Homogenization of nonconvex unbounded singular integrals

We study periodic homogenization by Γ-convergence of integral functionals with integrands W ( x, ξ ) having no polynomial growth and which are both not necessarily continuous with respect to the space variable and not necessarily convex with respect to the matrix variable. This allows to deal with homogenization of composite hyperelastic materials consisting of two or more periodic components whose the energy densities tend to inﬁnity as the volume of matter tends to zero, i.e., W ( x, ξ ) = P j ∈ J 1 V j ( x ) H j ( ξ ) where { V j } j ∈ J is a ﬁnite family of open disjoint subsets of R N , with | ∂V j | = 0 for all j ∈ J and | R N \ S j ∈ J V j | = 0, and, for each j ∈ J , H j ( ξ ) → ∞ as det ξ → 0. In fact, our results apply to integrands of type W ( x, ξ ) = a ( x ) H ( ξ ) when H ( ξ ) → ∞ as det ξ → 0 and a ∈ L ∞ ( R N ; [0 , ∞ [) is 1-periodic and is either continuous almost everywhere or not continuous. When a is not continuous, we obtain a density homogenization formula which is a priori diﬀerent from the classical one by Braides–Müller. Although applications to hyperelasticity are limited due to the fact that our framework is not consistent with the constraint of noninterpenetration of the matter, our results can be of technical interest to analysis of homogenization of integral functionals.


Introduction
In this paper we prove homogenization theorems (see Theorems 2.8, 2.19 and 2.33) in the sense of De Giorgi's Γ-convergence (see Definition 2.1) for functionals of typeˆΩ variable and not necessarily convex with respect to the second variable.
Our homogenization results can be summarized as follows (see §1.1 for details on the classes of integrands).
Theorem. If W ∈ I p per ∪ J p per ∪ K p per then (1.1) Γ-converges as ε → 0 to the homogenized functional Ω W hom (∇u(x)) dx .
If W ∈ I p per ∪ J p per then W hom is given by the classical density homogenization formula of Braides-Müller, i.e., W hom (ξ) = HW (ξ) where Y := ]− 1 2 , 1 2 [ N and W 1,p 0 (kY ; R m ) denotes the space of p-Sobolev functions from kY to R m which are null on the boundary of kY .
If W ∈ K p per then W hom is given by a priori different formula from the classical one, i.e., W (y, ξ + ∇ϕ(y)) dy : ϕ ∈ Aff 0 (Q ρ (x); R m ) , where Q ρ (x) := x + ρY and Aff 0 (Q ρ (x); R m ) denotes the space of continuous piecewise affine functions from Q ρ (x) to R m which are null on the boundary of Q ρ (x).
The distinguishing feature of our homogenization results is that they can be applied with integrands W : R N × M N ×N → [0, ∞] having a singular behavior of the type i.e., when W (x, · ) is compatible with one of the two basic facts of hyperelasticity, namely the necessity of an infinite amount of energy to compress a finite volume into zero volume (see Corollaries 2.13, 2.17, 2.22 and 2.36). However, our results are not consistent with the noninterpenetration of the matter.
The plan of the paper is as follows. In the next section we state our main results, see Theorems 2.8, 2. 19  2) and new relaxation theorems (see Theorems 3.8 and 3.15) whose statements and proofs are given in Section 3. In the appendix, we recall some standard and less standard results on relaxation of singular integrands (see §A. 1

Notation, hypotheses and classes of integrands
Troughout the paper, the symbol ffl stands for the mean value integral, i.e., Several general hypotheses are stated troughout the paper. For the convenience of the reader we summarize it below.
Other more particular hypotheses are stated troughout the paper. For the convenience of the reader, we list the main ones below.
(P) for every bounded open subset U of R N with |∂U | = 0 and every δ ∈ ]0, δ 0 ] with δ 0 > 0 small enough, there exists a compact K δ ⊂ U such that such that W is defined by Several different classes of integrands are defined troughout the paper (see Definitions 2.6, 2.9, 2.10, 2.14, 2.18, 2.20, 2.25, 2.34, 2.40). For the convenience of the reader, we summarize it below. The Borel measurable functions W belonging to I p or J p (and so to I p per or I p per ) are continuous almost everywhere with respect to the space variable (see Lemma 2.5). We consider subclasses S 1 (see Definition 2.10) and S 2 (see Definition 2.14) of I

Main results
Consider the family of integral functionals I ε : is a Borel measurable function, where M m×N denotes the space of real m × N matrices with m, N ≥ 1 two integers, which satisfies the following two assumptions: and some C > 0; (A 1 ) W is 1-periodic, i.e., for every ξ ∈ M m×N and every i ∈ {1, . . . , N }, In [14] (see also [16,Theorem 14.5]) Braides (and independently Müller in [24]) proved the following homogenization theorem (see Theorem 2.2) in the sense of De Giorgi's Γ-convergence whose definition is given below. Definition 2.1. We say that I ε Γ-converges to I hom : W 1,p (Ω; R m ) → [0, ∞] with respect to the L p (Ω; R m )-convergence as ε → 0, and we write Γ-lim ε→0 I ε = I hom , if (For more details on the concept of Γ-convergence, we refer the reader to [20,16,15].) The interest of Theorem 2.2 is to establish a suitable variational framework to deal with nonconvex homogenization problems in the vectorial case: it is the point of departure of many works on the subject related to hyperelasticity. However, because of the p-growth assumption on the integrand W , Theorem 2.2 is not consistent with (1.2).
In the present paper we establish new homogenization results (see §2.1, §2.2 and §2.3) which are consistent with (1.2). (For other works on homogenization related to hyperelasticity we refer the reader to [21,6,7,10] and the references therein.)

Homogenization with singular integrands which are continuous almost everywhere with respect to the space variable
In [2] it was proved the following homogenization theorem whose distinguishing feature is to be consistent with (1.2) even though it is not consistent with the noninterpenetration of the matter, see [2, §4] for more details.
However, since the condition (A 2 ) implies the continuity of W with respect to its first variable, Theorem 2.3 cannot be applied with W of the form where γ 1 , γ 2 > 0 and E 1 is a On the other hand, we have Lemma 2.5. If (P) is verified then λ is continuous almost everywhere.
Proof. Arguing by induction, it is easily seen that from (P) we can deduce that there exists a disjointed sequence {K n } n≥1 of compact subsets of R N such that K i for all n ≥ 2 , and for every n ≥ 1, where δ 0 > 0 is given by (P) and B n denotes the open ball in R N centered at the origin and of radius n. It is sufficient to prove that Using the second equality in (2.6) and the fact that by using the first inequality in (2.6). Consequently lim n→∞ |B n \ ∞ i=1 K i | = 0, which gives (2.7). Definition 2.6. We denote by I p the class of Borel measurable functions W : R N × M m×N → [0, ∞] satisfying (A 0 ), (A 3 ) and the following additional condition: Let us set I p per := W ∈ I p : W satisfies (A 1 ) .
Remark 2.7. The following are elementary properties whose the proofs are left to the reader.
Hence, if (A 4 ) holds and if λ| K is continuous for K ⊂ R N then ZW ( · , ξ)| K is continuous for all ξ ∈ M m×N and dom ZW ( (e) As a consequence of (c) and (d), if W satisfies (A 4 ) then W ( · , ξ) and ZW ( · , ξ) are continuous almost everywhere for all ξ ∈ M m×N because λ is continuous almost everywhere by Lemma 2.5.
Here is the first homogenization result of the paper.
A typical example of a function belonging to the class H is given by where h : R → [0, ∞] is a measurable function for which there exist δ, δ > 0 such that h(t) ≤ δ for all |t| ≥ δ. For example, given s > 0 and T ≥ 0 (possibly very large), this latter condition is satisfied with δ = 2T and δ = max Definition 2.10. We denote by S 1 the class of 1-periodic and Borel measurable functions W : If W ∈ S 1 then it is compatible with (1.2) and can be as in (2.5). In fact, we have Then ZW has p-growth, i.e., W satisfies the condition (A 3 ).
The following result is a direct consequence of Theorem 2.8 (which will be proved in Section 4) and Lemma 2.11. Another application of Theorem 2.8 can be obtained by introducing the following class of integrands. Definition 2.14. We denote by S 2 the class of 1-periodic and Borel measurable functions W : such that W is defined by Proof of (A 3 ). The condition (A 3 ) follows from the following Theorem 2.12. Indeed, since H 1 , H 2 ∈ H p , for i = 1, 2 there exist α i , β i > 0 such that for every ξ ∈ M N ×N , if |det ξ| ≥ α i then H i (ξ) ≤ β i (1 + |ξ| p ). Setting α := min{α 1 , α 2 } and β := max{β 1 , β 2 }, it is then clear that W satisfies ( H).
Proof of (A 4 ). We are going to prove that (A 4 ) is verified with λ = (γ − 1)1 E 1 where γ > 1 is given by (2.9). (Clearly (γ − 1)1 E 1 ∈ L, see Remark 2.4.) Fix x 1 , x 2 ∈ R N and ξ ∈ M N ×N . By definition of S p per we have dom H 1 = dom H 2 (see Remark 2.15) and so dom W (x 1 , · ) = dom W (x 2 , · ). Hence, without loss of generality we can assume that Then, it is easy to see that which proves the condition (A 4 ).
As a direct consequence of Theorem 2.8 and Lemma 2.16 we have the following result.

Homogenization with a sum of singular integrands
We introduce the following class of integrands.
Here is the second homogenization result of the paper.
The following lemma makes clear the link between S 3 and J p per .
Then, by definition, W is given by (2.11) with H j ∈ H for all j ∈ J. Firstly, (A 0 ) and (A 1 ) are clearly verified. Secondly, since every H j belongs to H, for each j ∈ J there exists α j , β j > 0 such that for every ξ ∈ M N ×N , if |det ξ| ≥ α j then H j (ξ) ≤ β j (1 + |ξ| p ). Setting α := min{α j : j ∈ J} and β := max{β j : j ∈ J}, it is then clear that W satisfies ( H) and (A 3 ) follows from Theorem 2.12.
As a direct consequence of Theorem 2.19 and Lemma 2.21 we have the following result.

Homogenization with singular integrands which are not continuous with respect to the space variable
Consequently, in such a case we have GL = ZL (see also Remark 2.37).
Definition 2.25. We denote by K p per the class of Borel measurable func- and the following additional conditions: Remark 2.26. If (A 5 ) holds then GW has p-growth.
Borel measurable, then such a W satisfies (A 6 ).

Remark 2.29. One always has H[GW ] ≥ H[GW ]
. On the other hand, we have: Proof of Lemma 2.30 Fix any k ≥ 1 and any ϕ ∈ W 1,p 0 (kY ; R m ). As Aff 0 (kY ; R m ) is strongly dense in W 1,p 0 (kY ; R m ) we can assert that there exists {ϕ n } n ⊂ Aff 0 (kY ; R m ) such that: But, by using (2.12) and (2.13) we see that: From (2.14) and (2.15) it follows that for all k ≥ 1 and all ϕ ∈ W 1,p 0 (kY ; R m ) , which gives the result. for all bounded open set U ⊂ R N . Without loss of generality we can also assume that a is lsc at x. Let ξ ∈ M m×N and let {ξ n } n ⊂ M m×N be such that |ξ n − ξ| → 0. We have to prove that and multiplying by a(x) we obtain and, recalling that ZH(ξ) < ∞, it follows that As I is finite we can assert that there exists n δ ≥ 1 such that But H is usc and |ξ n − ξ| → 0, hence θ i ≤ H(ξ + ζ i ) for all i ∈ I, and so and taking (2.21) into account we get Thus, by using (2.23), we deduce that for all ρ ∈ ]0, ρ δ [ and all n ≥ n δ . Taking (2.16) into account, by letting ρ → 0 and then n → ∞ we conclude that and (2.17) follows by letting δ → 0.
Here is the third homogenization result of the paper.
Here is the link between S 4 and K p per . Lemma 2.35. Let m = N . The class S 4 is a subclass of K p per , i.e., S 4 ⊂ K p per . Proof. Let W ∈ S 4 . Then, by definition, W is given by (2.24) with H ∈ H usc and a ∈ L ∞ (R N ; [0, ∞[) which is lsc and 1-periodic and such that a(·) ≥ η > 0. It is thus clear that (A 0 ) and (A 1 ) are verified. So it remains to prove that (A 5 ), (A 6 ) and (A 7 ) hold. Firstly, since H ∈ H usc ⊂ H, by Theorem A.4 we deduce that ZH has p-growth.
Secondly, by Remark 2.28 we can assert that (A 6 ) is satisfied. Finally, GW has p-growth because (A 5 ) is verified and, since a is lsc, H is usc and ZH is finite, we can assert that GW (x, · ) is usc for a.a. x ∈ R N , see Remark Braides-Müller. To make clear the link between these two formulas, we begin with the following proposition whose proof is given below.

Theorem 3.3. If W is finite then
The following result is a slight generalization of Theorem 3.3 (see [2]).
As E and E 0 are not given by explicit formulas, it is of interest to know under which conditions on W we have: ∞] whose we wish to give a representation formula. When W has p-growth, such integral representation problems was studied by Dacorogna (see [17,Theorem 5], see also [18,Theorem 9.1]) and Acerbi and Fusco (see [1,Statement III.7]) who proved the following theorem.

Relaxation with singular integrands which are continuous almost everywhere with respect to the space variable
In [2] it was proved the following relaxation theorem whose distinguishing feature is to be consistent with (1.2). Theorem 3.7 was used in [2] to establish Theorem 2.3. However, due to the assumption (A 2 ), in Theorem 3.7 the integrand W is necessarily continuous with respect to its first variable, and so this latter theorem cannot be used to prove Theorem 2.8. The following relaxation theorem improves Theorem 3.7 by allowing to the integrand W not to be necessarily continuous with respect to its first variable and will play an essential role in the proof of Theorem 2.8. and let ZE, ZE 0 : W 1,p (U ; R m ) → [0, ∞] be given by: We need the following lemma whose proof is given below.
In particular, from (3.8) and Remark 2.7 (d) we see that ZW (x, ξ i ) < ∞ for all i ∈ I and all x ∈ R N . Let λ ∈ L be given by (A 4 ). Then, for each i ∈ I and each δ ∈ ]0, δ 0 ] with δ 0 > 0 small enough, there exists a compact Fix any δ ∈ ]0, δ 0 ]. By Remark 2.7 (d) we see that for every ) . Hence, for each i ∈ I and each k ≥ 1, there exists a finite family On the other hand, as for every i ∈ I, λ| K i,δ is continuous with I finite and K i,δ compact, we deduce that there exists η > 0 such that for every i ∈ I, if x, y ∈ K i,δ and |x − y| < η then |λ(x) − λ(y)| < δ . (3.12) Fix any k > 1 η . As ZW (x, ξ i ) < ∞ for all i ∈ I and all x ∈ R N , for each (3.14) For every n ≥ 1, from Vitali's covering theorem we can assert that: • there exists a finite or countable family Since |∂K i,δ | = 0 for all i ∈ I, we can define {ψ n } n≥1 ⊂ W 1,∞ 0 (U ; R m ) by It is then easy to see that for all n ≥ 1, and so ψ n → 0 in L ∞ (U ; R m ). Thus {φ + ψ n } n≥1 ⊂ W 1,p (U ; R m ) (resp. {φ + ψ n } n≥1 ⊂ W 1,p 0 (U ; R m )) and φ + ψ n → φ in L p (U ; R m ). Hence, to prove (3.6) it is sufficient to show that for every n ≥ 1,ˆU W (x, ∇φ(x) + ∇ψ n (x)) dx ≤ˆU ZW (x, ∇φ(x)) dx . (3.16) Let n ≥ 1. Using the fact that |∂K i,δ | = 0 for all i ∈ I we see that Using (A 4 ) we see that for every i ∈ I, for all x ∈ R N , and taking (3.15) and (3.14) into account we deduce that .
Using again (A 4 ) we can assert that for every i ∈ I, every j ∈ J k i and every x ∈ U k i,j , .
Remark 3.10. By analysing the proof of Lemma 3.9 we see that this lemma is also valid with " ZW ", defined in (2.25), instead of "ZW ". Thus, by the same method we can also establish the following analogue of Theorem 3.8.
By the same reasoning, in replacing "W 1,∞ 0 " by "Aff 0 " and "ZW " by " ZW " defined in (2.25), we can also prove the following result.

Relaxation with a sum of singular integrands
The following relaxation theorem is a variant of Theorem 3.8.
resp. E 0 (φ) ≤ˆU ZW (x, ∇φ(x)) dx . (3.21) Proof. Let φ ∈ Aff(U ; R m ) (resp. φ ∈ Aff 0 (U ; R m )). By definition, there exists a finite family {U i } i∈I of open disjoint subsets of U such that |∂U i | = 0 for all i ∈ I, |U \ i∈I U i | = 0 and, for every By assumption, there exist a finite family {V j } j∈J of open disjoint subsets of R N , with |∂V j | = 0 for all j ∈ J and |R N \ j∈J V j | = 0, and a finite with U i,j := U i ∩ V j . Fix any δ > 0. Given any i ∈ I and any j ∈ J we For every n ≥ 1, by Vitali's covering theorem, there exists a finite or countable family It is then easy to see that for all n ≥ 1, and so ψ n → 0 in L ∞ (U ; R m ). Thus {φ + ψ n } n≥1 ⊂ W 1,p (U ; R m ) (resp. {φ + ψ n } n≥1 ⊂ W 1,p 0 (U ; R m )) and φ + ψ n → φ in It follows that and (3.21) follows by letting δ → 0.

Singular integrands which are continuous almost everywhere with respect to the space variable
In this section we prove Theorem 2.8 by following the same lines as in the proof of [2, Theorem 3.4]. We will need Theorems 3.8 and 2.2 and the following classical property of the Γ-convergence.
Proposition 4.1. The Γ-limit is stable by substituting I ε by its relaxed functional I ε , i.e., where, for each ε > 0, I ε : W 1,p (Ω; R m ) → [0, ∞] is given by Proof of Theorem 2.8. By Proposition 4.1 it suffices to prove Theorem 2.8 with "I ε " instead of "I ε ". Fix any ε > 0 and consider W ε : ξ). As W ∈ I p per and ZW ε (x, ξ) = ZW ( x ε , ξ) for all (x, ξ) ∈ R N × M m×N it is easy to see that W ε ∈ I p . Applying Theorem 3.8 to W ε we deduce that for every ε > 0, where ZW is clearly p-coercive, 1-periodic and has p-growth. From Braides-Müller's homogenization theorem (see Theorem 2.2) it follows that I hom = Γ-lim ε→0 I ε with I hom defined by (2.2) and W hom : Fix any k ≥ 1, any ξ ∈ M m×N and consider W ξ : R N × M m×N → [0, ∞] given by W ξ (x, ζ) := W (x, ξ + ζ). As W ∈ I p per and ZW ξ (x, ζ) = ZW (x, ξ + ζ) for all (x, ζ) ∈ R N × M m×N it is easy to see that W ξ ∈ I p . Applying again Theorem 3.8 to W ξ with U = kY we see that (3.2) holds with W = ZW ξ . Consequently, for every k ≥ 1 and every ξ ∈ M m×N , we have and the proof of Theorem 2.8 is complete. In the same way, by using Theorem 3.11 (see Remark 3.10) instead of Theorem 3.8, we can also establish the following result.

Sum of singular integrands
In this section we prove Theorem 2.33 by using Theorems 3.15 and 2.2 and Proposition 4.1.
Proof of Theorem 2.19. It is the same proof than the one of Theorem 2.8 where we replace "Theorem 3.8" by "Theorem 3.15", "I p " by "J p " and "I p per " by "J p per ".

Singular integrands which are not continuous with respect to the space variable
In this section we prove Theorem 2.33.
Proof of (4. The set function m * u is called the Vitali envelope of m u , see §A.4 for more details. First of all, it is easy to see that m u is subadditive. On the other hand, as u ∈ Aff(Ω; R m ), there exists a finite family {U j } j∈J of open disjoint subsets of Ω such that |∂U j | = 0 for all j ∈ J, |Ω \ j∈J U j | = 0 and, for every j ∈ J, ∇u ≡ ξ j in U j with ξ j ∈ M m×N . Hence, given any A ∈ O(Ω), we have |A \ j∈J (A ∩ U j )| = 0, and so, by subadditivity of m u , it follows that m u (A) Using (A 5 ) we see that for all A ∈ O(Ω). But, given any x ∈ Ω such that lim ρ→0 |Qρ(x)| exists and x ∈ j∈J U j , we have x ∈ U j 0 for some j 0 ∈ J and Q ρ 0 (x) ⊂ U j 0 for all ρ ∈ ]0, ρ 0 [ and some ρ 0 > 0, and so, for each ρ ∈ ]0, ρ 0 [, ∇u ≡ ξ j 0 in Q ρ (x). Hence, by a change of variable, we see that for every ρ ∈ ]0, ρ 0 [, for a.a. x ∈ Ω. From (4.15) we deduce that m * u (Ω) = GI ε (u). (So, in particular, we have m * u (Ω) < ∞ because GW has p-growth.) Thus, to establish (4.2) it remains to prove that (4.17) (Ω; R m ). From (4.17) and (4.18) we see that On the other hand, we have But, since diam(Q i ) ∈ ]0, δ[ for all i ∈ I, by using Poincaré's inequality we deduce that there exists K > 0, which depends only on p and N , such that for every i ∈ I, and so, taking (4.20) into account, we get Using (A 0 ) and (4.19), from (4.21) we deduce that which shows that ψ δ → u in L p (Ω; R m ) because lim δ→0 m δ u (Ω) = m * u (Ω) < ∞, and (4.15) follows from (4.19) by letting δ → 0 (and by noticing that I ε (u) ≤ lim δ→0´Ω W y ε , ∇ψ δ (y) dy). This completes the proof of Theorem 2.33.

Proposition A.5. Given ξ ∈ M m×N and a bounded open set
Fix any n ≥ 1 and k ≥ 1. By Vitali's covering theorem there exists a finite or countable family {a i + α i Y } i∈I of disjoint subsets of A, where a i ∈ R N and 0 < α i < 1 k , such that |A \ i∈I (a i + α i Y )| = 0 (and so i∈I α N i = |A|). Define ϕ n,k ∈ Aff 0 (A; R m ) by Clearly ϕ n,k L ∞ (A;R m ) = 1 k ϕ n L ∞ (Y ;R m ) , hence lim Proof. Given ξ ∈ M m×N there exists {k n ;φ n } n such that: For each n ≥ 1 and ε > 0, denote the k n Y -periodic extension ofφ n by ϕ n , consider A n,ε ⊂ A given by On the other hand, for every n ≥ 1 and every ε > 0, we havê and consequently by (A.4). As lim ε→0 |A \ A n,ε | = 0 for any n ≥ 1, and by using (A.5), we see that:
Finally, to conclude we prove that (A.14) and (A.15) imply θ * = 0. For this, we are going to prove the following two assertions: if d − θ − ≤ 0 then θ * ≤ 0 ; (A. 16) under (A.14), if d − θ − ≥ 0 then θ * ≥ 0 . (A.17) Proof of (A.16). Fix A ∈ O(Ω). Fix any δ > 0. Then d − θ − < δ, and so in particular lim ρ→0 d − θ − (x, ρ) < δ for all x ∈ A. Hence, for each x ∈ A there exists {ρ x,n } n ⊂ ]0, δ[ with ρ x,n → 0 as n → ∞ such that d − θ − (x, ρ x,n ) < δ for all n ≥ 1. Taking Remark A.14 into account, it follows that for each x ∈ A and each n ≥ 1 there is Q x,n ∈ Cub(A, x, ρ x,n ) such that for each x ∈ A and each n ≥ 1, Moreover, since diam Q x,n = diam(Q x,n ) ≤ ρ x,n for all x ∈ A and all n ≥ 1, we have inf diam Q x,n : n ≥ 1 = 0 (where Q x,n denotes the closed cube corresponding to the open cube Q x,n ). Let F 0 be the family of closed cubes of Ω given by F 0 := Q x,n : x ∈ A and n ≥ 1 .
Proof of (A.17). Fix A ∈ O(Ω). By Egorov's theorem, there exists a sequence {B n } n of Borel subsets of A such that: and, choosing x i ∈ Q i ∩ B n for each i ∈ I n and noticing that i∈I\In Q i ⊂ A \ B n , it follows that Taking (A.23) into account, we conclude that for all δ > 0 and all n ≥ 1, which gives θ * (A) ≥ 0 by letting δ → 0 and using (A.22) and then by letting n → ∞ and using (A.21).
As λ − and λ + are absolutely continuous with respect to the Lebesgue measure, it is easy to see that: Hence Θ * = λ − and Θ * = λ + by Lemma A.17.
We only need to prove that Θ Fix any n ≥ 1. As Θ is subadditive, see the assumption (b), we have Thus, using the assumption (a), we get But, ν(Q i \ Q i ) = 0 for all i ∈ I n because ν is absolutely with respect to the Lebesgue measure, hence Step 3: end of the proof.