On convergence of associative copulas and related results

Abstract: Triggered by a recent article establishing the surprising result that within the class of bivariate Archimedean copulasCar di erent notions of convergence standard uniform convergence, convergencewith respect to the metric D1, and so-called weak conditional convergence coincide, in the current contribution we tackle the natural question, whether the obtained equivalence also holds in the larger class of associative copulas Ca. Building upon the fact that each associative copula can be expressed as ( nite or countably in nite) ordinal sum of Archimedean copulas and the minimum copula M we show that standard uniform convergence and convergence with respect to D1 are indeed equivalent in Ca. It remains an open question whether the equivalence also extends to weak conditional convergence. As by-products of some preliminary steps needed for the proof of themain result we answer two conjectures going back to Durante et al. and show that, in the language of Baire categories, when working with D1 a typical associative copula is Archimedean and a typical Archimedean copula is strict.


Introduction
Various di erent notions of convergence in the family of bivariate copulas C have been considered in the literature: The standard uniform metric d∞ is probably the most common choice; since, however, d∞ is not capable of distinguishing independence and complete dependence (or, in the words of [24], d∞ does not 'distinguish between di erent types of statistical dependence') the stronger metric D was introduced in [36]. Letting K A (·, ·), K B (·, ·) denote Markov kernels (regular conditional distributions) of the copulas A and B, respectively, the metric D is de ned by Sticking to Markov kernels/conditional distributions and considering weak convergence of almost all conditional distributions of copulas results in the notion of weak conditional convergence which was introduced recently in [17]. Weak conditional convergence implies convergence with respect to D , and it is straightforward to construct examples illustrating that the reverse implication is wrong (again see [17]). When working within the subclass of absolutely continuous copulas other natural notions of convergence are the total variation distance TV (which, in the absolutely continuous setting coincides with the L -distance of the densities) and the Kullback-Leibler divergence KL (see, e.g., [35] and the references therein). In case the copula densities are greater than almost everywhere then according to [26] we have the following interrelation, where a ⇒ b indicates the convergence with respect to a implies convergence with respect to b: As mentioned above, the topologies induced by d∞, D and weak conditional convergence on C do not coincide: the topology induced by weak conditional convergence is strictly ner than the one induced by D , which, in turn is strictly ner than the one induced by d∞. According to [17], however, on large classes like the family of Archimedean copulas as well as on the family of Extreme Value copulas, all three topologies do coincide. Considering that the family of Archimedean copulas is dense in the family of associative copulas with respect to d∞ (see [18,19]) and that every associative copula admits a so-called ordinal sum representation (see [23]) in terms of nitely or countably in nitely many Archimedean copulas and M it is natural to ask, whether the three notions of convergence are also equivalent on the family Ca of all associative copulas. In the current contribution we provide a partial a rmative answer and show that on Ca convergence with respect to d∞ and D are indeed equivalent. The question, whether weak conditional convergence is equivalent too, remains open -we have neither been able to prove equivalence nor to construct a counterexample.
Although the main result of the paper is the equivalence of d∞ and D on Ca several auxiliary results on denseness of the class of strict and the class of non-strict Archimedean copulas in (Ca, D ) produce nice by-products in so far as we are able to answer two conjectures going back to Durante et al. in [6] and show that, in the language of Baire categories, when working with D a 'typical' associative copula is Archimedean and a 'typical' Archimedean copula is strict.
The remainder of this paper is organized as follows: Section 2 contains notation and preliminaries which will be used in the sequel. In Section 3 we rst prove the fact that the family of all strict and the family of all non-strict Archimedean copulas are dense in (Ca, D ) by following the procedure studied in [19] for d∞, and then show the afore-mentioned Baire category results. Finally, Section 4 focuses on convergence of associative copulas and establishes the main result in several steps. Some examples and graphics illustrate the chosen approach.

Notation and preliminaries
In the sequel we will let C denote the family of all bivariate copulas. For each copula C the corresponding doubly stochastic measure will be denoted by . For more background on copulas and doubly stochastic measures we refer to [8,27]. As copulas can be seen as binary operations on [ , ] associativity is of particular interest, that is, C ∈ C is called associative if for all x, y, z ∈ [ , ] we have Ca denotes the class of all associative copulas. Associative copulas are closely related to triangular norms. In fact, according to [25] a copula C is a triangular norm if and only if C ∈ Ca whereas a triangular norm T is a copula if and only if it is Lipschitz continuous with Lipschitz constant (see, for example, [1,18]).
The standard uniform metric d∞ on C is de ned by |C (x, y) − C (x, y)|.
It is well known that the metric space (C, d∞) is compact and that pointwise and uniform convergence of a sequence of copulas (Cn) n∈N are equivalent (see [8,37]). For every metric space (S, d) the Borel σ-eld on S will be denoted by B(S). In what follows Markov kernels will play a prominent role. A Markov kernel from R to R is a mapping K : R × B(R) → [ , ] such that for every xed E ∈ B(R) the mapping x → K(x, E) is (Borel-)measurable and for every xed x ∈ R the mapping E → K(x, E) is a probability measure. Given two real-valued random variables X, Y on a probability space (Ω, A, P) and letting 1 E denote the characteristic function of a set E ⊆ R we say that a Markov kernel K is a regular conditional distribution of Y given X if K(X(ω), E) = E(1 E • Y|X)(ω) holds P-almost surely for every E ∈ B(R). It is well-known that for X, Y as above, a regular conditional distribution of Y given X always exists and is unique for P X -a.e. x ∈ R whereby P X denotes the push-forward of P via X, i.e., P X (E) = P(X − (E)) for every E ∈ B(R) (see, e.g., [15,20]). In case (X, Y) has distribution function C ∈ C we will let denote (a version of) the regular conditional distribution of Y given X and refer to it as Markov kernel of C. De ning the x-section of a set G ∈ B([ , ] ) by Gx := {y ∈ [ , ] : (x, y) ∈ G} we have the following disintegration formula (see [15,20]) hence, in particular, [ , ] for every E ∈ B([ , ]), whereby λ denotes the Lebesgue measure on R. For more information on conditional expectation and general disintegration we refer to [15,20]. Following [8] the rightside upper Dini derivative∂ of a copula C with respect to the rst coordinate is de ned bȳ As shown in [7], the map y →∂ + C(x, y) is non-decreasing which we will use subsequently to clarify the relation between Markov kernel and rightside upper Dini derivative: Lemma 2.1. Let C be an arbitrary copula. Then for λ -a.e. (x, y) ∈ ( , ) .
Proof. Letting ∂ i denote the partial derivative with respect to the i-th coordinate it is well-known (see, e.g., [8,33]) that for every y ∈ [ , ] there exists a set Λy ∈ B([ , ]) of full measure such that for all x ∈ Λy we have that ∂ C(x, y) exists and∂ holds. Then Λ := y∈( , )∩Q Λy ∈ B([ , ]) is a set of full measure and for all x ∈ Λ, y ∈ ( , ) ∩ Q we havē As for x ∈ Λ, y ∈ ( , ) we have K C (x, [ , y]) ≥∂ + C(x, y) it su ces to show that is a λ -null set which can be done as follows: First, note that (x, y) → K C (x, [ , y]) as well as (x, y) → K C (x, [ , y)) are measurable functions. Second, for x ∈ Λ, y ∈ [ , ] we have K C (x, [ , y]) >∂ + C(x, y) ⇔ which is measurable. As for x ∈ Λ we have Γx is at most countably in nite, applying disintegration yields λ (Γ) = .
The following consequence of Lemma 2.1 is immediate: The class of all completely dependent copulas will be denoted by C cd .
For two copulas C , C the so-called Markov product or star product C * C is de ned by The star-product is always a copula (see [5]) and according to results of [38] a Markov kernel of C * C is given by Subsequently we will make use of the following properties of the star product: First, C ∈ C is completely dependent if and only if there exists a copula B such that the identity B * C = M holds, and second, if B ∈ C and if the sequence of copulas (Cn) n∈N converges uniformly to another copula C then (B * Cn) n∈N converges uniformly to B * C. For more properties of the class C cd and the star product we refer to [22,36] and [5,14], respectively. Considering Markov kernels allows to de ne stronger metrics than the standard uniform one: In [36] it was shown that D , D , D∞ are three metrics generating the same topology on C. In what follows we will primarily work with D and refer to [9,36] for more information on D and D∞. The metric space (C, D ) is complete and separable but not compact (again see [36]). An even stronger notion of convergence involving Markov kernels is introduced and studied in [17]: Let C, C , C , . . . be copulas with corresponding Markov kernels K C , K C , K C , . . . . Then (Cn) n∈N is said to converge weakly conditional to C if and only if for λ-almost every x ∈ [ , ] the sequence (K Cn (x, ·)) n∈N of probability measures on B([ , ]) converges weakly to the probability measure K C (x, ·) (in short, Cn wcc −−→ C, where 'wcc' stands for 'weak conditional convergence'). The following interrelation of the afore-mentioned modes of convergence holds: weak conditional convergence ⇒ convergence w.r.t. D ⇒ convergence w.r.t. d∞.
For counterexamples for the reverse implications again see [17] and [36].

The interrelation of Archimedean and associative copulas with respect to D
Triggered by [19], where the authors prove that the class of Archimedean copulas is dense in the class of associative copulas with respect to the uniform metric d∞, in what follows we show the result w.r.t. D and, as a by-product, answer two open questions posed in [6] regarding Baire category results for Archimedean copulas. In Section 4 we then exploit the proven denseness to derive surprising convergence results of associative copulas with respect to D .
An Archimedean copula A = Aφ is a copula induced by a convex and strictly decreasing function φ : is the respective one-sided limit. We refer to A as the Archimedean copula induced by φ; in case of φ( +) = ∞, A is called strict and non-strict otherwise. In the sequel Car denotes the class of all Archimedean copulas, C s ar the subclass of strict Archimedean copulas and C ¬s ar the subclass of nonstrict Archimedean copulas. Since generators are only unique up to a multiplicative constant we will from now on also assume (without explicit reference) that the generator is normalized in the sense that φ( ) = holds. With this convention there is a one-to-one correspondence between generators and induced copulas.
In what follows we let . By convexity, φ is di erentiable outside a countable subset of ( , ), i.e. D + φ(x) = D − φ(x) holds for all but at most countably many x ∈ ( , ) and D + φ is non-decreasing and right-continuous. According to [16,31] we additionally have D − φ(x) = D + φ(x−) for every x ∈ ( , ). Letting Cont(D + φ) ⊆ ( , ) denote the set of all continuity points of D + φ in ( , ) it follows that [ , ] \ Cont(D + φ) is at most countably in nite and thus has Lebesgue measure . Setting D + φ( ) = −∞ in case of strict φ as well as D + φ( ) = (for strict and non-strict φ) allows to view D + φ as non-decreasing and right-continuous function on the full unit interval. The Kendall distribution function of an Archimedean copula A with generator φ is given by (see, e.g., [13]) Following [10,27] for every t ∈ [ , ] we de ne the t-level If φ is strict then according to [10] is (a version of) the Markov kernel of A, for non-strict φ a version is given by (3.4) According to [17] all three afore-mentioned modes of convergence coincide within Car, that is, a sequence of Archimedean copulas (An) n∈N converges uniformly to some A ∈ Car if and only if (An) n∈N converges to A with respect to D if and only if (An) n∈N converges to A weakly conditional (as n → ∞). Archimedean copulas play a central role in our study since, together with the Fréchet-Hoe ding upper bound M, they form the building blocks of associative copulas: According to [23] (see also [1]) every associative copula C admits an ordinal sum representation in the following sense: there exists some index set Notice that, contrary to literature, the above de nition explicitly exposes (shrunk versions of) M along diagonal blocks. De ning the a ne transformation We will also work with the pre-image of T i and also denote it by T − i since no confusion will arise. In this paper a nite associative copula is by de nition an associative copula with nite index set I. Moreover, a subinterval (a i , b i ) appearing in the ordinal sum representation of an associative copula C will also be simply called subinterval of C.
We start with the following Lemma stating that any non-strict Archimedean copula can be approximated arbitrarily well by strict Archimedean copulas with respect to D and vice versa: Proof. We only prove denseness of strict Archimedean copulas, the non-strict case follows in a similar manner. Fix ε > and let A¬s have non-strict generator φ¬s. In [19] it is shown that for every n ∈ N there exists a strict generator φ n s with φ n s = φ¬s on ( n , ] such that the corresponding strict Archimedean copula A n s ful lls d∞(A¬s, A n s ) ≤ n . Letting n → ∞ we obtain pointwise convergence of φ n s → φ¬s on ( , ] and using [17, Theorem 4.2] we even have weak conditional convergence of (A n s ) n∈N implying D (A n s , A¬s) → as n → ∞.
Next we show that an associative copula can be approximated arbitrarily well w.r.t. D by a nite ordinal sum consisting only of strict Archimedean copulas or M.
Choosing η ∈ ( , ε) we obtain the assertion for arbitrary Archimedean copulas. Moreover, applying Theorem 3.1 to all those Archimedean components that are non-strict completes the proof.
Loosely speaking, we found that selecting only nitely many 'large' components of an associative copula yields a nite associative copula that is close to the original one with respect to D . However, we do not necessarily have a nite ordinal sum purely consisting of Archimedean components as there can still exist subintervals (a i , b i ) of C containing M in our approximation. The following lemma resolves this issue: Proof. We show that for generators of the form φn(x) = φ(x) n for some (strict and) continuously di erentiable Archimedean generator φ we have convergence of the induced sequence (An) n∈N to M w.r.t. D (see Remark 3.4 for the proof based on a general (strict) generator φ): and the corresponding copula is given by An(x, y) = φ − ( n φ(x) n + φ(y) n ). It is well-known (see, e.g. [18,Proposition 8.5]) that the sequence (An) n∈N converges uniformly to M. In fact, this follows from either of the following two relations: The Markov kernel Kn corresponding to An is given by First suppose < x < y < . As φ is continuously di erentiable φ (x) φ (An(x,y)) tends to as n → ∞. Concerning the second factor we have In case of x > y we have An(x, y) → y for n → ∞ and the rst factor of the Markov kernel converges to the constant φ (x) φ (y) whereas for the second factor we have φ(x) φ(An(x,y)) < . Indeed, choose δ > such that < y < x−δ then there exists n ∈ N such that for all n ≥ n we have An(x, y) < x − δ which directly yields that for these n holds. As consequence the second factor converges to as n → ∞ which shows that for x ∈ ( , ) we have weak convergence of Kn(x, ·) n→∞ − −−− → K M (x, ·). Hence (An) n∈N even converges weakly conditional to M which implies converges w.r.t. D (again see [36]). An(x, u) ∈ Cont(D + φ)} it follows that Ux is the countable intersection of sets of full measure and as such has full measure itself. The rest of the proof is identical to the one of Lemma 3.3 above.
2. Notice that by Lemma 4.3 for x > y we even have weak conditional convergence of general sequences of Archimedean copulas uniformly converging to M.
Example 3.5. The (normalized) generator φ(x) = /x − produces the copula commonly abbreviated by Aφ = Π Σ−Π which is a member of numerous families of Archimedean copulas. Moreover, the generators φn(x) = φ(x) n = ( /x − ) n de ne the family (4.2.12) in Nelsen [27] and the sequence of induced copulas (An) n∈N not only converges uniformly to M, but also weakly conditional by Lemma 3.3.   As next step we show that the ordinal sum C = ( , c, A , c, , A ) of two strict Archimedean copulas can be approximated arbitrarily well with respect to D by one strict Archimedean copula. In [19] the authors de ne for every ε > a strict Archimedean copula Aε having the property that d∞(Aε , C) < ε. We prove that for this particular glueing construction even lim ε→ D (Aε , C) → holds. For each ε > and c ∈ ( , ) setting de nes the generator of a strict Archimedean copula Aε ful lling d∞(C, Aε) < ε. Here, φ , φ are the generators of A , A and T , T are the transformations from [ , c] and [c, ] into [ , ], respectively. The linear function g de ned on (c − ε , x ] for some x < c + ε connects the two (appropriately transformed) generators of A , A in a way such that the resulting map is again an Archimedean generator. We will make use of this construction, however, the upcoming proof is based on the order of the copulas C and Aε in di erent sections of [ , ] rather than on the formula of φε.  . , A , . , , A ). Following the construction in [19] and considering ε > yields a strict Archimedean copula Aε ful lling d∞(C, Aε) < ε.
Lemma 3.8. Let C = ( , c, A , c, , A ) be the ordinal sum of the two strict Archimedean copulas A , A with respect to some c ∈ ( , ) and, for ε > , let Aε be the strict Archimedean copula with the afore-mentioned generator φε. Then Proof. In order to simplify notation, within integrals we will (sometimes) set F x C (y) = K C (x, [ , y]) and  We start with the following segmentation of [ , ] and estimate the absolute value within the central horizontal and vertical ε-strips very roughly: The integrals I and II are identical to as in the considered squares Aε = C and therefore Kε = K C by Corollary 2.2. Regarding III, for every ( y − Aε c + ε , y dλ(y) ≤ ε.
Combining Theorem 3.6 and Lemma 3.8 and using induction on the number of segments shows the following result: Car is dense in (Ca, D ).
We conclude this section by answering two open problems posed in [6] regarding Baire category results for Archimedean copulas with respect to D . Following [6] [30]. In [6] the authors considered the metric spaces (C, d∞) and (C, D ) and derived several Baire category results for di erent subclasses of C. In particular, they showed that C s ar is co-meager in (Car, d∞) as well as that (Car, d∞) is co-meager in (Ca, d∞), and then conjectured that the same results should hold with respect to the stronger metric D . The proof of the rst conjecture is a direct consequence of results from [17] whereas denseness of Car in (Ca, D ) (as established above) is the key to con rm the second conjecture: (Car, D ).

Corollary 3.10. C s ar is co-meager in
Proof. Since in Car convergence with respect to d∞ is equivalent to D -convergence both metrics induce the same topology on Car. It therefore follows immediately that (Car, D ) contains the same co-meager subsets as (Car, d∞). In particular, C s ar is co-meager in (Car, D ).

Corollary 3.11.
Car is co-meager (hence of second category) in (Ca, D ).
Proof. As mentioned before, denseness of Car in (Ca, D ) Considering that Car is dense in Ca w.r.t. D , A k cannot contain any non-empty open ball of Ca, i.e. A k is nowhere dense in Ca. As direct consequence Ca \ Car is the union of nowhere dense subsets and the proof is complete.
In this sense (see [3]), with respect to D a typical associative copula is Archimedean and a typical Archimedean copula is strict.

Convergence in the class of associative copulas
In this section we prove the surprising result that within the class of associative copulas, convergence w.r.t. d∞ is equivalent to convergence w.r.t. D .
As already mentioned for C ∈ Ca there exist some nite or countably in nite index set I ⊆ N, {(a i , b i )} i∈I a collection of subintervals forming a 'partition' of [ , ] and a sequence of copulas (A i ) i∈I with A i ∈ Car or whenever (x, y) ∈ (a i , b i ) and C(x, y) = M(x, y) otherwise; see Section 3. It follows that a Markov kernel of C is given by Note that the level curves of associative copulas are convex.
In the sequel we always assume that our points of interest lie in some subinterval (a i , b i ) with A i ∈ Car and handle the subintervals of C containing M separately later on. Following [10] and de ning the set E C s,t for xed s, t ∈ [ , ] by holds and therefore the following formulas follow immediately: For s, t ∈ (a i , b i ) with t ≤ s we have Denoting by L t the t-level set we further have Moreover, setting s = b i immediately yields a representation for the Kendall distribution function for t ∈ (a i , b i ) which will be of use in the sequel. We now focus on convergence to M, or more generally C ∈ C cd , and show the following surprising result: A t * A(y, y) + B t * B(y, y) − · A t * B(y, y) dλ(y).
Considering that for C ∈ C cd the identity C t * C = M holds and that the star product is continuous in each argument with respect to d∞ (see [8]) it follows that From the rst part we already know that limn→∞ [ , ] C t n * Cn(s, s) dλ(s) = . Letting δn denote the diagonal of C t n * C and considering Lipschitz continuity of diagonals it follows that δn converges pointwise to the identity function id [ , ] . Letting B δn denote the Bertino copula with diagonal δn (see [11,12]) yields B δn ≤ C t n * Cn as B δn is the smallest symmetric copula with diagonal δn. Using the fact that δn → id [ , ] we have B δn → M for n → ∞. Hence considering C t n * Cn ≥ B δn it follows that C t n * Cn → M uniformly and the proof is complete.
The second assertion is surprising in so far that the * -product is not jointly continuous w.r.t. d∞ (but w.r.t. D ).
The following Corollary, which is immediate from Theorem 4.1, states the promised equivalence of d∞and D -convergence whenever the limiting associative copula is the Fréchet-Hoe ding upper bound M: Taking the absolute value and dividing by |D + φn(y)| we obtain As direct consequence of (4.2) we have limn→∞ D + φn(x) D + φn(y) = . Considering limn→∞ An(x, y) = y there exists some n ∈ N such that for all n ≥ n we have An(x, y) ≤ y + ε, hence Remark 4.4. As result of the above Lemma, in case of Archimedean copulas converging uniformly to M, we have weak conditional convergence whenever x > y in [ , ]. It remains an open question whether the result also holds above the diagonal (i.e. for x < y) which would be key for proving that within Ca we even have equivalence of d∞-convergence, D -convergence and weak conditional convergence.
As next step we consider a sequence (An) n∈N of Archimedean copulas uniformly converging to some associative copula C whereby we focus on studying convergence within Archimedean blocks of C. That is, we focus on (x, y) ∈ (a i , b i ) , for some i ∈ I, with corresponding copula A i ∈ Car in the 'partition' {[a i , b i ]} i∈I underlying our associative limit C. However, since it is not necessary that A i appears in the sequence (An) n∈N we deviate from this notation; subsequently we denote by (a * , e * ) the subinterval of interest corresponding to its Archimedean content A * with generator φ * in the ordinal sum representation of C.
Proof. It is well-known that limn→∞ d∞(An , C) = implies F K An (x) → F K C (x) pointwise in continuity points of F K C . Hence, (1) is a direct consequence of the before established fact that F K C (x) = T − * (F K A * (T * (x))) and (2) follows immediately. According to [13], an Archimedean generator φ ful lls and y ∈ ( , ). Therefore, where the exchange of limit and integral is justi ed in [4,Proposition 2]. Considering x > y, property (2) and using change of coordinates yields the case x < y follows similarly. This shows (4) and (5) is a consequence of properties (2) and (4).
We now show that for a sequence of Archimedean copulas converging to an associative limit C, within the Archimedean blocks of C we have weak convergence of the corresponding Markov kernels on a dense set above the level curve f a * of C. Notice that f a * (x) = T − * (f * (T * (x))), x ∈ (a * , b * ) is the a * -level curve of C. Hence the following Theorem speci cally includes the case of strict Archimedean blocks in C whereas nonstrict blocks are tackled in Theorem 4.9.
Theorem 4.6. Let A , A , . . . be Archimedean copulas with generators φ , φ , . . . and let C be an associative copula such that (An) n∈N converges uniformly to C. Then for every subinterval (a * , b * ) of C containing an Archimedean copula A * ∈ Car there exists some Λ ∈ B((a * , b * )) of full measure and for x ∈ Λ some set Ux ⊆ (f a * (x), b * ) which is dense in (f a * (x), b * ) such that for every x ∈ Λ and y ∈ Ux we have = φ * (T * (x)).
Using the afore-mentioned results we can now nally prove the equivalence of convergence with respect to d∞ and convergence with respect to D within the class of associative copulas: Theorem 4.9. Let C, C , C , . . . be associative copulas. If the sequence (Cn) n∈N converges uniformly to C then the sequence even converges with respect to D .
Proof. As proved in Section 3 the class C s ar of strict Archimedean copulas is dense in (Ca, D ). Consequently, for every Cn there exists some An ∈ C s ar such that D (Cn , An) < n . Thus, and it su ces to show that D (An , C) → for n → ∞ which can be done as follows: According to [6, Lemma 3] d∞(An , C) ≤ · D (An , Cn)+ d∞(Cn , C) and the latter converges to for n → ∞ whence (An) n∈N converges uniformly to C. Write C = ( a i , b i , B i ) i∈I for some nite or countably in nite index set I. Fix ε > then as in Lemma 3.2 we can choose a nite set J ⊆ I such that j∈J λ((a j , b j )) ≥ − ε and consider the following partition of [ , ] (as before we set F x n (y) = Kn(x, [ , y]) = K An (x, [ , y]) and µn = µ An , similarly for the limit C): Clearly II ≤ ε; for calculating III divide the integration area in the sets L and U denoting the lower and upper part, respectively, i.e. U = i∈I (a i , b i ) × (b i , ) and L = i∈I (a i , b i ) × ( , a i ) (as depicted in Figure 5). Using disintegration we get and as U is a µ C -continuity set it follows that limn→∞ µn(U) = µ C (U) = . The lower area L follows analogously whence there is n ∈ N such that III ≤ ε for all n ≥ n . It remains to show that the same holds true for I: We split the index set J into J denoting the subset of indices Observe that now we are in the setting of Theorem 4.6 and hence also have convergence to for n → ∞. To calculate I.II we proceed similarly, x a subinterval (ā * ,b * ) := (a j , b j ) for some j ∈ J of C containing a copy of M and consider for n → ∞ (again using Dominated Convergence). It directly follows that there exists n ∈ N such that for all n ≥ n we have I ≤ ε. Altogether, there exists n ∈ N such that for all n ≥ n = max(n , n ) we have D (An , C) = I + II + III ≤ · ε, which completes the proof.