼ T if t i,j = ⇔ s i,j = . Let α = (α , . . . , αp) and β = (β , . . . , βq) be non-negative vectors such that i α i = j β j = . Suppose we are given α, β and a contingency table T. Then, the following natural questions arise. Is there a contingency table S such that S T and Similarly, is there a contingency table S satisfying the above condition as well as S ∼ T ?

Absolutely continuous laws with given marginals
We begin with a de nition. Let ν ∈ P. Say that ν is dominated by µ on rectangles, written ν R µ, if Then, ν µ implies ν R µ, but not conversely. As an example, take Since ν is supported by the diagonal, ν fails to be absolutely continuous with respect to µ. However, ν R µ for Next result gives conditions for ν µ and ν R µ to be equivalent. (Such equivalence is also brie y discussed in Section 5). Let µ and µ denote the marginals of µ. Lemma 4. Suppose X and Y are separable metric spaces. Let ν ∈ P. Then, ν µ ⇐⇒ ν R µ provided (at least) one of the following conditions holds: where f is a density of µ with respect to γ and ∂{f = } is the boundary of the set {f = }.
Proof. Let ν and ν denote the marginals of ν. If ν R µ, as assumed throughout this proof, then ν µ and ν µ . Furthermore, ν is dominated by µ on the open sets, that is, ) : x ∈ X} and g : X → Y. We rst suppose g continuous. Then, G is closed, so that ν(G) = as well. Hence, both µ and ν can be written as Thus, ν µ follows from ν µ . Next, suppose g measurable. By Lusin's theorem, given ϵ ∈ ( , ), there is a closed set F ⊂ X such that µ (F c ) < ϵ and g is continuous on F. Since ν µ , it can be assumed ν (F) > . De ne By what already proved, since ν F R µ F and g is continuous on F, one obtains ν F µ F . Hence, ν µ follows from ν µ and the arbitrariness of ϵ. The proof is exactly the one obtains γ C ∩ {f > } = . Since µ{f = } = , one also obtains ν {f = } = , where H denotes the interior of H. Hence, In connection with (iii) of Lemma 4 it is worth noting that, since ν γ, the condition ν ∂{f = } = is automatically true whenever γ ∂{f = } = .
We next turn to problem (a). Let α be a p.m. on A and β a p.m. on B. For all functions f : Moreover, suppose X and Y are Polish spaces and P is given the topology of weak convergence of p.m.'s.
By a classical result of Strassen [8], if Λ ⊂ P is convex and closed, for all bounded continuous f : X → R and g : Y → R.
Basing on this fact, it is tempting to let Λ = {ν ∈ P : ν µ} in condition (1). But such a Λ is not closed, and in fact Strassen's result does not apply to problem (a). As a trivial example, take Λ = {ν ∈ P : ν µ} and Since β = δ is not absolutely continuous with respect to µ = m, problem (a) admits no solutions. Nevertheless, if βn is uniform on ( , /n), then α × βn ∈ Λ and βn → β weakly. Therefore, for all bounded continuous f and g. Even though Strassen's result does not work as it stands, the underlying ideas can be adapted to problem (a). In fact, Λ = {ν ∈ P : ν µ} is sequentially closed if P is given the topology of setwise convergence, that is, the topology on P generated by the maps λ → λ(H) for all H ∈ A ⊗ B. Similarly, Λ = {ν ∈ P : ν R µ} is sequentially closed in such topology. This suggests to require condition (1), with Λ = {ν ∈ P : ν µ} or Λ = {ν ∈ P : ν R µ}, replacing continuous functions with measurable functions. Proof. If ν ∈ Γ(α, β) ∩ Λ and f and g are bounded measurable, then where the equality is because ν ∈ Γ(α, β) and the inequality for ν ∈ Λ. Conversely, suppose condition (1) holds for all bounded measurable f and g. De ne for all λ ∈ Λ and all bounded measurable f and g, and let S = X f ,g : f and g bounded and measurable .
Then, S is a linear space of real bounded functions on the set Λ. By condition (1), Hence, by de Finetti's coherence principle, there is a nitely additive probability P on the power set of Λ such that Λ X(λ) P(dλ) = for all X ∈ S; see e.g. Berti et al. [1] and Berti et al. [2]. Let R be the eld generated by A × B, for all A ∈ A and B ∈ B, and Such ν is a nitely additive probability on R. In view of [7,Theorem 2], since one between α and β is perfect, ν is actually σ-additive on R. Take ν to be the (only) σ-additive extension of ν to σ(R) = A ⊗ B. Given A ∈ A, let Then, Theorem 5 provides only a partial solution to problem (a), for one only obtains ν R µ (and not ν µ) for some ν ∈ Γ(α, β). Under the conditions of Lemma 4, however, ν R µ amounts to ν µ and Theorem 5 yields a full solution. If µ has at least one discrete marginal, for instance, there exists ν ∈ Γ(α, β) such that ν µ if and only if condition (1) holds, with Λ = {λ ∈ P : λ R µ}, for all bounded measurable f and g.
The argument which leads to Theorem 5 allows to obtain some other results. Next Theorems 6 and 7 are examples of this claim.
Say that Λ ⊂ P is uniformly dominated by µ if, for each ϵ > , there is δ > such that sup λ∈Λ λ(C) ≤ ϵ whenever C ∈ A ⊗ B and µ(C) < δ. This notion of absolute continuity is well known, mainly with regard to Vitali-Hahn-Saks theorem and related topics. A straightforward example of Λ uniformly dominated by µ is Λ = {λ ∈ P : λ ≤ r µ} for some constant r. Theorem 6. Let α be a p.m. on A and β a p.m. on B. Suppose Λ is uniformly dominated by µ and condition (1) holds for all bounded measurable f : X → R and g : Y → R. Then, there is ν ∈ Γ(α, β) such that ν µ.
An open problem is whether condition (1) generally implies Γ(α, β) ∩ Λ ≠ ∅ when Λ is taken to be Λ = {λ ∈ P : λ µ}. This is actually the case under the conditions of Lemma 4. Furthermore, concerning nitely additive solutions to problem (a), the following result is available. A and β a p.m. on B. Suppose condition (1) holds, with Λ = {λ ∈ P : λ R µ}, for all bounded measurable f : X → R and g : Y → R. Then, there is a nitely additive probability ν on A ⊗ B, with marginals α and β, such that ν µ.

Theorem 7. Let α be a p.m. on
Proof. We rst prove a claim.
Claim: Let P i be a nitely additive probability on the eld F i , i = , , and let F ⊂ F . Then, P can be extended to a nitely additive probability P on F such that P P if and only if P (P |F ), where P |F is the restriction of P on F .
In fact, the "only if" part is trivial. Conversely, suppose P (P |F ) and de ne D = {B ∈ F : P (B) ∈ { , }}. Fix A ∈ F and B ∈ D with A ⊂ B. If P (B) = , then P (A) ≤ P (B). If P (B) = , then A ⊂ B implies P (A) = . Since A ∈ F and P (P |F ), one obtains P (A) = , and again P (A) ≤ P (B). By [3, Theorem 3.6.1], there is a nitely additive probability P on F such that P = P on F and P = P on D. Such a P does the job.
We next prove Theorem 7. De ne ν as in the proof of Theorem 5. Such ν is a nitely additive probability, de ned on the eld R generated by rectangles, with marginals α and β. It is straightforward to verify that ν (µ|R). Thus, it su ces to apply the previous claim with Incidentally, unlike Theorem 5, Theorems 6 and 7 do not request α or β to be perfect. It may be that perfectness can be dropped from Theorem 5 as well, but we have not a proof of this fact. So far, µ, α and β are all xed. We now take a di erent point of view, we x µ only while α and β are allowed to vary subject to the condition α µ and β µ (recall that µ and µ are the marginals of µ). Such condition can not be bypassed, being necessary for problem (a) to admit a solution.
As a last result on problem (a) we now show that, under the conditions of Lemma 4, the converse of the above implication is true as well.
Note that, under the assumptions of Theorem 8, condition (2) amounts to The above condition, in fact, is clearly equivalent to µ × µ R µ.

Equivalent laws with given marginals
A general approach to problem (b), introduced in Berti et al. [1,2], is the following.
Recall that a determining class for a measurable space (Ω, F) is a class H of real bounded measurable functions on Ω such that whenever P and Q are p.m.'s on F. For instance, H = {I A : A ∈ F } is a determining class if F is a eld such that F = σ(F ). Or else, H = {bounded continuous functions on Ω} is a determining class if Ω is a metric space and F the Borel σ-eld.
Fix a determining class F for (X, A) and a determining class G for (Y, B). It is also assumed that F and G are linear spaces. Further, given α and β, de ne Such L is a linear space of bounded random variables on the measurable space (X × Y, A ⊗ B) and has the property that ν ∈ Γ(α, β) ⇐⇒ ν ∈ P and Eν(X) = for each X ∈ L .
Thus, problem (b) can be stated as: Is there ν ∈ P satisfying ν ∼ µ and Eν(X) = for each X ∈ L ? Next result gives a tool for answering this question.
On one hand, Theorem 10 formally solves problem (b). On the other hand, Theorem 10 is not very helpful in real problems, since the proposed condition is quite hard to be checked. There are some exceptions, however. In [2, Example 15], a usable condition for solving problem (b) is obtained through Theorem 10. We now (slightly) improve such condition. We also provide a new and simpler proof.
Theorem 11 provides a su cient condition for problem (b) to admit a solution. Note that α ∼ µ and β ∼ µ are necessary for solving problem (b). Thus, the real requirement of Theorem 11 is µ × µ µ. Note also that, under the conditions of Theorem 8, µ × µ µ reduces to µ × µ R µ. Among other things, Theorem 11 allows to settle the following conjecture. Let us consider the condition for all p.m.'s α on A and β on B, satisfying α ∼ µ (4) and β ∼ µ , there is ν ∈ Γ(α, β) such that ν ∼ µ.
As for problem (a), one might be also interested in a nitely additive solution to problem (b). In this case, the conditions of Theorem 11 may be weakened.
Theorem 13. Let α and β be p.m.'s on A and B, respectively. If there is a nitely additive probability ν on A ⊗ B, with marginals α and β, such that ν ∼ µ.
Hence, α * × β * ∈ Λ, and this implies for all bounded measurable f : X → R and g : Y → R. By Theorem 7, there is a nitely additive probability ν * on A ⊗ B, with marginals α * and β * , such that ν * µ. Therefore, it su ces to let In fact, ν ∼ µ follows from λ ∼ µ and ν * µ, while it is straightforward to verify that ν has marginals α and β.

Concluding remarks
This section collects some miscellaneous material, connected to parts of the paper. Some problems to be investigated are mentioned as well.
A further problem, connected to mass transportation, concerns conditions for Λ or Λ to admit an optimal transport plan. -Equivalent martingale measures. As noted in Example 2, the case where X and Y are required to be martingales with respect to a common ltration is not covered by problem (b). To capture this case, problems (a)-(b) should be generalized as follows. Let (Ω, F, P) be a probability space and P i a p.m. on the sub-σ-eld F i ⊂ F, where i = , . Is there a p.m. Q on F such that Q P, or Q ∼ P, and Q = P i on F i for each i ? This question looks intriguing but also quite hard to be answered in general.
-Domination on rectangles. Let R be the eld generated by the measurable rectangles; ν R µ just means that ν is dominated by µ on R but not necessarily on σ(R) = A ⊗ B. Nevertheless, to our knowledge, domination on rectangles has not been explicitly investigated so far. Lemma 4 provides conditions under which ν R µ implies ν µ, but possibly some other conditions can be singled out. However, condition (ii) of Lemma 4 can not be improved by asking µ to be supported by countably many graphs. In fact, next example exhibits a situation where ν is not dominated by µ even if ν R µ and µ ∪n Gn = where each Gn is the graph of a measurable function. Then, µ ∪n Gn = and ν ∪n Gn = where Gn = {(x, fn(x)) : x ∈ [ , )}. Hence, µ is supported by countably many graphs and ν is not dominated by µ. However, ν R µ. To prove the latter fact, since µ ∼ µ ∼ m, we need to show that Therefore, ν R µ.
-An open problem. Let Λ = {λ ∈ P : λ µ}. For such a Λ, as already noted, we do not know whether condition (1) (required for all bounded measurable f and g) implies Λ = Γ(α, β) ∩ Λ ≠ ∅. In problem (a), one is looking for a (countably additive) extension ν of ν * to A ⊗ B. In problem (b), ν is also required to be strictly positive whenever µ is strictly positive. Now, since problems (a)-(b) are of the extension type, allowing for nitely additive probabilities makes easier to solve them. This is con rmed by Theorems 7 and 13. Note also that, up to technical details, condition (1) is essentially a coherence condition in de Finetti's sense. Indeed, in the proof of Theorem 5, condition (1) is actually used as a coherence condition. And, a coherent map can be coherently extended to any larger domain.
Thus, problem (a) admits a solution for all admissible α and β if and only if the same happens to problem (b).