Bivariate measure-inducing quasi-copulas

It is well known that every bivariate copula induces a positive measure on the Borel $\sigma$-algebra on $[0,1]^2$, but there exist bivariate quasi-copulas that do not induce a signed measure on the same $\sigma$-algebra. In this paper we show that a signed measure induced by a bivariate quasi-copula can always be expressed as an infinite combination of measures induced by copulas. With this we are able to give the first characterization of measure-inducing quasi-copulas in the bivariate setting.


Introduction
Copulas, introduced in 1959 by Sklar, are one of the main tools for modeling dependence of random variables in statistical literature.These are multivariate functions that link the cumulative distribution function of a random vector and its one-dimensional marginal distributions.Copulas have found widespread use in various practical applications such as finance [13], biology [21], environmental sciences [4,10] and many others.
Quasi-copulas, a generalization of copulas, were introduced by Alsina, Nelsen, and Schweizer [1] in order to characterize operations on distribution functions that can be derived from operations on random variables.Their importance is explained by the following property: a point-wise infimum respectively supremum of a given set of copulas is always a quasi-copula.Thus, quasi-copulas are indispensable in the theory of imprecise probabilities, which model situations when the exact dependence structure between random variables, i.e. copula, is not known and is therefore replaced by a set of copulas.
The set of all quasi-copulas has been studied intensively in recent years and compared to the set of all copulas.The lattice theoretical properties of both sets were investigated in [17,11,2,19], while topological properties, particularly from the perspective of Baire categories, were studied in [8,9].From a measure-theoretic point of view one of the major differences between copulas and quasi-copulas is that, while every n-variate copula C induces a positive measure µ C on the Borel σ-algebra on [0, 1] n (see [14]), there exist n-variate quasi-copulas (for all n ≥ 2) that do not induce a signed measure on the same σ-algebra [16,12].This has stimulated numerous investigations of the mass distribution of quasi-copulas [15,5,24,23,7] and related concepts [6,18,20], aimed at a better understanding of the behaviour of quasi-copulas.As evidenced by several very recent papers, this is an active area of research, where there is still much to be done.In fact, a full characterization of quasi-copulas that do induce a signed measure on the Borel σ-algebra on [0, 1] n is still an open problem, see [3,Problem 4].
In this paper we study bivariate quasi-copulas that induce a signed measure on the Borel σ-algebra on [0, 1] 2 .We show that the signed measure µ Q induced by such a quasi-copula Q can always be expressed with measures induced by bivariate copulas.This is done by making use of a recent result in [7] giving a characterization of those quasi-copulas that can be expressed as linear combinations of two copulas.Note that any such quasi-copula automatically induces a signed measure, but not all measure-inducing quasi-copulas can be expressed as linear combinations of copulas, see [7, Example 13].However, the same paper also initiated the study of quasi-copulas using infinite series of copulas and this idea is key to our result.In particular, our main theorem reads as follows.
Theorem 1.For a bivariate quasi-copula Q the following two conditions are equivalent.
(ii) There exists a sequence of bivariate copulas C n and a sequence of real numbers γ n such that (a) the series of functions ∞ n=1 γ n C n converges uniformly to Q and (b) the series of induced measures ∞ n=1 γ n µ Cn converges in the total variation norm to some finite signed measure.
This result can be seen as a characterization of bivariate quasi-copulas that induce a signed measure on [0, 1] 2 , so it gives an answer to the open problem [3,Problem 4] mentioned above in the bivariate case.However, we do not consider the bivriate case to be completely resolved, since it would still be beneficial to obtain a characterization of measure-inducing quasi-copulas in operationally simpler terms.
The paper is structured as follows.In Section 2 we recall some basic notions from measure theory and some results on bivariate quasi-copulas that we will need in our proofs.The main part of the paper is devoted to the proof of Theorem 1. Assuming a quasi-copula Q induces a signed measure µ Q , we construct in Section 3 a sequence of measure-inducing quasi-copulas Q N that converge to Q and whose induced measures converge to µ Q .In addition, all these quasi-copulas are linear combinations of copulas.In Section 4 we convert the sequence Q N into a series of multiples of copulas C n , and finally prove Theorem 1.An example that demonstrates our result is given in Section 5.

Preliminaries
Throughout the paper we will denote the unit interval by I = [0, 1] and the unit square by I 2 .A function Q : I 2 → I is a (bivariate) quasi-copula if it satisfies the following conditions: Let (X, A) be a measurable space equipped with a finite signed measure µ.Let µ = µ + − µ − be the Jordan decomposition of measure µ, i.e., µ + and µ − are finite positive measures with disjoint supports.If S + and S − denote the supports of µ + and µ − , respectively, then µ + (A) = µ(A ∩ S + ) and µ − (A) = −µ(A ∩ S − ) for all A ∈ A. The positive measure |µ| = µ + + µ − is called the total variation measure of µ.It satisfies the inequality |µ(A)| ≤ |µ|(A) for all A ∈ A. The total variation norm of µ is given by ( 1) The vector space of all finite signed measures on (X, A) equipped with the total variation norm is a Banach space.For two finite measures µ 1 and µ 2 on A we write In particular, we have −µ − ≤ µ ≤ µ + for any finite signed measure µ.
The Borel σ-algebras on I and I 2 will be denoted by B(I) and B(I 2 ), respectively.Note that B(I 2 ) = B(I) ⊗ B(I) is the smallest σ-algebra on I 2 that contains all rectangles of the form R = [0, x] × [0, y] for some x, y ∈ I.It also contains all rectangles that are either closed of open on any of their four sides.In particular, it contains all half-open rectangles of the form R = (x 1 , x 2 ] × (y 1 , y 2 ] for some x 1 , x 2 , y 1 , y 2 ∈ I with x 1 < x 2 and y 1 < y 2 . A bivariate quasi-copula Q is said to induce a signed measure on B(I 2 ) if there exist a signed measure is the volume of R with respect to Q. Since quasi-copulas are continuous functions, the equality µ Q (R) = V Q (R) holds also if the rectangle R is either open or closed on any of its sides, because the µ Q measure of a vertical or horizontal segment is 0. For example, It is well known that any copula C induces a (positive) measure µ C on B(I 2 ) and this measure is stochastic in the sense that µ C (A × I) = µ C (I × A) = λ(A), where λ denotes the Lebesgue measure on I.
We recall a result from [7] that will be crucial for our constructions.For a positive integer m we will denote [m] = {1, 2, . . ., m}.
(i) There exist bivariate copulas A and B and real numbers α and β such that n ] for all k, l ∈ [2 n ], and x + = max{x, 0} for all x ∈ R.
Informally speaking, what the coefficient α Q does, is approximately detect at most how much positive mass quasi-copula Q accumulates on any vertical/horizontal strip relative to the width/height of the strip.With increasing n the detection is more and more accurate.

Measure induced by a quasi-copula
We assume throughout this section that Q is a bivariate quasi-copula that induces a signed measure µ Q on B(I 2 ).We will denote by µ Q is the zero measure.The goal of this section is to construct a sequence of bivariate quasi-copulas Q N with the following properties: (a) sequence Q N converges to Q uniformly, (b) for every N , Q N induces a signed measure µ N , (c) for every N , Q N is a linear combination of two copulas, (d) sequence µ N converges to µ Q in the total variation norm.
Let us give an outline of the construction, which is split into several steps to make it easier to follow.The main idea is to start with condition (c), using Theorem 2. We need to approximate quasi-copula Q, which need not satisfy α Q < ∞ (see condition (ii) of Theorem 2), with a quasi-copula Q N , that does satisfy α Q N < ∞.To this end we identify sets of the form K N × I and I × L N which cause α Q to be greater than cN for some fixed normalising constant c.These sets are defined in Subsection 3.1.Quasi-copula Q N (for every N ≥ 1) is constructed in Subsection 3.3 by "smoothing out" the mass distribution of Q on the set (K N × I) ∪ (I × L N ) and leaving it unchanged elsewhere.For this to work, the sets K N and L N need to be constructed carefully, because, to ensure property (a), they need to have small Lebesgue measure, i.e., in the limit when N goes to infinity the measure must tend to 0, and, to ensure property (d), the sets (K N × I) ∪ (I × L N ) need to have small |µ Q | measure in the same sense.Both of these properties are verified in Subsection 3.2.We then prove property (b) by explicitly constructing signed measures µ N and showing with a direct calculation that they are induced by quasi-copulas Q N .This is done in Subsection 3.4, where property (d) is also verified.Finally, in Subsection 3.5 we prove that quasi-copulas Q N satisfy condition (ii) of Theorem 2, which then give us property (c) and consequently also property (a).

Construction of sets K N .
We start with the construction of sets K N , the sets L N will be defined later.For an integer n ≥ 0 we introduce a family of open intervals Note that for every S ∈ P n we have λ(S) = 1 2 n , so that (3) S ∈ J n,N if and only if S ∈ P n and To make sense of what follows, we make the following remark.Intuitively, the sets J n,N ×I are the "bad" strips, which make α Q from condition (ii) of Theorem 2 large and possibly infinite.We will later "smooth out" the mass distribution of Q on the bad strips in order to make α Q finite.For this to work, we actually need to slightly enlarge the sets J n,N , so that the smoothing will also affect the boundary of the bad strips.
For every n ≥ 1 and S ∈ P n there exists a uniquely determined S ∈ P n−1 such that S ⊆ S. For all N ≥ 1 let and define inductively for all n ≥ 1 A.
We give an example to demonstrate the sets introduced above.
Example 3. Let Q be a quasi-copula with mass distributed as depicted in Figure 1 left, where the unit square I 2 is divided into 16 × 16 small squares of dimensions and the corresponding sets J n,N and K n,N (right).Each dark gray square contains a mass of 1  16 and each light gray square contains a mass of − 1 16 .
The corresponding (nonempty) sets J n,N and K n,N , where 0 ≤ n ≤ 4 and 1 ≤ N ≤ 2, are depicted on Figure 1

right, along with the values µ
Next, we give some basic properties of the sets defined above.Lemma 4. For all n ≥ 1 we have Proof.We prove this by induction on n.From equations ( 2) and ( 4) if follows that J 1,N = S∈J 1,N S ⊆ S∈J 1,N S = K 0,N , so the claim holds for n = 1.Suppose it holds for some n ≥ 1 and let S ∈ J n+1,N .Then, by equation ( 5), either Since S ∈ P n+1 and K k,N is a union of some subfamily of P k , it follows that S ⊆ K k,N , because P n+1 is a refinement of P k for any k ≤ n.By equation ( 2) we conclude that J n+1,N ⊆ n k=0 K k,N , which finishes the proof.□ Lemma 5.For any N ≥ 1 the sets K n,N with n ≥ 0 are disjoint.
Proof.If S ∈ J n+1,N , then S ∈ P n , while n−1 k=0 K k,N is a union of some subfamily of By equation ( 5) this implies that the sets K n,N and n−1 k=0 K k,N are disjoint.The claim now follows.□ Lemma 5 implies that for every N ≥ 1 we have a disjoint (double) union ( 6) Next lemma shows that the sequence of sets K N is decreasing in N with respect to inclusion.
Proof.Let A ∈ K n,N +1 for some n ≥ 0. Then A = S for some S ∈ J n+1,N +1 .Clearly, J n+1,N +1 ⊆ J n+1,N , so that S ∈ J n+1,N .From equation ( 2) and Lemma 4 it follows that B∈K k,N B, so there exists 0 ≤ k ≤ n and B ∈ K k,N such that S ∩B ̸ = ∅.Since S ∈ P n+1 and B ∈ P k with k ≤ n, we infer S ⊆ B and consequently even A = S ⊆ B, because k < n + 1.We have thus shown that Note that all subsets of I considered above are open, so they are Borel measurable.We will now show that the Lebesgue measure λ of K N is small.
N for all N ≥ 1. Proof.By Lemma 5 the double union in ( 6) is disjoint, so we have From equations ( 4) and ( 5) it follows that Using property (3) we obtain .
By Lemma 4 we have n k=1 J k,N ⊆ n−1 k=0 K k,N , so we can further estimate

□
We can now give one of the key points of this construction.Let The first crucial property of the set K is that its Lebesgue measure is 0.
Proof.By Lemma 6 the sequence of sets K N is decreasing in N .Hence, the conclusion follows directly from equation ( 7) and Lemma 7. □ Furthermore, the restriction of measure µ Q to the set K × I is the zero measure.
Lemma 9.The measures µ ′ defined for all B ∈ B(I 2 ) by is the zero measure.
Proof.Let (x, y) ∈ I 2 be arbitrary and let Taking into account the monotonicity of K N , see Lemma 6, and the fact that µ Q is a finite signed measure, we obtain Using equation ( 6) and the fact that the union in ( 6) is disjoint by Lemma 5, we obtain (8) For any rectangle (a, b}×[0, y], where } stands for either ) or ], the continuity, groundedness and 1-Lipschitz property of quasi-copulas imply regardless of whether (a, b} is empty or not.Applying this inequality to the rectangle ([0, x] ∩ A) × [0, y] in (8) and using also Lemma 8 we obtain and consequently µ ′ (R) = 0. To prove that µ ′ is the zero measure on B(I 2 ) we use a standard argument.By the above the collection ) and µ ′ is the zero measure.□ As a consequence to Lemma 9, we obtain the second crucial property of the set K, which will be essential for the convergence of measures constructed later.
Lemma 10.We have Proof.The sequences K N ×I is decreasing in N by Lemma 6, and we have ∞ N =1 (K N ×I) = K × I by equation (7).This implies lim By Lemma 9, the measure µ ′ defined in that lemma is a zero measure.Denote the support of µ + Q by P + .Then Switching the first and second coordinate in the above constructions, we can analogously define for all n ≥ 0 and N ≥ 1 the sets By symmetry the versions of Lemmas 7-10 for the sets L N and L also hold.In particular, is the zero measure, and the following lemma holds.
Lemma 11.We have We now turn our attention to the construction of a sequence of quasi-copulas Q N .For every positive integer N denote the complements 6) and ( 9) the sets K N and L N are open, so the sets K ′ N and L ′ N are closed and contain 0 and 1.The restriction is a sub-quasi-copula according to [22,Definition 2.3].We can extend sub-quasi-copula N to a quasi-copula Q N using the formula from the proof of [22, Theorem 2.3], which we recall now.Let (x, y) be an arbitrary point in I 2 and denote Then Then the value of the extension at (x, y) is given by In particular, if R is a closed rectangle such that only the vertices of R lie in K ′ N × L ′ N , then Q N is bilinear on R (e.g.R 5 on Figure 2).If only the left and right side of R lie in K ′ N × L ′ N , then Q N is linear in x on R but not necessary in y (e.g.R 2 and R 8 on Figure 2), if only the bottom and top side of R lie in K ′ N × L ′ N , then Q N is linear in y on R but not necessary in x (e.g.R 4 and R 6 on Figure 2), and if the whole R lies in K ′ N × L ′ N , then Q N on R is not linear in any coordinate in general (e.g.R 1 , R 3 , R 7 , and R 9 on Figure 2).So the extension Q N is as linear as possible.This means that it spreads mass on I\(K ′ N ×L ′ N ) = (K N ×I)∪(I×L N ) uniformly in certain directions, which will be important later on.

Construction of measures µ N .
We aim to prove that quasi-copulas Q N induce signed measures on B(I 2 ).We now construct these measures.From equations ( 6) and ( 9) it follows that the sets K N and L N are countable unions of disjoint open intervals, say (11) K where each (a N i , b N i ) and (c N i , d N i ) is a member of ∞ n=0 P n .We are assuming here without loss of generality that the unions are infinite.If they are finite the arguments are essentially the same.
For a set E ∈ B(I) with λ(E) > 0, define a probability measure λ E on B(I) by for all A ∈ B(I), and finite signed measures φ E and ψ E on B(I) by ( 12) × E for all A ∈ B(I).Furthermore, introduce finite signed measures µ N and µ N on B(I 2 ) by ( 13) and ( 14) , where × denotes both the product of measures and the Cartesian product of sets.The idea behind the definition of µ N is that it should correspond to the measure induced by the the extension Q N on the set (K N × I) ∪ (I × L N ).In particular, the first sum corresponds to the regions where Q N is linear in the x coordinate (regions R 2 and R 8 on Figure 2), the second sum corresponds to the regions where Q N is linear in y coordinate (regions R 4 and R 6 on Figure 2) and the third sum corresponds to the regions where Q N is bilinear (region R 5 on Figure 2).
We first need to show that µ N is well defined.
Lemma 12.For every N ≥ 1 the sum in equation (13) converges in the total variation norm, so µ N is a finite signed measure on B(I 2 ).
Proof.For every  1) and (12) we estimate ( 16) Combining inequalities (15) and (16) gives , and similarly we obtain Since the vector space of all finite measures equipped with the total variation norm is a Banach space, this implies that the sum in equation ( 13) converges in the total variation norm and µ N is a finite signed measure.□ We can now prove that quasi-copulas Q N induce signed measures.For every N ≥ 1 define a finite signed measure where µ N and µ N are given in equations ( 13) and ( 14).We show with a direct calculation that measure µ N is induced by Q N .
Lemma 13.For every N ≥ 1 quasi-copula Q N induces signed measure µ N .
Proof.For any (x, y) ∈ I 2 we use equations ( 13) and ( 14) to get (18) . We consider four cases, depending on where the point (x, y) lies.
Case 1: x / ∈ K N and y / ∈ L N .On Figure 2 this corresponds to (x, y) ∈ R 1 ∪R 3 ∪R 7 ∪R 9 .In this case x is not contained in any (a N i , b N i ) and y is not contained in any (c N j , d N j ) so equation ( 18) becomes and this union is disjoint, we conclude By equation (10) this is equal to Q N (x, y) because x ∈ K ′ N and y ∈ L ′ N .
Case 2: x ∈ K N and y / ∈ L N .On Figure 2 this corresponds to (x, y) ∈ R 2 ∪ R 8 .We may assume without loss of generality that x ∈ (a N 0 , b N 0 ) since the order of the intervals in equation ( 11) is arbitrary.On the other hand, y is not contained in any (c N j , d N j ).By splitting we infer that equation ( 18) can be written as Note that a N 0 / ∈ K N , so we may use Case 1 to evaluate the first term in the above expression.Using equation (19), the first term is equal to µ N [0, a N 0 ] × [0, y] = Q(a N 0 , y).Furthermore, the last term and the first of the two sums Finally, by continuity of Q we can express By equation (10) this is equal to Q N (x, y) because x ∈ K N and y ∈ L ′ N .Case 3: x / ∈ K N and y ∈ L N .On Figure 2 this corresponds to (x, y) ∈ R 4 ∪ R 6 .This case is treated similarly as Case 2. Assuming y ∈ (c N 0 , d N 0 ), we obtain which is again equal to Q N (x, y) by equation (10).
Case 4: x ∈ K N and y ∈ L N .On Figure 2 this corresponds to (x, y) ∈ R 5 .We may assume without loss of generality that x ∈ (a N 0 , b N 0 ) and y ), we infer that equation ( 18) can be written as Note that c N 0 / ∈ L N , so by Case 2, using equation ( 20), the first term in the above expression is equal to Furthermore, the last term and the first of the two sums ∞ i=0 are equal to 0 because (c N 0 , y] ⊆ L N .Hence, . We can simplify the last sum, also using x ∈ (a N 0 , b N 0 ), to obtain ( 21)

Note that
so we may combine the second and third row of equation ( 21) to get By equation (10) this is again equal to Q N (x, y) because x ∈ K N and y ∈ L N .□ Next step is to establish the convergence of finite signed measures µ N .
Lemma 14.The sequence of measures µ N converges to µ Q in the total variation norm.
Proof.From the proof of Lemma 12, see in particular inequality (17), it follows that When N tends to infinity, the right-hand side converges to 0 by Lemmas 10 and 11, so the sequence of measures µ N converges to the zero measure in the total variation norm.Furthermore, note that ).So we can use equation ( 1) and an analogous calculation as in inequality (16) to obtain . Using a similar argument as for µ N , we infer that the sequence µ Q − µ N converges to the zero measure, so that µ N converges to µ Q in the total variation norm.We conclude that the sequence of finite measures We have so far shown that quasi-copulas Q N induce signed measures µ N and the measures µ N converge in the total variation norm to measure µ Q , which is induced by Q.
The final property of quasi-copulas Q N that we need is that every Q N is a linear combination of two copulas.To prove this we will show that each Q N satisfies condition (ii) of Theorem 2.
Lemma 15.For every N ≥ 1 there exist bivariate copulas A N and B N and real numbers α N and β N such that Proof.Fix N ≥ 1.By Theorem 2 it suffices to prove that Hence, by equations ( 13) and ( 14), 12) and ( 14) we infer We now consider two cases.
Case 1: ∈ n m=0 K m,N by equation ( 6).This implies that there exists m with 0 ≤ m ≤ n and A ∈ K m,N such that ( k−1 n ) ⊆ A because A ∈ P m and m ≤ n.By equation (11) we may assume without loss of generality that A = (a N 0 , b N 0 ).Hence, ( k−1 2 n , k 2 n ) does not intersect any (a N i , b N i ) with i ≥ 1.It now follows from inequality (23) and from or equivalently Since A ∈ K m,N with m ≤ n and the set K m,N is disjoint from m−1 l=0 K l,N , Lemma 4 and equation (2) imply that A / ∈ J m,N .By condition (3) we have Case 2: ) is a member of ∞ n=0 P n .Hence, we have three options for each (a N i , b N i ), namely, (i) the interval . By the case assumption and equation (11), option (ii) cannot happen, so the interval Thus, inequality (23) implies , otherwise Lemma 4 and equation ( 6) would imply that This, together with inequality (25), gives (26) ). Combining inequalities (24) and (26) with inequality (22) gives Since n and k were arbitrary, we infer sup In addition, let (30) Then the series in equation ( 27) is expressed as If we omit the parenthesis in series (31), the resulting series is not convergent in the total variation norm.For example, evaluating it on the set I 2 we obtain the series Before we omit the parenthesis in series (31), we need to split its terms into sums of terms with small enough norm, so that after omitting the parenthesis the "oscillation" will tend to 0. For every N ≥ 1 we choose a positive integer We rewrite the series in equation ( 31) as and remove the parenthesis to obtain the series (33) We now prove the following.
Lemma 17.The series of finite signed measures (33) converges to µ Q in the total variation norm.
Proof.The difference between µ Q and any partial sum of series (33), is of the form for some m ≥ 1, k ∈ [mM m ], and δ ∈ {0, 1}.We can estimate its total variation norm as follows .
Using M m > |ξ m | and the fact that µ Em is a probability measure, we obtain .
When m tends to infinity, the first term converges to 0, the second term converges to 0 because it is the norm of a single term of a converging series (31), and the last term converges to 0 because it is the norm of the tail of a converging series (31).This proves that the partial sums of series (33) converge to µ Q in the total variation norm.□ We are now finally ready to prove Theorem 1.
Proof of Theorem 1.
(i) =⇒ (ii): Assume that a quasi-copula Q induces a signed measure µ Q on B(I 2 ).Let C 1 , C 2 , C 3 , . . .be the sequence of copulas , . . .defined in equations ( 28) and (30), and let γ 1 , γ 2 , γ 3 , . . .be the sequence of real numbers , . . .defined in equations ( 29) and (30).By Lemma 17 the series ∞ n=1 γ n µ Cn converges in the total variation norm to µ Q , which proves condition (b).For all k ≥ 1 and (x, y) ∈ I 2 we have Hence, the convergence of the series ∞ n=1 γ n µ Cn in the total variation norm implies that the series ∞ n=1 γ n C n converges uniformly to Q.This proves condition (a).We note that if condition (i) of Theorem 1 holds, then the series in condition (ii)(b) converges to µ Q .This implies that a measure induced by a bivariate quasi-copula can always be expressed as a converging sum of multiples of measures induced by bivariate copulas, i.e., as what we can call an infinite linear combination of measures induced by copulas.
Corollary 18.Any measure µ induced by some measure-inducing bivariate quasi-copula can be expressed as µ = ∞ n=1 γ n µ n , where each µ n is a measure induced by some bivariate copula, each γ n is a real number, and the series converges in the total variation norm.
As another corollary to Theorem 1 we obtain the following interesting property.Theorem 19.Let Q be a bivariate quasi-copula that induces a signed measure µ Q on B(I 2 ).Then is a joint distribution function of two absolutely continuous random variables X and Y with ranges in I 2 .
Proof.Let C n and γ n be the sequences from Theorem 1, so that µ Q = ∞ n=1 γ n µ Cn .Function H is clearly a distribution function of two random variables X and Y with support in I.The cumulative distribution of X is given by Let µ F X be the positive measure induced by F X , i.e., µ for all A ∈ B(I).Suppose A ∈ B(I) satisfies λ(A) = 0. Let P + and P − be the supports of measures µ + Q and µ − Q , respectively.Then Since µ Cn is a positive measure, µ Cn (A × I) ∩ P + ≤ µ Cn (A × I) = λ(A) = 0 and similarly µ Cn (A × I) ∩ P − = 0 for all n ≥ 1.Hence, µ F X (A) = 0.This shows that measure µ F X is absolutely continuous with respect to Lebesgue measure.By Radon-Nikodym theorem there exists a B(I)-measurable function f X such that F X (x) = µ F X ([0, x]) = [0,x] f X dλ.Therefore, X is absolutely continuous random variable with density f X .Similarly, Y is also absolutely continuous.

Example
We conclude this paper with an example illustrating Theorem 1 and Corollary 18.In [7, Example 13] the authors construct an example of a quasi-copula Q, that induces a finite signed measure, but cannot be written as a linear combination of two copulas.This implies that its induced measure µ Q cannot be written as a linear combination of two measures induced by copulas.On the other hand, by Corollary 18, measure µ Q can be expressed as an infinite linear combination of measures induced by copulas.We now find such a representation of µ Q .
First, we briefly recall the definition of quasi-copula Q, for some additional details see [7,Example 13].For a positive integer n let Q n be a discrete quasi-copula defined on an equidistant mesh {0, 1 2n+1 , 2 2n+1 , . . ., 1} 2 ⊆ I 2 , which has mass distributed in a checkerboard pattern (of positive and negative values) within the central diamond-shaped area (with no mass outside this area), so that there are exactly (n + 1) 2 squares with positive mass 1 2n+1 and n 2 squares with negative mass − 1 2n+1 .For example, the spread of mass of Q 3 is given by the matrix If we denote by Q n the bilinear extension of Q n to I 2 , then quasi-copula Q is defined as an ordinal sum of quasi-copulas { Q n } ∞ n=1 , with respect to the partition J = {J n } ∞ n=1 , where J n = [a n−1 , a n ] and a n = 1 2 + 1 4 + . . .+ 1 2 n = 1 − 1 2 n .Since the length of J n is 1 2 n , the summand Q n contributes a total mass of 1  2 n to the mass of Q. Denote the product copula by Π.Note that for every positive integer n the function C n = 1 2n ((2n + 1)Π − Q n ) is clearly grounded and has uniform marginals.In fact, it is a copula since its mass is nowhere negative and its total mass is equal to 1.We can express (35) For a positive integer n denote by C n the ordinal sum with respect to partition J , where the n-th summand is C n and all other summands are Π.In addition, denote by P the ordinal sum with respect to partition J , where all the summands are Π.We claim that (36) µ Q = µ P + 2(µ P − µ C 1 ) + 4(µ P − µ C 2 ) + 6(µ P − µ C 3 ) + . . .+ 2n(µ P − µ Cn ) + . . .Indeed, the series on the right converges in the total variation norm because the support of measure 2n(µ P − µ Cn ) is contained in J n × J n , so by equation ( 35) and the series ∞ n=1 4n(n+1) 2   2 n (2n+1) 2 is convergent with sum ≈ 2.842.In addition, on each J n ×J n the sum of the series in (36) coincides with µ Q by equation (35), because only the two terms µ P and 2n(µ P − µ Cn ) are nonzero there.This implies that the sum of the series in (36) is indeed equal to µ Q .
Using the trick from the proof of Lemma 17, we obtain from equation (36) the converging sum expression for µ Q , namely (ii) =⇒ (i): By condition (b) the sum of the series ∞ n=1 γ n µ Cn is a finite signed measure on B(I 2 ).Denote this measure by µ.As above, this implies that the series ∞ n=1 γ n C n converges uniformly to the function (x, y) → µ [0, x] × [0, y] .On the other hand, this series converges to Q by condition (a).Hence, Q(x, y) = µ [0, x] × [0, y] for all (x, y) ∈ I 2 .□

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Remark 20.Theorem 19 essentially states that the signed measures µ 1 (A) = |µ Q |(A × I) and µ 2 (A) = |µ Q |(I × A) are absolutely continuous with respect to the Lebesgue measure on I.The conclusion of Theorem 19 holds also if we replace measure |µ Q | in formula (34) by either µ + Q or µ − Q .