Discrete spin systems on random lattices at the bulk scaling

We study by Γ-convergence the stochastic homogenization of discrete energies on a class of random lattices as the lattice spacing vanishes. We consider general bounded spin systems at the bulk scaling and prove a homogenization result for stationary lattices. In the ergodic case we obtain a deterministic limit.


1.
Introduction. In this paper we study the discrete-to-continuum limit of a system of bounded spins on stochastic lattices at the bulk scaling. In particular we focus on the properties of these discrete systems at zero temperature where the abstract methods of Γ-convergence have already proved to be an effective tool for the asymptotic analysis (see [2], [3], [4], [5], [10], [11]). More specifically we consider systems parameterized over the points of a lattice L ⊂ (R d ) Z d that we see as the outcome of a random variable on the probability space (Ω, F, P), namely ω → L(ω) = {L(ω)(i)} i∈Z d . On the geometry of the lattices we make two assumptions preventing arbitrarily big empty regions and cluster of points. We then scale the lattices by a small parameter ε that will be eventually sent to 0 in the continuum limit. On these admissible lattices we define a spin field u : εL(ω) → K where K ⊂ R m is a bounded set and consider non negative energies of the form F ε (u) = εx,εy∈D ε d f ε (x, y, u(εx), u(εy)). ( Under boundedness and long-range decay assumptions on the densities f ε we prove a compactness and integral representation result for a Γ-limit of F ε . In this analysis the assumptions on the geometry of the lattice play an important role. In particular they allow us to relate our spin systems to systems on a periodic lattice and to take advantage of the results in [5] to show that the limit functional fulfills the assumption of an integral representation theorem proved in [14]. As a result the compactness and integral representation Theorem 3.2 follows. In Theorem 3.4 we extend the proven Γ-convergence result to the case when the spin field is constrained to realize a certain prescribed mean value. As a corollary we obtain the convergence of minimum problems under mean value constraints which turns out to be a key element in order to prove a stochastic homogenization theorem for spin systems on stationary random lattices. It is at this point that we need to combine the abstract methods of Γ-convergence with the subadditive ergodic theorem in [1].

MARCO CICALESE AND MATTHIAS RUF
This argument has been first used in the framework of stochastic homogenization in [18] and in the context of discrete-to-continuum limits under bulk scaling in [6] for functionals defined on Sobolev spaces. In this paper we apply it in the context of Lebesgue spaces. Even if in this framework the proofs benefit from the analysis done in [5], [6] and [18], a number of technical issues deserve some care (see for instance the measurability Lemma A.1). Finally, it is worth noticing that in the simple case K = {±1} and f ε (x, y, u, v) = c ε (x, y)|u − v|, our results fill a gap in the asymptotic theory of Ising systems on stochastic networks. Indeed, for this choice of K and f ε , the main result in this paper together with that in [7], where the surface scaling (ε d−1 in place of ε d ) of the energy functional in (1) is considered, can be seen as the first two terms in the development by Γ-convergence (see [13]) of long-range Ising systems. As a final remark, it is worth noticing that, the results contained in this paper are the first necessary step for the extension to the stochastic setting of the discrete-to-continuum limits of frustrated spin systems recently considered in [16] and [15].
2. Stochastic lattices and discrete energies. In this section we introduce the stochastic framework we will use in the rest of the paper.
2.1. Basic notation. Given x ∈ R d we denote by |x| the Euclidean norm of x. If B ⊂ R d is a Borel set we denote by |B| its Lebesgue measure. Given an open set D ⊂ R d we denote by B(D) the Borel σ-algebra on D, by A(D) the class of all bounded open subsets of D and by A R (D) the family of those sets in A(D) such that |∂A| = 0. We denote by dim H (·) the Hausdorff dimension. The triple (Ω, F, P) denotes a fixed probability space Ω with a complete σ-algebra F and a probability measure P. In the proofs C denotes a generic constant that may change from line to line.

2.2.
Random networks. The random lattices in terms of which we will define our discrete systems need to satisfy some geometric properties. These are listed below and characterize what we call admissible sets of points.
To an admissible set of points we associate the Voronoi tessellation (C(x)) x∈Σ defined by Two points in Σ are said to be nearest neighbors if the Voronoi cells having them as centers share a (d − 1)-dimensional edge. Hence we define the set of nearest neighbors of Σ as The following two properties are proved in [7].
Lemma 2.2 (geometric properties of admissible sets). Let Σ be an admissible set of points with constants r, R as in Definition 2.1. Then there exists a constant M > 0 depending only on r and R such that, for all x ∈ Σ, By stochastic lattices we mean a random variable realizing almost surely sets which are admissible with respect to the same constants r, R in Definition 2.1. To describe the stochastic properties of these lattices we need some additional definition. Definition 2.3 (group action). Let (τ z ) z∈Z d , τ z : Ω → Ω, be an additive group action on Ω. We say that it is measure preserving if Let us specify the assumptions on the random variable generating the network.
is admissible in the sense of Definition 2.1 and the constants r, R can be chosen independent of ω P-almost surely. The stochastic lattice L is said to be stationary if there exists a measure preserving group action (τ z ) z∈Z d on Ω such that, for P-almost every ω ∈ Ω, If in addition (τ z ) z∈Z d is ergodic, then L is called ergodic.
Given a realization L(ω) of a stochastic lattice, we denote by N N (ω) the corresponding nearest neighbors. Remark 1. We stress that our geometric assumptions on the random lattice rule out many examples of stationary point processes used in stochastic geometry, e.g. homogeneous Poisson point processes.
and we introduce the notion of discrete subadditive stochastic process. Definition 2.5. A function µ : I → L 1 (Ω) is said to be a discrete subadditive stochastic process if the following properties hold P-almost surely: (i) for every I ∈ I and for every finite partition (I k ) k∈K ⊂ I of I we have µ(I, ω) ≤ k∈K µ(I k , ω).
Our homogenization results are based on the following pointwise ergodic theorem (see [1]), which has already been used in the pioneering paper [18] in the context of stochastic homogenization of elliptic integral functionals and in [6] in the case of discrete hyperelastic systems. Theorem 2.6. Let µ : I → L 1 (Ω) be a discrete subadditive stochastic process and let I ∈ I. If µ is stationary with respect to a measure preserving group action (τ z ) z∈Z d ; i.e., ∀I ∈ I, ∀z ∈ Z d : µ(I + z, ω) = µ(I, τ z ω) almost surely, then there exists Φ : Ω → R such that, for P-almost every ω, lim k→+∞ µ(kI, ω) |kI| = Φ(ω).

2.3.
Discrete spin-type energies. Having settled the stochastic environment, we are now able to define our discrete spin energies on these stochastic lattices. We restrict ourselves to pairwise interaction energies, but it is worth noticing that the core of these results can be (modulo details) generalized to multi-body interactions as well. We let ε > 0 be a small parameter (describing the average distance between particles), D ∈ A R (R d ) be a regular reference domain and let L(ω) be admissible. The energies we have in mind are defined on functions u : εL(ω) → R m . We consider u as generalized spin fields parameterized over the ε lattice εL(ω), with ε eventually going to zero in what is usually referred to as a discrete-to-continuum limit. As our proofs rely on an integral representation result, as usual in this setting we need to define a localized version of the energies. Given A ∈ A R (D), we consider where the nearest neighbor interactions and long-range interactions are respectively given by functionals of the form Note that the random character of our energies depends on the underlying random geometry of the lattice but not in the interaction coefficients f ε,nn , f ε,lr , which are instead considered deterministic.
As usual in the discrete-to-continuum analysis, we embed the problems in a suitable Lebesgue space (depending on the scalings). In general, note that we can identify each function u : εL(ω) → R m with its piecewise constant interpolation on the Voronoi cells of εL(ω). Therefore, given a bounded set K ⊂ R m let us introduce the class Since the intersection of two Voronoi cells has zero Lebesgue-measure, this class is well defined and C ε (ω, K) ⊂ L ∞ (D). As a result we may define F ε (ω)(u, A) : Following some ideas in [6], we now use the square lattice r √ d Z d to conveniently rewrite the energies F ε .
For fixed ξ ∈ r Z d we can now rewrite the energy contribution using 3. The bulk scaling. As pointed out in the introduction this paper deals with the case, where f ε,nn and f ε,lr are of order 1 with respect to ε. In particular, in contrast to [6], there is no gradient structure in the energy. As a result, the variational limit of the energies F ε (·)(ω) is given by volume integrals of the form We divide this section into three paragraphs. At first we show how to derive an integral representation result for the variational limit. Then we prove the convergence of minimum problems under mean-value constraints. Finally we use the first two results to prove a stochastic homogenization theorem under the assumption of stationarity of the random lattices. In this paragraph, for a fixed bounded set K we consider non negative interaction energies fulfilling the following assumptions: Hypothesis 1. There exist C > 0 and a decreasing function J lr : such that The next proposition, whose simple proof we omit, suggests to conveniently embed C ε (ω, K) ⊂ L ∞ (D, co(K)) and equip L ∞ (D, co(K)) with the weak*-topology so that L ∞ (D, co(K)) is a separable metric space.
3.1. Integral representation. As in [5] the key ingredient is the following integral representation theorem (see [14]).
which is absolutely continuous with respect to the d-dimensional Lebesgue measure, (iv) for all B ∈ B(D) the functional F (·, B) is lower semicontinuous with respect to the strong (respectively weak) convergence of L p (D, R m ). Then there exists a positive measurable function f : D × R m → [0, +∞], with the property that f (x, ·) is lower semicontinuous (respectively convex and lower semicontinuous) for a.e. x ∈ D, such that For u ∈ L ∞ (D, co(K)), A ∈ A R (D) let us introduce i.e. the Γ-lim inf and the Γ-lim sup of the energies F ε (·, A)(ω) with respect to the weak*-topology on L ∞ (D, co(K)) (we refer to [9] for an introduction to Γconvergence). With the help of the auxiliary square lattice introduced in Section 2.3, one can use the same technique developed in [6] and repeat the arguments in [5] to prove the following proposition. We provide a sketch of its proof for readers' convenience.
Proof. Given ξ ∈ r Z d , we letξ ∈ ξ + [−r , r ] d be such that (i): Fix u ∈ L ∞ (D, co(K)). Since the discrete energy is non-negative, the lower bound is trivial. For the upper bound, let u ε ∈ C ε (ω, K) be such that u ε * u (see Proposition 1). Take A ⊂ R d such that A ⊂⊂ A . From Lemma 2.2 we deduce that, for ε small enough, ε d #R ξ nn,ε (A) + #R ξ lr,ε (A) ≤ C|A |, so that, by the monotonicity and integrability assumptions on J lr , we have Letting A ↓ A we deduce the upper bound since |∂A| = 0. (ii) holds by the lower semicontinuity property of Γ-lower/upper limits. (iii) can be proven by standard arguments using (iv) and the upper bound in (i), therefore we only prove (iv). To reduce notation we take into account only the long-range term. We may assume that the right-hand side is finite and we let u ε , v ε ∈ C ε (ω, K) both converge weakly* to u such that Given δ > 0, by the decay assumptions on J lr we may find M δ > 0 such that We define a function z ε ∈ C ε (ω, K) setting Since A ∈ A R (D) we still have that z ε * u. Fix |ξ| ≤ M δ . Then, for ε small enough, we have Letting ε → 0, as in the proof of (i), we infer that Since |∂A| = 0, the inequality (iv) follows letting δ → 0.
The proof of (v) and (vi) is straightforward.
From Proposition 2 we deduce our first main theorem.
A local version of the theorem holds. For all A ∈ A R (D) and all u ∈ L ∞ (D, co(K)) we have Proof. In order to apply Theorem 3.1, we need to define an appropriate functional on L p (D) for some p ∈ [1, +∞). This can be done as follows: by compactness of Γconvergence on separable metric spaces (see Proposition 1.42 in [9]) we can construct a diagonal sequence such that there exists a Γ-limitF (ω)(u, A) of F ε (ω)(u, A) for all (u, A) ∈ L ∞ (D, co(K)) × A R (D) (in order to pass from countably many sets A to A R (D) one can argue as in the proof of Theorem 10.3 in [12]). We extend this limit to all open sets by Referring to Proposition 2 we can apply the De Giorgi-Letta-criterion (see e.g. Theorem 1.62 in [19]) and deduce that F (ω)(u, ·) is the trace of a Borel measure that is absolutely continuous with respect to the Lebesgue measure (Proposition 2 (i)), so we can extend F (ω)(u, ·) to all Borel sets B ∈ B(D). Let us denote by p K : R m → co(K) the projection map onto the compact, convex set co(K). Consider the functional F K (ω)(·, B) : One can show that F K (ω)(u, B) satisfies the assumptions of Theorem 3.1 with respect to strong convergence in L 1 (D). The convexity and lower semicontinuity of the integrand follows from well known properties of weak* lower semicontinuous integral functionals and the local version is a direct consequence of the above construction.

Remark 3.
In general the integrand f (ω; x, ·) in Theorem 3.2 is not continuous up to the boundary of co(K). A counterexample can be constructed easily using the discontinuous, convex function on the unit ball given by the first lemma in [20]. However f (ω; x, ·) is continuous on one-dimensional segments contained in co(K). Thus we can fully characterize it by its values on the relative interior of co(K). In particular only countably many of these values are needed since the function is continuous on the relative interior of co(K). In the case co(K) is a convex polytope, then the results given in [20] show that continuity holds.

3.2.
Convergence of minimum problems. In order to prove a (stochastic) homogenization result we will use a blow-up argument to characterize the limit integrands. To relate the blow-up to our discrete energy functionals we need to prove a result regarding the convergence of minimum problems under average constraints. We first introduce a notion of discrete mean value. Given an open set A ∈ R d and u ∈ L 1 (A) we define u A as the average of u in A, that is u A = 1 |A| A u(x) dx.
Since the co-domain of our functions will be relaxed in the limit, we do not require our constrained functionals to be defined only on those functions fulfilling a precise mean value constraint. Instead we introduce a threshold δ > 0 and, for fixed z ∈ co(K), ε > 0 and ω ∈ Ω we define the functional F z,δ ε (ω) : The following convergence result holds true: Theorem 3.4. Let L be an admissible stochastic lattice and let f ε,nn and f ε,lr satisfy Hypothesis 1. Then, for P-almost every ω, the following holds: for every sequence (ε j ) converging to 0, let the subsequence (ε j ) and the function f be as in Theorem 3.2. For every z ∈ co(K), δ > 0 and every A ∈ A R (D) the functionals F z,δ εj (ω)(·, A) Γ-converge with respect to the weak* L ∞ (D, co(K))-topology to the functional F z,δ (ω)(·, A) : L ∞ (D, co(K)) → [0, +∞] defined by Proof. We start proving the lower bound inequality: if u j * u in L ∞ (D, co(K)) with equibounded energy, then we have u j ω,εj A ∈ B δ (z). Using Lemma 2.2 and the regularity of A it is straightforward to check that u A ∈ B δ (z). The claim follows by noticing that F z,δ εj (ω)(u, A) ≥ F εj (ω)(u, A) and Theorem 3.2. To prove the upper bound, note that if u ∈ L ∞ (D, co(K)) is such that u ∈ B δ (z) we can take the same recovery sequence as for the unconstrained functional and conclude. It remains the case when u A ∈ ∂B δ (z). In this case we argue by approximation as follows. We have u A = z + δν with ν ∈ S m−1 . Then the set has positive measure. For η > 0 small enough take A η ⊂ A 1 such that |A η | ≤ η. We define u η ∈ L ∞ (D, co(K)) via We have, for η small enough, Since F (ω)(u η , A) → F (ω)(u, A) as η → 0 we deduce the claim from lower semicontinuity.
By standard arguments in the theory of Γ-convergence we obtain the following corollary about the convergence of minimum problems.

Corollary 1.
Under the assumptions of Theorem 3.4, for almost every ω, for z ∈ co(K), δ > 0 and A ∈ A R (D), it holds that then its limit is a minimizer of F z,δ (ω)(·, A).

Stochastic homogenization.
This section contains the main result of the paper, namely Theorem 3.7 about the stochastic homogenization of the bulk scaling of discrete spin type energies on random lattices. As a first result needed in the proof of the stochastic homogenization Theorem 3.7 we prove a lemma that ensures that we can recover the continuum limit energy density by a suitable blow-up procedure. Our proof follows the same arguments of the one of Theorem 1 in [17]. At first let us recall the definition of nicely shrinking sets.
Definition 3.5. A family (Q η ) η>0 of Borel sets shrinks nicely to x ∈ R d if there exists a constant c > 0 such that for all x ∈ D\N, z ∈ co(K) and every family Q η shrinking nicely to x.
Proof. To reduce notation we set and At first we prove the statement in the case f is Lipschitz continuous on co(K) uniformly in x ∈ D. Applying Jensen's inequality we have that for almost every x. It follows that f (x, u(y)) − f (y, u(y)) dy |f (x, q) − f (y, q)| dy.
From this estimate, using Lebesgue's differentiation theorem combined with a covering argument (see the proof of Proposition 1.1 in [17] for details) one concludes that there exists a null set N ⊂ D such that for all x ∈ D\N , z ∈ co(K) and every family Q η shrinking nicely to x. Letting δ → 0 in the above equality yields the claim.
For a general function f let N be the null set such that f (x, ·) it is not convex and lower semicontinuous. We define a decreasing sequence f n : D × co(K) → R by It is well known that f n (x, ·) is Lipschitz continuous with Lipschitz constant bounded by n. By lower semicontinuity for x ∈ D\N the supremum defining f n can be taken over a countable dense set G ⊂ co(K) ensuring the measurability of f n (·, z) for every z ∈ co(K). Using Lebesgue's differentiation theorem one can construct a set N n of measure zero such that for all x ∈ D\N n , q ∈ G and every family Q η shrinking nicely to x. Using the uniform Lipschitz continuity of f n it is not hard to verify that (10) holds also if we replace q by any z ∈ co(K). Set N − = N ∪ n≥1 N n and fix x ∈ D\N − , z ∈ co(K) and a family Q η shrinking nicely to x. Denoting by m n δ (z, Q η ) the corresponding infimum value of the approximating sequence, by monotonicity and (10) we have By Remark 3 it holds that where ri aff denotes the relative interior. Let v ∈ ri aff (co(K)). Since f (x, ·) is continuous in v we have lim which yields by (11) lim sup In order to get the converse inequality we define g n : D × co(K) → R as This family is increasing, convex and Lipschitz continuous with Lipschitz constant bounded by n in z and, again by Remark 3 we can restrict the infimum over a countable dense set ensuring measurability in x. In this case, by lower semicontinuity, monotonicity and Proposition 1.27 in [9], we have for all z ∈ co(K). By the first part of the proof, for every g n , we can find null sets N n ⊂ D such that (9) holds for all x ∈ D\N n , z ∈ co(K) and every family Q η shrinking nicely to x. Since g n ≤ f we deduce that Using the fundamental theorem of Γ-convergence we can pass to the limit in n which implies together with (12) that for all x ∈ N := N − ∪ n≥1 N n , z ∈ co(K) and every family Q η shrinking nicely to x. Letting δ → 0 and using again the fact that f (x, ·) is lower semicontinuous yields the claim of the proposition.
For a bounded set A ⊂ R d , ρ > 0, z ∈ R m and ε > 0 we set In this section we consider spin-type systems of a particular form. Let us suppose that f nn , f lr : R n × K × K → [0, +∞) are measurable in the first variable, upper semicontinuous in the remaining couple of variables and such that Theorem 3.7. Let L be a stationary admissible stochastic lattice and assume Hypothesis 1 holds with the additional upper semicontinuity and structure assumption in (13). For P-almost every ω ∈ Ω and for all z ∈ co(K) there exists The functionals F ε (ω) Γ-converge with respect to the weak* L ∞ (D, co(K))-topology to the functional F hom (ω) : L ∞ (D, co(K)) → [0, +∞) defined by If L is ergodic, then f hom (·, z) is constant almost surely and it is given by Remark 4. The upper semicontinuity assumption can be dropped if K is at most countable. In fact it is only needed to prove measurability of the stochastic process defined in (19).
Proof of Theorem 3.7. Let ε j → 0. By Theorem 3.2, for almost every ω ∈ Ω there exists a subsequence (not relabeled) such that for all A ∈ A R (D) and u ∈ L ∞ (D, co(K)). We divide the proof into several steps.
Step 1. (characterization of the energy density) Theorem 3.2 allows us to apply Lemma 3.6 to obtain a null set N (ω) ⊂ D depending only on ω such that for all x 0 ∈ D\N , z ∈ co(K) and a family of open cubes with rational vertices shrinking nicely to x 0 . Using Corollary 1 and setting t j = ε −1 j we deduce that Suppose we have shown that for every δ > 0 there exists a function φ z,δ : Ω → R such that almost surely for every open cube Q ⊂ D with rational vertices. Then we can take the limit with respect to δ because φ z,δ is increasing when δ → 0. Taking a countable dense set G ⊂ co(K) we can construct a set Ω of full probability such that for fixed ω ∈ Ω we have Observing that z → f hom (ω; z) is convex and lower semicontinuous, by Remark 3 we have that (15) holds for all z ∈ co(K). Thus the Γ-limit exists and the integrand is given by f hom (ω; z). It remains to show (14).
Step 2. (truncation of the interaction range) Let us fix z ∈ co(K) and δ > 0. As a next step we limit the range of interactions as well as the possible positions of interacting points: To this end, for ξ ∈ r Z d and U ⊂ R d , we set where [A, B] := {tx + (1 − t)y : x ∈ A, y ∈ B, 0 ≤ t ≤ 1}. Note that I ξ nn/lr (U ) = R ξ nn/lr,1 (U ) if U is convex. Moreover for L ∈ N ∪ {+∞} we define the (truncated) energy by Given a set Q ⊂ R d we let . By the decay assumptions on the long-range interactions of Hypothesis 1, it follows that (14) is proved if we show that for every L ∈ N there exists a function φ L z,δ : for every open cube Q ⊂ D with rational vertices and all ω ∈ Ω L , where Ω L is a set of full measure. At this point note that if t k is large enough, then there exists at least one function u ∈ C z,δ 1 (ω, t k Q) to test the two infimum problems. It remains to prove (18).
Step 3. (definition of the stochastic process) As observed in the previous step, since z and δ are fixed there exists n ∈ N such that C z,δ 1 (ω, nI) = ∅ for all I ∈ I. Let us define a discrete stochastic process µ L : I → L 1 (Ω) byμ where γ > 0 is a suitable constant to be chosen. Testing the infimum problem defining µ L with any function u ∈ C z,δ 1 (ω, nI) we infer from Hypothesis 1 that there exists a constant C > 0 such that in particularμ L (I) ∈ L ∞ (Ω). The proof of the measurability is more involved and can be found in the appendix. For the proof of stationarity we can assume without loss of generality that r = 1 l for some l ∈ N. Then, using (3) and (13) it is straightforward to check that µ L (I − z)(ω) =μ L (I)(τ z ω), which yields stationarity. In order to establish subadditivity, let J be the set of finite unions of elements in I and let I 1 , I 2 ∈ J be disjoint. Given θ > 0 there exist functions u 1 ∈ C z,δ 1 (ω, nI 1 ) and u 2 ∈ C z,δ 1 (ω, nI 2 ) such that µ L (ω; nI i ) ≥ F L 1 (ω)(u i , nI i ) − θ. The function u defined on L(ω) by is admissible in the infimum problem defining µ L (ω; n(I 1 ∪ I 2 )). Note that by the special structure of the sets I ξ nn (U ) and I ξ lr (U ) two lattice points x α ∈ nI 1 and x α+ξ ∈ nI 2 can interact only if the ray [x α , x α+ξ ] has length at most L + 2r and intersects a (d − 1)-dimensional side of n(I 1 ∩ I 2 ). Using Lemma 2.2 it follows that µ L (ω; n(I 1 ∪ I 2 )) ≤ F L 1 (u 1 , nI 1 ) + F L 1 (u 2 , nI 2 ) + C L Per(n(I 1 ∩ I 2 ), R d ) ≤ µ L (ω; nI 1 ) + µ L (ω; nI 2 ) + C L Per(n(I 1 ∩ I 2 ), R d ) + 2θ.
Step 4. (proof of (18) and the case of ergodic lattices) Applying Theorem 2.6 to the processμ L (I)(ω) we deduce that there exists a function φ L z,δ : Ω → R such that φ L z,δ (ω) = lim k→+∞ inf 1 |k Q| F L 1 (ω)(u, k Q) : u ∈ C z,δ 1 (ω, k Q) for almost every ω ∈ Ω and every Q ∈ nI. The extension to arbitrary sequences and to all open cubes with rational vertices is straightforward since the differences are lower order terms. Finally note that the function φ L z,δ (ω) is invariant under the group action τ z , thus ergodicity implies that it is constant together with its limit as L → +∞. of measurable sets. By the measurability of the mapping ω → L(ω) we have proven the claim of this substep. is measurable. However the set over which we take the infimum in order to define µ L (ω; I) is not countable and still depends on ω. To overcome this issue we first fix l ∈ N. Now take a countable dense subset G of K and pick one element g 0 ∈ G. We set v zi (g 1 , . . . , g l ) := g i if i ≤ l, g 0 otherwise, where g i ∈ G. These are countably many vectors. Using upper semicontinuity of f nn and f lr with respect to the second two variables and the fact that we took the open ball B δ (z) for the constraint we deduce by density that µ L (nI)(ω) = lim l→+∞ inf (g1,...,g l )∈G l {F L 1 (ω)(v(g 1 , . . . , g l ), nI) + M (v(g 1 , . . . , g l ))(ω)} and we conclude the proof if we show that for fixed u ∈ (K) Z d the function F L 1 (ω)(u, nI) is measurable. This is shown in the appendix of [7].