Duality properties of metric Sobolev spaces and capacity †

: We study the properties of the dual Sobolev space H − 1 , q ( X ) = (cid:0) H 1 , p ( X ) (cid:1) (cid:48) on a complete extended metric-topological measure space X = ( X , τ, d , m ) for p ∈ (1 , ∞ ). We will show that a crucial role is played by the strong closure H − 1 , q pd ( X ) of L q ( X , m ) in the dual H − 1 , q ( X ), which can be identiﬁed with the predual of H 1 , p ( X ). We will show that positive functionals in H − 1 , q ( X ) can be represented as a positive Radon measure and we will charaterize their dual norm in terms of a suitable energy functional on nonparametric dynamic plans. As a byproduct, we will show that for every Radon measure µ with ﬁnite dual Sobolev energy, Cap p -negligible sets are also µ -negligible and good representatives of Sobolev functions belong to L 1 ( X , µ ). We eventually show that the Newtonian-Sobolev capacity Cap p admits a natural dual representation in terms of such a class of Radon measures.


Introduction
In this paper we investigate the properties of the duals of the metric Sobolev spaces H 1,p (X), where X = (X, τ, d, m) is an extended metric-topological measure space and p ∈ (1, +∞).
In the simpler case when (X, d) is a complete and separable metric space, τ is the topology induced by the metric and m is a positive and finite Borel (thus Radon) measure on X, H 1,p (X) can be defined as the natural domain of the L p (X, m)-relaxation of the pre-Cheeger energy form For every function f ∈ H 1,p (X) one can define the Cheeger energy and the Sobolev norm f H 1,p (X) := f p L p + CE p ( f ) 1/p , thus obtaining a Banach space.It is therefore quite natural to study its dual, which we will denote by H −1,q (X).
In such a general situation, however, when we do not assume any doubling and/or Poincaré assupmptions, H 1,p (X) may fail to be reflexive or separable and it is not known if the generating class of bounded Lipschitz functions is strongly dense.
As a first contribution, we will show that it could be more convenient to consider the smaller subspace H −1,q pd (X) of H −1,q (X) obtained by taking the strong closure of L q (X, m).Linear functionals in H −1,q pd (X) are characterized by their behaviour on Lip b (X) (or on even smaller generating subalgebras) and their dual norm can also be computed by the formula which is well adapted to be applied to general Borel measures µ on X.
In Sections 3 and 4 we will show that H −1,q pd (X) has three important properties: (a) it can be identified with the predual of H 1,p (X) (thus showing in particular that H 1,p (X) is the dual of a separable Banach space); (b) every positive Borel measure µ satisfying for every f ∈ Lip b (X) can be extended in a unique way to a functional L µ ∈ H −1,q pd (X); (c) every positive functional on Lip b (X) such that the supremum in (1.1) is finite can be represented by a positive Radon measure.
This last property relies on a representation formula of the dual of the Cheeger energy by (nonparametric) dynamic plans (Theorem 4.6) which is interesting by itself.As a further important application of this result, in the final section 5 we will show that negligible sets in E with respect to the Newtonian capacity Cap p are also µ-negligible for every positive Borel measure with finite dual energy.As a byproduct, we can express the duality of µ with a function f in H 1,p (X) in the integral form µ, f = X f dµ, where f is any good representative of f in the Newtonian space N 1,p (X).

Topological and measure theoretic notions
Let (Y, τ Y ) be a Hausdorff topological space.We will denote by C b (Y, τ Y ) the space of τ Y -continuous and bounded real functions defined on Y; B(Y, τ Y ) is the collection of the Borel subsets of Y; we will often omit the explicit indication of the topology τ Y , when it will be clear from the context.
We will denote by M + (Y) the collection of all the finite positive Radon measures on Y.
It is worth mentioning that every Borel measure in a Polish, Lusin, Souslin, or locally compact space with a countable base of open sets is Radon [22,Ch. II,Sect. 3].In particular the notation of M + (Y) is consistent with the standard one adopted e.g., in [4,6,24], where Polish or second countable locally compact spaces are considered.
If (Y, τ Y ) is completely regular, the weak (or narrow) topology τ M + on M + (Y) can be defined as the coarsest topology for which all maps Let (Z, τ Z ) be another Hausdorff topological space.A map f : Every Lusin µ-measurable map is also µ-measurable.Whenever f is Lusin µ-measurable, its pushforward induces a Radon measure in Z.Given a power p ∈ (1, ∞) and a Radon measure µ in (Y, τ Y ) we will denote by L p (Y, µ) the usual Lebesgue space of class of p-summable µ-measurable functions defined up to µ-negligible sets.

Extended metric-topological (measure) spaces
Let (X, τ) be a Hausdorff topological space.An extended distance is a symmetric map d : X × X → [0, ∞] satisfying the triangle inequality and the property d(x, y) = 0 iff x = y in X: we call (X, d) an extended metric space.We will omit the adjective "extended" if d takes real values.
Let d be an extended distance on X.For every f : X → R and A ⊂ X we set We adopt the convention to omit the set A when A = X.We consider the class of τ-continuous and and for every κ > 0 we will also consider the subsets A particular role will be played by Lip (X1) the topology τ is generated by the family of functions Lip b (X, τ, d); (X2) the distance d can be recovered by the functions in Lip b,1 (X, τ, d) through the formula We will say that (X, τ, d) is complete if d-Cauchy sequences are d-convergent.All the other topological properties usually refer to (X, τ).
The previous assumptions guarantee that (X, τ) is completely regular.When an e.m.t.space (X, τ, d) is provided by a positive Radon measure m ∈ M + (X, τ) we will say that the system X = (X, τ, d, m) is an extended metric-topological measure (e.m.t.m.) space.Definition 2.2 yields two important properties linking d and τ: first of all since it is the supremum of a family of continuous maps by (2.3).On the other hand, every d-converging net (x j ) j∈J indexed by a directed set J is also τ-convergent: It is sufficient to observe that τ is the initial topology generated by Lip b (X, τ, d) so that a net (x j ) is convergent to a point x if and only if In many situations it could be useful to consider smaller subalgebras which are however sufficiently rich to recover the metric properties of an extended metric topological space (X, τ, d).

Definition 2.3 (Compatible algebras of Lipschitz functions).
Let A be a unital subalgebra of Lip b (X, τ, d) and let us set A κ := A ∩ Lip b,κ (X, τ, d).
We say that A is compatible with the metric-topological structure (X, τ, d) if In particular, A separates the points of X.
It is not difficult to show that any compatible algebra A is dense in L p (X, m) [21, Lemma 2.27].If we do not make a different explicit choice, we will always assume that an e.m.t.m. space X is endowed with the canonical algebra A (X) := Lip b (X, τ, d).

The asymptotic Lipschitz constant
For every f : X → R and x ∈ X, denoting by U x the directed set of all the τ-neighborhoods of x, we set lip f (x) := lim Notice that Lip( f, {x}) = 0 and therefore lip f (x) = 0 if x is an isolated point of X.We can also define lip f as where the convergence of y, z to x in (2.4) is intended with respect to the topology τ.In particular, It is not difficult to check that x → lip f (x) is a τ-upper semicontinuous map and f is locally d-Lipschitz in X iff lip f (x) < ∞ for every x ∈ X.When (X, d) is a length space, lip f coincides with the upper τ-semicontinuous envelope of the local Lipschitz constant (2.5).
We collect in the next useful lemma the basic calculus properties of lip f .

Continuous curves and rectifiable arcs
We briefly recap some useful results concerning the extended metric-topological structure of the space of rectifiable arcs in an e.m.t.space (X, τ, d).We refer to [21, § 3] for a more detailed discussion and for the related proofs.
The extended distance d : and the function It is possible to prove that ∼ is an equivalence relation [21, § 3.2, Cor.3.5] and d A satisfies In particular d A satisfies the triangle inequality, is invariant with respect to ∼ and γ ∼ γ if and only if d A (γ, γ ) = 0. We collect a list of useful properties [21, § 3.2]: Lemma 2.5.(a) The space RA(X) := BVC([0, 1]; X)/ ∼ endowed with the quotient topology τ A is an Hausdorff space.We will denote by q : BVC([0, 1]; X) → RA(X) the quotient map.
(b) If the topology τ is induced by the distance δ then the quotient topology τ A is induced by δ A (considered as a distance between equivalence classes of curves).
(d) For every γ, γ ∈ BVC([0, 1]; X) we have and all the curves γ equivalent to γ can be described as so that the functions R, , the evaluation maps e 0 , e 1 , and the integral γ f are invariant w.r.t.reparametrizations.We will still denote them by the same symbols.
We conclude this section with a list of useful properties concerning the compactness in RA(X) and the measurability of some importants maps, see [21,Thm. 3.13].
and for every bounded and continuous function f ∈ C b (X, τ) we have In particular, we have lim In particular the family of measures {ν γ } γ∈RA(X) is Borel.d) is complete and Γ ⊂ RA(X) satisfies the following conditions: 1) sup γ∈Γ (γ) < +∞; 2) there exists a τ-compact set K ⊂ X such that e[γ] ∩ K ∅ for every γ ∈ Γ; then Γ is relatively compact in RA(X) w.r.t. the τ A topology.
Notice that the third condition in the statement (f) of Theorem 2.6 implies the second one whenever inf γ∈Γ (γ) > 0.

Metric Sobolev spaces and their duals
In this section we will always assume that X = (X, τ, d, m) is a complete e.m.t.m. space and A is a compatible sub-algebra of Lip b (X, τ, d).We also fix a summability exponent p ∈ (1, ∞) with conjugate q = p/(p − 1).
For every k ≥ 0 and f ∈ L p (X, m) we also set We denote by H 1,p (X) the subset of L p (X, m) whose elements f have finite Cheeger energy Remark 3.2 (The notation CE and H 1,p ).We used the symbol CE instead of Ch (introduced by [6]) in the previous definition to stress three differences: • the dependence on the strongest lip f instead of |D f |, • the factor 1 instead of 1/p in front of the energy integral.
In this paper we will mainly adopt the "strong" approach to metric Sobolev spaces and we will use the notation H 1,p (X) to stress this fact.We refer to [5,6] for the equivalent weak definition of W 1,p (X) by test plan.In the final section 5 we will also use a few properties related to the intermediate (but still equivalent) Newtonian point of view, see [8,17].
It is not difficult to check that CE p : L p (X, m) → [0, +∞] is a convex, lower semicontinuous and phomogeneous functional; it is the greatest L p -lower semicontinuous functional "dominated" by pCE p .Notice that when m has not full support, two different elements f 1 , f 2 ∈ Lip b (X, τ, d) may give rise to the same equivalence class in L p (X, m).In this case, CE p can be equivalently defined starting from the functional defined on the quotient space Lip b (X, τ, d)/ ∼ m .Whenever CE p ( f ) < ∞ one can show [5,6] that the closed convex set admits a unique element of minimal norm, the minimal relaxed gradient denoted by |D f | .|D f | is also minimal in S p ( f ) with respect to the natural order structure, i.e., The Cheeger energy CE p admits an integral representation in terms of the minimal relaxed gradient: and enjoys the following strong approximation result (see [5,6] in the case of bounded Lipschitz functions, [7] for the "metric" algebra generated by truncated distance functions and [21, Thm.12.1] for the general case): Theorem 3.3 (Density in energy of compatible algebras).Let A be a compatible sub-algebra of Lip b (X, τ, d) and let I be a closed (possibly unbounded) interval of R. For every f ∈ H 1,p (X) taking values in I there exists a sequence ( f n ) ⊂ A with values in I such that
We collect a list of useful properties [6] of the minimal p-relaxed gradient.
Theorem 3.4.For every f, g ∈ H 1,p (X) we have and In particular for every constant c ∈ R 3.2.Legendre transform of the Cheeger energy and the dual of the Sobolev space H 1,p (X) Let us now study a few important properties of the Legendre transform of the p-Cheeger energy and its relation with the dual of the Sobolev space H 1,p (X) when p ∈ (1, ∞); recall that we denote by q = p = p/(p − 1) the conjugate exponent of p.Let us first recall a simple property of p-homogeneous convex functionals (see e.g., [21,Lemma A.7]. Lemma 3.5 (Dual of positively p-homogeneous functionals).Let C be a convex cone of some vector space V, p > 1, and φ, ψ : C → [0, ∞] with ψ = φ 1/p , φ = ψ p .We have the following properties: (a) φ is convex and positively p-homogeneous (i.e., φ(κv) = κ p φ(v) for every κ ≥ 0 and v ∈ C) in C if and only if ψ is convex and positively 1-homogeneous on C (a seminorm, if C is a vector space and ψ is finite and even).
(b) Under one of the above equivalent assumptions, setting for every linear functional z : we have where in the first infimum we adopt the convention inf A = +∞ if A is empty.
We want to study the dual functionals related to CE p,κ and pCE p,κ .The simplest situation is provided by L p − L q -duality: u) for every w ∈ L q (X, m), By Fenchel-Moreau duality Theorem (see e.g., [9, Theorem 1.11], [13, Chap.IV]) it is immediate to check that The situation is more complicated if one wants to study the dual of CE p with respect to the Sobolev duality.Just to clarify all the possibilities we consider three normed vector spaces: • The separable and reflexive Banach space V := L p (X, m); • The vector space A p associated to a compatible algebra A endowed with the norm pCE 1/p p,1 .
• The Banach space W = H 1,p (X) with the norm CE 1/p p,1 .Notice that we do not know any information concerning the separability and the reflexivity of the Banach space H 1,p (X) nor the (strong) density of A in W. Since both A and W = H 1,p (X) are dense in V = L p (X, m), if we identify V with L q (X, m) we clearly have On the other hand, every element L ∈ W can be considered as a bounded linear functional on A p and thus induces an element L restr of (A p ) just by restriction, but it may happen that this identification map is not injective.Finally, since pCE p,1 may be strictly greater than CE p,1 on A p , in general not all the bounded linear functionals on A p may admit an extension to W.
Taking all these facts into account, now we want to address the question of the unique extension of a given bounded linear functional L on A p to an element of the dual Sobolev space W .We begin with a precise definition.Definition 3.6 (The spaces H −1,q (X), H −1,q pd (X) and A q ).We define: • H −1,q (X) as the dual W of H 1,p (X); • H −1,q pd (X) as the subset of H −1,q (X) whose elements L satisfy the following condition: for every choice of f, f n ∈ H 1,p (X), n ∈ N, and every constant C > 0 Mathematics in Engineering Volume 3, Issue 1, 1-31.
• A q as the set of linear functionals L on A satisfying the following two conditions: there exists a constant D > 0 such that and for every sequence f n ∈ A and every constant C > 0 It would not be difficult to check that if H 1,p (X) is reflexive then A is strongly dense in H 1,p (X) and H −1,q (X) = H −1,q pd (X) A q .In the general case, only a partial result holds and we will show that H 1,p (X) can be identified with the dual of H −1,q pd (X), i.e., H −1,q pd (X) is a predual of H 1,p (X) (this property justifies the index pd in the notation).Let us start with a first identification: Proposition 3.7 (A q H −1,q pd (X)).The following properties hold: (a) A q is a closed subspace of (A p ) : in particular, it is a Banach space with the norm (b) A linear functional L on A belongs to A q if and only if for every ε > 0 there exists a constant In this case (3.8b) holds in the stronger form where lim inf is replaced by lim sup.
(c) Every linear functional L ∈ A q admits a unique extension L in H −1,q pd (X).The map L → L is a surjective isometry between A q and H −1,q pd (X), which is therefore a closed subspace of H −1,q (X).In particular, if L, L ∈ H −1,q pd (X) coincide on A then L = L .Proof.(a) It is sufficient to prove that A q is closed in the Banach space (A p ) .Let L be an element of the closure and for every ε > 0 choose elements Since ε > 0 is arbitrary we obtain that L ∈ A q .
(b) If L satisfies (3.9) and f n ∈ A is a sequence as in (3.8b) we have since ε is arbitrary we deduce that lim sup n→∞ L, f n = 0, thus (3.8b) in the stronger form.
In order to prove the converse implication, we argue by contradiction by assuming that there exists ε > 0 and a sequence f n ∈ A such that Mathematics in Engineering Volume 3, Issue 1, 1-31.
By possibly replacing f n with f n pCE p,1 ( f n ) −1/p , it is not restrictive to assume that pCE p,1 ( f n ) = 1; by (c) In order to define L we fix f ∈ H 1,p (X) and any sequence f n ∈ A such that f n → f in L p (X, m) with E p := sup pCE p ( f n ) < ∞.By (3.9), for every ε > 0 there exists κ > 0 such that which shows that the sequence n → L, f n satisfies the Cauchy condition and thus admits a limit which we denote by L, f .This notation is justified by the fact that the limit does not depend on the sequence f n : in fact, if f n is another sequence converging to f in L p (X, m) with equibounded energy, (3.7) shows that lim n→∞ L, f n − f n = 0.It is also easy to check that the map H 1,p (X) f → L, f is a linear functional.
In order to show that L is bounded, for every f ∈ H 1,p (X) we select an optimal sequence f n such that On the other hand for every f ∈ A with pCE p ( f ) ≤ 1 by choosing the constant sequence f n ≡ f we get X) so that the extension map ι : L → L is an isometry.
It remains to prove that the image of ι coincides with H −1,q pd (X).Since it is clear that H −1,q pd (X) ⊂ ι(A q ), it is sufficient to show the converse inclusion, i.e., that every element L = ι(L) satisfies (3.7).By linearity, it is not restrictive to check (3.7) for f = 0.If f n ∈ H 1,p (X) has equibounded Cheeger energy and lim n→∞ f n L p = 0, by the very definition of the Cheeger energy and the definition of L we can find another sequence g n ∈ A such that Since L ∈ A q and lim n→∞ g n L p = 0 we have lim n→∞ L, g n = 0 so that lim n→∞ L, f n = 0.
Let us now express the dual functionals by a infimal convolution.
Proof.(3.10) is a particular case of the duality formula for the sum of two convex functions ϕ, ψ : which holds in every Banach space W whenever there exists a point w 0 ∈ W such that φ(w 0 ) < ∞ and ψ is finite and continuous at w 0 by Fenchel-Rockafellar Theorem ( [20], see also [9,Theorem 1.12]).
Here W = H 1,p (X), φ(g) := 1 p CE p,α (g), ψ(g) := β p g p L p .We collect in the next proposition a further list of useful properties.We will denote by J p : L p (X, m) → L q (X, m) the duality map and by A p : H 1,p (X) → P(H −1,q (X)) the subdifferential of the Cheeger energy with respect to the Sobolev duality Since CE p is continuous in H 1,p (X), A p u ∅ for every u ∈ H 1,p (X) [13, Chap. 1, Prop.5.3] (notice that A p is different from the subdifferential of CE p w.r.t. the L p -L q duality pair).The sum Proposition 3.9.We have the following properties (a) For every L ∈ H −1,q pd (X) and every κ ≥ 0 we have (c) For every L ∈ H −1,q pd (X) and κ > 0 there exists a unique solution u which satisfies (d) For every L ∈ H −1,q (X) and κ > 0 there exists a unique function f κ := R κ (L) solving the minimum problem The map R κ : (e) For every L ∈ H −1,q (X) we have Proof.(a) (3.13) (which implies (3.12)) follows by an easy approximation argument combining the definition of CE p and the continuity property (3.7) and it follows by the same argument at the end of the proof of claim (c) of Proposition 3.7.
(b) Since H −1,q pd (X) is a closed subspace of H −1,q (X) and clearly contains L q (X, m), it is sufficient to prove that L q (X, m) is dense in H −1,q pd (X).For every n ∈ N we consider the functional G n := CE * p,1+n p and we want to show that lim sup n↑∞ G n (L) = 0; (3.17) by using (3.10) (with α := 1, β := n p ), (3.17) is in fact equivalent to the density of L q (X, m) in H −1,q pd (X).By the first formula of (3.10), for every ε > 0 we can find g n ∈ H 1,p (X) such that and G n (L) ≥ 0, we obtain so that CE p,1 (g n ) is uniformly bounded and g n L p → 0 as n → ∞.By (3.7) we conclude that lim n→∞ L, g n = 0 and therefore (3.18) yields lim sup n→∞ G n (L) ≤ ε.Since ε > 0 is arbitrary, we obtain (3.17).
(c) The existence of a solution u κ ∈ H 1,p (X) to (3.14) follows by (3.7) and the Direct method of the Calculus of Variations.Let us take a minimizing sequence Since f n is uniformly bounded in H 1,p (X), up to extracting a suitable subsequence (still denoted by f n ), it is not restrictive to assume that f n is converging to a function f ∈ H 1,p (X) weakly in L p (X, m) and We prove that f n is a Cauchy sequence: by the uniform convexity of the L p (X, m)-norm, for every ε > 0 there exist S < S < S such that for every h 1 , h 2 ∈ L p (X, m) By (3.20) we can find n ∈ N such that for every n ≥ n and For every m, n ≥ n we thus get and therefore f n + f m 2 L p ≥ S so that (3.21) yields f n − f m L p ≤ ε for every n, m ≥ n.We deduce that lim n→∞ f n − f L p = 0; since f n is uniformly bounded in H 1,p (X), (3.7) yields lim n→∞ L, f n = L, f and the lower semicontinuity of the Cheeger energy yields CE p ( f ) ≤ lim inf n→∞ CE p ( f n ).By (3.19) we conclude that 1  p CE p,κ ( f ) − L, f = M so that f is the unique minimizer of (3.14).
In order to prove the continuity of R κ , let L n ∈ H −1,q (X) be a sequence strongly converging to L and let ) is uniformly bounded, we obviously get a uniform bound for CE * p,κ (L n − f n ) and f n L q .Let f ∈ L q (X, m) be any weak L q limit point of f n , e.g., attained along a subsequence f n( j) .Since we deduce that so that f = R κ (L).Since R κ (L) is the unique weak limit point of the sequence f n in L q , we conclude that f n R κ (L) in L q (X, m).The same variational argument also shows that lim sup n→∞ f n L q ≤ f L q so that the convergence is strong.
Finally, if L ∈ H −1,q pd (X), f κ is the (unique) minimizer of (3.10) and u κ is the (unique) minimizer of (3.14), we get (e) Since the map κ → CE * p,κ (L) is nonincreasing, we have lim κ↓0 CE * p,κ (L) = sup κ>0 CE * p,κ (L) ≤ CE * p (L).On the other hand, for every f ∈ H 1,p (X) and ε > 0, choosing κ > 0 sufficiently small so that κ p f p L p ≤ ε we get Since the inequality holds for every ε > 0 and every f ∈ H 1,p (X), we obtain the converse inequality Proposition 3.9 yields the following interesting duality result, which is also related to the theory of derivations discussed in [12].Corollary 3.10 (H 1,p (X) is the dual of a Banach space).H 1,p (X) can be isometrically identified with the dual of H −1,q pd (X).In particular, if L q (X, m) is a separable space, H 1,p (X) is the dual of a separable Banach space.
Proof.Let z be a bounded linear functional on H −1,q pd (X).Since L q (X, m) is continuously and densely imbedded in H −1,q pd (X), for every f ∈ L q (X, m) z, f ≤ z H −1,q pd (X) f L q , so that there exists a unique By (3.6) and the strong density of L q (X, m) in H −1,q pd (X) .
It follows that ι is an isometry from the dual of H −1,q pd (X) and H 1,p (X).Since ι is clearly surjective, we conclude.
Remark 3.11 (H 1,p (X) as Gagliardo completion [16]).Recall that if (A, • A ) is a normed vector space continuously imbedded in a Banach space (V, • V ), the Gagliardo completion A V,c is the Banach space defined by When supp(m) = X, we can identify A with a vector space A with the norm induced by pCE p imbedded in V := L p (X, m); it is immediate to check that H 1,p (X) coincides with the Gagliardo completion of A in V.When A (and therefore W) is strongly dense in V, we can identify the dual V of V as a subset of the dual W of W and we can define the set W pd as the closure of V in W .If V is uniformly convex, the same statements and characterizations given in Propositions 3.7 and 3.9 hold in this more abstract setting.In particular, W can be isometrically identified with the dual of W pd .

Radon measures with finite (dual) energy
The following result provides a useful criterion to check if a linear functional on A belongs to A q .Let us first recall that a subset F ⊂ L 1 (X, m) is weakly relatively compact in L 1 (X, m) if and only if it satisfies one of the following equivalent properties [13, Chap.VIII, Theorem 1.3]: a) for all ε > 0 there exists m ≥ 0 such that Proposition 3.12.Let L be a linear functional on A satisfying (3.8a).If for every sequence f n ∈ A satisfying Proof.We split the proof in two steps.Claim 1: if L is a linear functional on A satisfying (3.8a) and for every sequence We argue by contradiction and we assume that there exists a sequence f n ∈ A such that By possibly changing the sign of f n it is not restrictive to assume that L, f n ≥ c > 0 for every n ∈ N. Applying Mazur Lemma we find coefficients α n,m ≥ 0, n ∈ N, 0 ≤ m ≤ M(n) such that g n := M(n) m=0 α n,m lip f n+m is strongly converging in L p (X, m).Thus n → g p n is strongly converging in L 1 (X, m) and it is therefore equi-integrable.
We now consider fn := M(n) m=0 α n,m f n+m .By construction and lip fn ≤ α n,m lip f n+m = g n so that (3.23) yields lim inf n→∞ L, fn = 0, which contradicts the first inequality of (3.24).
Claim 2: it is sufficient to prove the implication (3.23) for sequences taking values in Let us fix ε > 0 and δ := ε/3E so that E p δ p ≤ ε p /3.For every choice of n ∈ N we can find an odd polynomial P n such that (see e.g., [21,Lemma 2.23]) We set notice that g n and h n belong to A as well.Since lip g n ≤ lip f n and g n takes values in [−1, 1], by assumption we have lim inf n→∞ | L, g n | = 0. On the other hand, we can choose n 0 sufficiently big so that for every n ≥ n 0 (3.8a) then yields | L, h n | ≤ Dε for n ≥ n 0 and therefore lim inf n→∞ | L, f n | ≤ Dε.Since ε > 0 is arbitrary, we conclude.
Our last criterium concerns positive functionals, i.e., satisfying We will see in Theorem 4.7 that they are always induced by a Radon measure.
Theorem 3.13.If L is a linear functional on A satisfying (3.8a) and (3.26), then L ∈ A q .
If π is a dynamic plan in M + (RA(X)), thanks to Theorem 2.6(e) and Fubini's Theorem [11, Chap.II-14], we can define the Borel measure µ π := Proj(π) ∈ M + (X) by the formula µ π is a Radon measure with total mass π( ) given by (4.1) [21, § 8] and it can also be considered as the integral w.r.t.π of the Borel family of measures ν γ , γ ∈ RA(X) [11,, in the sense that Recall that p, q ∈ (1, ∞) is a fixed pair of conjugate exponents.
Definition 4.2.We say that π ∈ M + (RA(X)) has barycenter in L q (X, m) if there exists h ∈ L q (X, m) such that µ π = hm, or, equivalently, if γ f dπ(γ) = f h dm for every bounded Borel function f : X → R, and we call Bar q (π) := h L q (X,m) the barycentric q-entropy of π.We will denote by B q (RA(X)) the set of all plans with barycenter in L q (X, m) and we will set Bar q (π) := +∞ if π B q (RA(X)).
Bar q : M + (RA(X)) → [0, +∞] is a convex and positively 1-homogeneous functional, which is lower semicontinuous w.r.t. the weak topology of M + (RA(X)), since it can also be characterized as the L q entropy of the projected measure µ π = Proj(π) with respect to m: where for an arbitrary σ ∈ M + (X) Notice that Bar q (π) = 0 iff π is concentrated on the set of constant arcs in RA(X).
D q provides an important representation for the dual of the p-Cheeger energy.
Let us give a first important application of the above result to the representation of positive functionals.
Theorem 4.7.Let A be a compatible subalgebra of Lip b (X, τ, d) and let L be functional on A satisfying Then L admits a unique extension L ∈ H −1,q pd (X) and there exists a unique µ ∈ M + (X) representing L as L, f = X f dµ for every f ∈ A .(4.8) Proof.By Theorem 3.13 and Proposition 3.7(c) we know that L is the restriction to A of a unique functional L ∈ H −1,q pd (X).It is easy to check that L is also positive on H 1,p (X) and applying Proposition 3.9 we can find a sequence w n ∈ L q (X, m), w n ≥ 0, strongly converging to L in H −1,q (X).Let µ n := w n m and ν n := R 1 (w n )m; applying Theorem 4.6 we can find optimal dynamic plans π ) is also uniformly bounded, the sequence π n satisfies the tightness criterium of [21,Lemma 8.5], so that it admits a subsequence (still denoted by π n ) weakly converging to π ∈ B q (RA(X)) ⊂ M + (RA(X)).
The Radon measure µ := (e 0 ) π is the weak limit of µ n : in particular, for every

Measures with finite energy and Newtonian capacity
In this last section we apply the previous result to prove new properties of the Newtonian capacity.We first recall the basic facts about the Newtonian approach [18,23], based on the notion of p-Modulus which has been introduced by Fuglede [15] in the natural framework of collection of positive measures, as in [1].We refer to [8,17] for a comprehensive presentation of this topic.As usual, p, q ∈ (1, ∞) denote a pair of conjugate exponents.
Γ is said to be Mod p -negligible if Mod p (Γ) = 0. We say that a property P on RA(X) holds Mod p -a.e. if the set of arcs where P fails is Mod p -negligible.We say that a property P on X holds quasi everywhere (q.e.) if the set of points E where P fails is m-negligible and satisfies Notice that if E is m-negligible then for Mod p -a.e.arc γ the set {t ∈ [0, 1] : R γ (t) ∈ E} is L 1negligible.It can be shown (see e.g., [8]) that Mod p is an increasing and subadditive functional which is continuous along increasing sequences.In fact, by [1] and [21, § 7.2], Mod p is also continuous along decreasing sequence of compact sets, so that it is a Choquet capacity for the compact paving in RA(X) [11,Chap. III,28].
It is not difficult to check that for every dynamic plan π ∈ B q (RA(X)) and every π-measurable set Γ ⊂ RA(X) π(Γ) ≤ Bar q (π) Mod 1/p p (Γ), which in particular shows that Borel Mod p -negligible sets are also π-negligible for every π ∈ B q (RA(X)).In fact, a much more refined result holds [1,21]: Theorem 5.2.If X is a complete e.m.t.m. space and τ is a Souslin topology for X, then every Borel or Souslin set Γ in RA(X) is Mod p -capacitable and satisfies In particular, Γ is Mod p -negligible if and only if π(Γ) = 0 for every π ∈ B q (RA(X)).
Recall that e i (γ), i = 0, 1, denote the initial and final points of a rectifiable arc γ ∈ RA(X).Definition 5.3 (Newtonian weak upper gradient).Let f : X → R be m-measurable and p-summable function.We say that f belongs to the Newtonian space N 1,p (X) if there exists a nonnegative g ∈ L p (X, m) such that f (e 1 (γ)) − f (e 0 (γ)) ≤ γ g for Mod p -a.e.arc γ ∈ RA(X). (5.1) In this case, we say that g is a N 1,p -weak upper gradient of f .
Functions with Mod p -weak upper gradient have the important Beppo-Levi property of being absolutely continuous along Mod p -a.e.arc γ (more precisely, this means f • R γ is absolutely continuous, see [23,Proposition 3.1]).Notice that functions in N 1,p (X) are everywhere defined.We say that f ∈ N 1,p (X) is a good representative of a function f ∈ L p (X, m) if m({ f f }) = 0.
Theorem 5.4.Let f n ∈ N 1,p (X), g n ∈ L p (X, m) be two sequences strongly converging to f, g ∈ L p (X, m) respectively in L p (X, m).If g n is a N 1,p -weak upper gradient of f then there exists a good representative f ∈ N 1,p (X) of f and a subsequence k → n(k) such that f n(k) → f quasi everywhere; moreover g is a N 1,p -weak upper gradient of f .
It is clear that a function f ∈ Lip b (X, τ, d) belongs to N 1,p (X) and lip f is a N 1,p -weak upper gradient (it is in fact an upper gradient).By Theorems 5.4 and 3.3 one can easily get that also every f ∈ H 1,p (X) admits a good representative f ∈ N 1,p (X) and |D f | is a N 1,p -weak upper gradient of f .Equivalently, f is absolutely continuous along Mod p -a.e.arc and g satisfies (5.1) Mod p -a.e.
In fact these two notions are essentially equivalent modulo the choice of a representative in the equivalence class [1,5,6,21]: Theorem 5.5.Let us suppose that X is a complete e.m.t.m. space.Every function f ∈ N 1,p (X) also belongs to H 1,p (X) and every N 1,p -weak upper gradient g satisfies g ≥ |D f | m-a.e., so that |D f | is also the minimal N 1,p -weak upper gradient of f .

Applications to the Newtonian capacity
The Newtonian capacity Cap p (E) of a subset E ⊂ X can be defined as Cap p (E) := inf CE p,1 (u) : u ∈ N 1,p (X), u ≥ 1 on E . ( By choosing u as the function taking the constant value 1 it is immediate to check that in our setting the capacity of a set is always finite and Cap p (E) ≤ m(X) < ∞ for every E ⊂ X.
It can be proved [8,Prop. 1.48] that E ⊂ X has 0 capacity if and only if E is m-negligible and Mod p (Γ E ) = 0, (5.3) so that a property P on X holds quasi everywhere if the set where P fails has 0 capacity.Notice that if fi ∈ N 1,p (X) coincide m-a.e., then f1 = f2 q.e. in X.Notice moreover that we can use q.e.inequality in (5.2), i.e., u ∈ N 1,p (X), u ≥ 1 q.e. on E ⇒ Cap p (E) ≤ CE p,1 (u).
We also recall that the capacity satisfies the following properties [8,  We want now to study the relation between the Newtonian capacity and measures µ ∈ M + (X) with finite energy, according to Definition 3.14.We will denote by Theorem 5.6.Let µ ∈ M + (X) be a measure with finite energy and let L µ ∈ H −1,q pd (X) be the linear functional associated to µ according to Corollary 3.15.
(a) If E is a universally measurable subset of X with 0 capacity then E is µ-negligible.
(b) If u ∈ H 1,p (X) is nonnegative and ũ ∈ N 1,p (X) is a good representative of u, then ũ ∈ L 1 (X, µ) and Proof.(a) Let E be a set with 0 capacity according to (5.3); since m(E) = 0, by considering the Lebesgue decomposition (5.4) it is sufficient to show that µ ⊥ (E) = 0.It is not restrictive to assume µ ⊥ (X) > 0; by Theorem 4.6 we can find a plan π ∈ B q (RA(X)) such that µ ⊥ = (e 0 ) π.
(b) Let us first assume that ũ ≤ M for some constant M > 0. We can find a sequence u n ∈ Lip b (X, τ, d) taking values in [0, M], converging to ũ q.e. and with u n → u, lip u n → |Du| in L 2 (X, m).The uniform bound, the q.e.convergence and the fact that L µ ∈ H −1,q pd (X) yield The case of a general nonnegative u follows by passing to the limit in the sequence u M := u ∧ M as M ↑ ∞ and using monotone convergence.(5.6) pCE p ( f ) := X lip f (x) p dm(x), f ∈ Lip b (X),initially defined only for bounded Lipschitz functions.Here lip f (x) defines the asymptotic Lipschitz constant lip f (x) = lim sup y,z→x y z | f (y) − f (z)| d(y, z) .

5. 1
. p-Modulus of a family of arcs and Newtonian Sobolev spaces Definition 5.1 (p-Modulus of a family of rectifiable arcs).The p-Modulus of a collection Γ ⊂ RA(X) is defined by Mod
b,1 (X, τ, d).It is easy to check that Lip b (X, τ, d) is a real and commutative sub-algebras of C b (X, τ) with unit.According to [2, Definition 4.1], an extended metrictopological space (e.m.t.space) (X, τ, d) is characterized by a Hausdorff topology τ and an extended distance d satisfying a suitable compatibility condition.