Quasi-continuous vector fields on RCD spaces

In the existing language for tensor calculus on RCD spaces, tensor fields are only defined m-a.e.. In this paper we introduce the concept of tensor field defined `2-capacity-a.e.' and discuss in which sense Sobolev vector fields have a 2-capacity-a.e. uniquely defined quasi-continuous representative.


Introduction
The theory of differential calculus on RCD spaces as proposed in [9,10] is built around the notion of L 0 (m)-normed L 0 (m)-module, which provides a convenient abstraction of the concept of 'space of measurable sections of a vector bundle'. In this sense, one thinks at such a module as the space of measurable sections of some, not really given, measurable bundle over the given metric measure space (X, d, m). More precisely, given that elements of L 0 (m) are 'Borel functions identified up to equality m-a.e.', elements of such modules are, in a sense, 'measurable sections identified up to equality m-a.e.'. Notice that this interpretation is fully rigorous in the smooth case, where given a normed vector bundle, the space of its Borel sections identified up to m-a.e. equality is a L 0 (m)-normed L 0 (m)-module. We also remark that in [9,10] the emphasis is more on the notion of L ∞ (m)-module rather than on L 0 (m) ones, but this is more a choice of presentation rather than an essential technical point, and given that for the purposes of this manuscript to work with L 0 is more convenient, we shall concentrate on this.
In particular, all the tensor fields on a metric measure space which are considered within the theory of L 0 (m)-modules are only defined m-a.e.. While this is an advantage in some setting, e.g. because a rigorous first order differential calculus can be built on this ground over arbitrary metric measure structures, in others is quite limiting: there certainly might be instances where, say, one is interested in the behaviour of a vector field on some negligible set. For instance, the question of whether the critical set of a harmonic function has capacity zero simply makes no sense if the gradient of such function only exists as element of a L 0 (m)-normed module.
Aim of this paper is to create a theoretical framework which allows to speak about 'Borel sections identified up to equality Cap-a.e.' and to show that Sobolev vector fields on RCD spaces, which are introduced via the theory of L 0 (m)-modules, in fact can be defined up to Cap-null sets and turn out to be continuous (in a sense to be made precise) outside sets of small capacities.
Here the analogy is with the well-known case of Sobolev functions on the Euclidean space: these are a priori defined in a distributional-like sense, and thus up to equality L d -a.e., but once the concept of capacity is introduced one quickly realizes that a Sobolev function has a uniquelydefined representative up to Cap-null sets which is continuous outside sets of small capacities.
More in detail, in this paper we do the following: o) We start recalling how to integrate w.r.t. an outer measure and that such integral is sublinear iff the outer measure is submodular, which is the case of capacity. This will allow to put a natural complete distance on the space L 0 (Cap) of real-valued Borel functions on X identified up to Cap-null sets. Given that Cap-null sets are m-null, L 0 (m) can be seen as quotient of L 0 (Cap); we shall denote by Pr : L 0 (Cap) → L 0 (m) the quotient map. We then recall the concept of quasi-continuous function which, being invariant under modification on Cap-null sets, is a property of (equivalence classes of) functions in L 0 (Cap). The space QC(X) of quasi-continuous functions is actually a closed subspace of L 0 (Cap) and coincides with the L 0 (Cap)-closure of continuous functions (then by approximation in the uniform norm one easily sees that in proper spaces one could also take the completion of the space of locally Lipschitz functions and, in R d , of smooth ones); we believe that this characterization of quasi-continuity is well-known in the literature but have not been able to find a reference -in any case, for completeness in the preliminary section we provide full proofs of all the results we need. In connection with the concept of capacity the space QC(X) is relevant for at least two reasons: a) The restriction of the projection operator Pr : L 0 (Cap) → L 0 (m) to QC(X) is injective. b) Any Sobolev function f ∈ W 1,2 (X) ⊂ L 0 (m) has a (necessarily unique, by a) above) quasi-continuous representativef ∈ QC(X), i.e. Pr(QC(X)) ⊃ W 1,2 (X). i) We propose the notion of L 0 (Cap)-normed L 0 (Cap)-modules (L 0 (Cap)-modules, in short), defined by properly imitating the one of L 0 (m)-module. At the technical level an important difference between the two notions is that the capacity is only an outer measure: while in some cases this is only a nuisance (see e.g. the proof of the fact that the natural distance on L 0 (Cap)-modules satisfies the triangle inequality), in others it creates problems whose solution is unclear to us (e.g. in defining the dual of a L 0 (Cap)-module -see Remark 2.3).
We then see that, much like starting from L 0 (Cap) and quotienting out up to m-null sets we find L 0 (m), starting from an arbitrary L 0 (Cap)-module M and quotienting via the relation ii) The main construction that we propose in this manuscript is that of tangent L 0 (Cap)module L 0 Cap (T X) on an RCD space X. Specifically, in such setting we prove that there is a canonical couple L 0 Cap (T X),∇ , where L 0 Cap (T X) is a L 0 (Cap)-module and∇ : Test(X) → L 0 Cap (T X) is a linear map whose image generates L 0 Cap (T X) and such that |∇f | coincides with the unique quasi-continuous representative of the minimal weak upper gradient |Df | of f , see Theorem 2.6. Here the space Test(X) of test functions is made, in some sense, of the smoothest functions available on RCD spaces; this regularity matters in the definition of L 0 Cap (T X) to the extent that |Df | belongs to W 1,2 (X) whenever f ∈ Test(X) (and this fact is in turn highly depending on the lower Ricci curvature bound: it seems hard to find many functions with this property on more general metric measure spaces).
The relation between (L 0 Cap (T X),∇) and the already known L 0 (m)-tangent module L 0 m (T X) and gradient operator ∇ is the fact that L 0 m (T X) can be seen as the quotient of L 0 Cap (T X) via the equivalence relation (0.1), where the projection operator sends∇f to ∇f for any f ∈ Test(X) (see Propositions 2.9, 2.10 for the precise formulation). iii) We define the notion of 'quasi-continuous vector field' in L 0 Cap (T X). Here a relevant technical point is that there is no topology on the 'tangent bundle' or, to put it differently, it is totally unclear what it means for a tangent vector field to be continuous or continuous at a point (in fact, not even the value of a vector field at a point is defined in our setting!). In this direction we also remark that the recent result in [7] suggests that it might be pointless to look for 'many' continuous vector fields already on finite dimensional Alexandrov spaces, thus a fortiori on RCD ones.
Thus, much like in R d quasi-continuous functions are the L 0 (Cap)-closure of smooth ones, we define the space of quasi-continuous vector fields QC(T X) as the L 0 Cap (T X)-closure of the space of the 'smoothest' vector fields available, i.e. linear combinations of those of the form g∇f for f, g ∈ Test(X). The choice of terminology is justified by the fact that the analogue of a), b) above hold (see Proposition 2.13 and Theorem 2.14) and, moreover: c) for v ∈ QC(T X) we have |v| ∈ QC(X) (see Proposition 2.12).
We conclude by pointing out that, while the concept of L 0 (Cap)-module makes sense for any p-capacity, for our purposes only the case p = 2 is relevant. This is due to the fact that the natural Sobolev space to which |Df | belongs for f ∈ Test(X) is W 1,p (X) with p = 2. Also, we remark that, albeit the definitions proposed in this paper are meant to be used in actual problems regarding the structure of RCD spaces (like the already mentioned one concerning the size of {∇f = 0} for f harmonic -see Example 2.17 for comments in this direction), in this manuscript we concentrate on building a solid foundation of the theoretical side of the story. The added value here is in providing what we believe are the correct definitions: once these are given, proofs of relevant results will come out rather easily.
Hence the following definition (via Cavalieri's formula) is well-posed. For an arbitrary set E ⊂ X we shall also put In the next result we collect the basic properties of the above-defined integral: Proposition 1.1 (Basic properties of µ and integrals w.r.t. it). The following properties hold: a) Let f, g : X → [0, +∞] be fixed. Then the following holds: In Then f n dµ → f dµ as n → ∞. c) Borel-Cantelli. Let (E n ) n∈N be subsets of X satisfying n∈N µ(E n ) < +∞. Then it holds that µ n∈N m≥n E m = 0.

Proof.
(a) (i) is trivial and (ii) follows by a change of variables. The 'if' implication in (iii) is trivial, for the 'only if' recall that t → µ {f > t} is non-increasing to conclude that if f dµ = 0 it must hold µ {f > t} = 0 for every t > 0. Then use the countable subadditivity of µ and the identity {f > 0} = n {f > 1/n} to deduce that µ {f > 0} = 0. To prove (iv) notice that {g > t} ⊂ {f > t} ∪ {f = g}, hence taking into account the subadditivity of µ and the assumption we get g dµ ≤ f dµ. Inverting the roles of f, g we conclude. In order to prove (v), so that µ {f = +∞} = 0, as required.
(b) Assume for a moment that f n (x) ↑ f (x) for every x ∈ X. Then the sequence of sets {f n > t} is increasing with respect to n and satisfies n {f n > t} = {f > t}. Hence the monotone convergence theorem (for the Lebesgue measure) gives The general case follows taking into account point (iv).
(c) The standard proof applies: let us put E := n∈N m≥n E m , then µ(E) ≤ µ m≥n E m for all n ∈ N, so that as required.
Example 1.2. Consider the closed interval S := [0, 1] in R (equipped with Euclidean distance and Lebesgue measure). Given any n ≥ 1, denote by P n ⊆ S the singleton {1/n}. One can check that 0 < Cap(P n ) < Cap(S), but Cap(S \ P n ) = Cap(S). In other words, χ S\Pn dCap = χ S dCap and Cap { χ S\Pn = χ S } > 0. This shows that -even for µ := Cap and f ≤ g -the converse of item iv) of Proposition 1.1 fails.
3. An analogue of the dominated convergence theorem cannot hold, as shown by the following counterexample. For any n ≥ 1, consider the singleton P n in R defined in Example 1.2.
Since the capacity in the space R is translation-invariant, one has that Cap(P n ) = Cap(P 1 ) > 0 for all n ≥ 1. Moreover, we have lim n χ Pn (x) = 0 for all x ∈ R and χ Pn ≤ χ [0,1] for all n ≥ 1, with χ [0,1] dCap = Cap [0, 1] < +∞. Nevertheless, it holds that χ Pn dCap ≡ Cap(P 1 ) does not converge to 0 as n → ∞, thus proving the failure of the dominated convergence theorem. To provide such a counterexample, we exploited the fact that the capacity is not σ-additive; indeed, we built a sequence of pairwise disjoint sets, with the same positive capacity, which are contained in a fixed set of finite capacity. The lack of a result such as the dominated convergence theorem explains the technical difficulties we will find in the proofs of Proposition 1.10 and Theorem 1.11.
Let us now introduce a crucial property of outer measures: Definition 1.4 (Submodularity). We say that µ is submodular provided The importance of the above notion is due to the following result (we refer to [8, Chapter 6] for a detailed bibliography): Theorem 1.5 (Subadditivity theorem). It holds that µ is submodular if and only if the integral associated to µ is subadditive, i.e.
Proof. The 'if' trivially follows by taking f := χ E and g := χ F , so we turn to the 'only if'. Notice that, up to a monotone approximation argument based on point (b) of Proposition 1.1, it suffices to consider the case in which f, g assume only a finite number of values and µ {f > 0}∪{g > 0} < ∞. Then up to replacing X with {f > 0} ∪ {g > 0} we can also assume that µ is finite. Thus assume this is the case and let A be the (finite) algebra generated by f, g, i.e. the one generated by the sets {f = a} and {g = b}, a, b ∈ R. Let A 1 , . . . , A n ∈ A be minimal with respect to inclusion among non-empty sets in A and ordered in such a way that a i ≥ a i+1 for every i = 1, . . . , n − 1, where a i := (f + g)(A i ) (i.e. a i is the value of f + g on A i ).
Define a finite measure ν on A by putting Notice that once such claim is proved the conclusion easily follows from (f + g) dµ = (f + g) dν (from the equality case of the claim (1.6) and the choice of enumeration of the A i 's), the inequalities f dν ≤ f dµ, g dν ≤ g dµ (from the claim (1.6)) and the linearity of the integral w.r.t. ν. Also, notice that the equality case in (1.6) is a direct consequence of Cavalieri's formula for both µ, ν, the defining property (1.5) and the fact that for h as stated it holds Let us now assume that for someī it holds h(Aī) =: bī ≤ bī +1 := h(Aī +1 ) and let us define another finite measureν as in (1.5) with the sets (A i ) i replaced by (Ã i ) i , whereÃ i := A i for i =ī,ī + 1,Ãī := Aī +1 andÃī +1 := Aī. We shall prove that h dν ≤ h dν and notice that this will give the proof, as with a finite number of such permutations the sets (A i ) i get ordered as in the equality case in (1.6). Using Cavalieri's formula and noticing that by construction it holds ν {h > t} =ν {h > t} for t / ∈ [bī, bī +1 ) we reduce to prove that we know from the submodularity of µ that ν(B ∪ Aī +1 ) ≤ν(B ∪ Aī +1 ), as required.
1.2. Capacity on metric measure spaces. We shall be interested in a specific outer measure: the 2-capacity (to which we shall simply refer as capacity) on a metric measure space (X, d, m). For the purposes of the current manuscript, by metric measure space we shall always mean a triple (X, d, m) such that (X, d) is a complete and separable metric space, m ≥ 0 is a Borel measure on (X, d), finite on balls. (1.8) In this setting, starting from [6] (see also [16,2]) there is a well-defined notion of Sobolev space W 1,2 (X, d, m) (or, briefly, W 1,2 (X)) of real-valued functions on X, and to any f ∈ W 1,2 (X, d, m) is associated a function |Df | ∈ L 2 (m), called minimal weak upper gradient, which plays the role of the modulus of the distributional differential of f . For our purposes, it will be useful to recall that W 1,2 (X, d, m) is a lattice (i.e. f ∨ g and f ∧ g are in W 1,2 (X) provided f, g ∈ W 1,2 (X)), that the minimal weak upper gradient is local, i.e.
Finally, the norm on W 1,2 (X) is defined as and with this norm W 1,2 (X) is always a Banach space whose norm is L 2 -lower semicontinuous, i.e.
where as customary f W 1,2 (X) is set to be +∞ if f / ∈ W 1,2 (X). Even if in general W 1,2 (X) is not Hilbert (and thus the map f → 1 2 |Df | 2 dm is not a Dirichlet form), the concept of capacity is well-defined as the definition carries over quite naturally (see e.g. [3], [12] and references therein for the metric setting, and [4] for the more classical framework of Dirichlet forms): Definition 1.6 (Capacity). Let E be a given subset of X. Let us denote Then the capacity of the set E is defined as the quantity Cap(E) ∈ [0, +∞], given by with the convention that Cap(E) := +∞ whenever the family F E is empty.
In the following result we recall the main properties of the set-function Cap: The capacity Cap is a submodular outer measure on X, which satisfies the following properties: Proof. We start claiming that for any f, g ∈ W 1,2 (X) it holds for m-a.e. x and thus and, similarly, by the locality property (1.9) the set |D(f ∨ g)|(x), |D(f ∧ g)|(x) coincides with the set |Df |(x), |Dg|(x) for m-a.e. x and thus Adding up these two identities and integrating we obtain (1.14). Now let E 1 , E 2 ⊂ X be given and notice that for Hence passing to the infimum over f i ∈ F Ei we conclude that In particular, this shows that Cap is finitely subadditive and thus to conclude that it is a submodular outer measure it is sufficient to show that if (E n ) n is an increasing sequence of subsets of X it holds that Cap n E n = sup n Cap(E n ). Since trivially Cap(E) ≤ Cap(F ) if E ⊂ F , it is sufficient to prove that Cap n E n ≤ sup n Cap(E n ) and this is obvious if sup n Cap(E n ) = +∞. Thus assume that S := sup n Cap(E n ) < +∞ and assume for the moment also that the E n 's are open. Let f n ∈ F En be such that f n 2 W 1,2 (X) ≤ Cap(E n ) + 2 −n ≤ S + 1. Thus in particular such sequence is bounded in L 2 (m) and thus for some n k ↑ +∞ we have that f n k ⇀ f in L 2 (m) for some f ∈ L 2 (m). Passing to the (weak) limit in k in the inequality f n k ≥ χ En k ≥ χ En ℓ valid for k ≥ ℓ, we conclude that f ≥ χ En ℓ for every ℓ, hence f ≥ χ n En . Since the E n 's are open, this means that f ∈ F n En . Therefore taking into account the semicontinuity property (1.11) we deduce that Now let us drop the assumption that the E n 's are open. Let ε > 0. We use the submodularity property (1.15) and an induction argument to find an increasing sequence (Ω n ) n of open sets such that Ω n ⊃ E n and Cap(Ω n ) ≤ Cap(E n ) + ε i≤n 2 −i . Then taking into account what already proved for open sets we deduce that Cap(E n ) + ε and the conclusion follows from the arbitrariness of ε > 0. The inequality in (i) trivially follows noticing that for f ∈ F E it holds f ≥ 1 m-a.e. on E, thus so that the conclusion follows taking the infimum over f ∈ F E . For the statement (ii) it is sufficient to recall that for any B ⊂ X bounded there is f Lipschitz with bounded support that is ≥ 1 on a neighbourhood of B and that such f belongs to W 1,2 (X).
1.3. The space L 0 (Cap). We have just seen that Cap is a submodular outer measure and in Subsection 1.1 we recalled how integration w.r.t. outer measures is defined. It makes therefore sense to consider the integral associated to Cap and that such integral is subadditive by Proposition 1.7 and Theorem 1.5. In light of this observation, the following definition is meaningful: Given any two functions f, g : X → R, we will say that f = g in the Cap-a.e. sense provided Cap {f = g} = 0. We define L 0 (Cap) as the space of all the equivalence classes -up to Cap-a.e. equality -of Borel functions on X.
We endow L 0 (Cap) with the following distance: pick an increasing sequence (A k ) k of open subsets of X with finite capacity such that for any B ⊂ X bounded there is k ∈ N with B ⊂ A k (for instance, one could pick A k := B k (x) for somex ∈ X), then let us define Notice that the integral A k |f − g| ∧ 1 dCap is well-defined, since its value does not depend on the particular representatives of f and g, as granted by item (iv) of Proposition 1.1. Moreover, we point out that the fact that d Cap satisfies the triangle inequality is a consequence of the subadditivity of the integral associated with the capacity. Remark 1.9. We point out that if Cap(X) < +∞ then the choice A k := X for all k ∈ N is admissible in Definition 1.8.
The next result shows that, even if the choice of the particular sequence (A k ) k might affect the distance d Cap , its induced topology remains unaltered. Proposition 1.10 (Convergence in L 0 (Cap)). The following holds: • Characterization of Cauchy sequences. Let (f n ) n ⊆ L 0 (Cap) be given. Then the following conditions are equivalent: ii) lim n,m Cap B ∩ |f n − f m | > ε = 0 for any ε > 0 and any bounded set B ⊂ X. • Characterization of convergence. Let f ∈ L 0 (Cap) and (f n ) n ⊆ L 0 (Cap). Then the following conditions are equivalent: Proof. We shall only prove the characterization of Cauchy sequences, as the other claim follows by similar means. i) =⇒ ii) Fix any 0 < ε < 1 and a bounded set proving that lim n,m d Cap (f n , f m ) = 0, as required.
Proof. Let (f n ) n be a d Cap -Cauchy sequence of Borel functions f n : X → R. Fix any k ∈ N. Let (f ni ) i be an arbitrary subsequence of (f n ) n . Up to passing to a further (not relabeled) subsequence, it holds that Let us call We proved this property for some subsequence of a given subsequence (f ni ) i of (f n ) n , hence this shows that Since any bounded subset of X is contained in the set A k for some k ∈ N, we immediately deduce that lim n Cap B ∩ |f n − f | > ε = 0 whenever ε > 0 and B ⊂ X is bounded. This grants that lim n d Cap (f n , f ) = 0 by Proposition 1.10, thus proving that L 0 (Cap), d Cap is a complete metric space and accordingly the statement.
We conclude this section with some other basic properties of the metric space L 0 (Cap), d Cap : Moreover, the space Sf(X) of simple functions, which is defined as Proof. As for the standard case of measures, let the subsequence satisfy d Cap (f nj , f nj+1 ) ≤ 2 −j for all j ∈ N. By the very definition of d Cap , we deduce that for every j, k ∈ N one has . Given any k ∈ N, we know from item b) of Proposition 1.1 (and the subadditivity of the integral associated to Cap) that Therefore item v) of Proposition 1.1 ensures that g ∞ (x) < +∞ for Cap-a.e. x ∈ A k , whence also for Cap-a.e. x ∈ X by arbitrariness of k ∈ N. Now observe that for all j ′ ≥ j and Cap-a.e. x ∈ X it holds that By letting j, j ′ → ∞ in (1.23) we deduce that f nj (x) j ⊂ R is Cauchy for Cap-a.e. x ∈ X, thus it admits a limitf (x) ∈ R. Again by item b) of Proposition 1.1 we know for every k ∈ N that This means that A k |f − f | ∧ 1 dCap = 0 for every k ∈ N, thus accordinglyf = f holds Cap-a.e. by item iii) of Proposition 1.1. We then finally conclude that f nj (x) → f (x) for Cap-a.e. x ∈ X.
For the second statement we argue as follows. Fix f ∈ L 0 (Cap) and ε > 0. Choose a Borel representativef : X → R of f . For any integer i ∈ Z, let us define E i :=f −1 [i ε, (i + 1) ε) . Then (E i ) i∈Z constitutes a partition of X into Borel sets, so thatḡ := i∈Z i ε χ Ei is a well-defined Borel function that belongs to Sf(X). Finally, it holds that f (x) −ḡ(x) < ε for every x ∈ X, which grants that d Cap (f, g) ≤ ε, where g ∈ L 0 (Cap) denotes the equivalence class ofḡ. Hence the statement follows. Remark 1.13. In general, Cap-a.e. convergence does not imply convergence in L 0 (Cap), as shown by the following counterexample. Consider P n as in Example 1.2 for any n ≥ 1. We have that the functions f n := χ Pn pointwise converge to 0 as n → ∞. However, it holds that Cap [0, 1] ∩ |f n | > 1/2 = Cap(P n ) ≡ Cap(P 1 ) > 0 does not converge to 0, thus we do not have lim n d Cap (f n , 0) = 0 by Proposition 1.10.

Quasi-continuous functions and quasi-uniform convergence.
Here we quickly recall the definition and main properties of quasi-continuous functions associated to Sobolev functions (see [3], [13], [4] for more on the topic and detailed bibliography). Definition 1.14 (Quasi-continuous functions). We say that a function f : X → R is quasicontinuous provided for every ε > 0 there exists a set E ⊂ X with Cap(E) < ε such that the function f | X\E : X \ E → R is continuous.
It is clear that if f,f agree Cap-a.e. and one of them is quasi-continuous, so is the other. Also, by the very definition of capacity, in defining quasi-continuity one could restrict to sets E which are open. In particular, if f is quasi-continuous there is an increasing sequence (C n ) n of closed subsets of X with lim n Cap(X \ C n ) = 0 such that f is continuous on each C n . Then N := n X \ C n is a Borel set with null capacity -in particular, we have m(N ) = 0 by item i) of Proposition 1.7 -and f is Borel on X \ N . This proves that any quasi-continuous function is m-measurable and Cap-a.e. equivalent to a Borel function.
We shall denote by QC(X) the collection of all equivalence classes -up to Cap-a.e. equality -of quasi-continuous functions on X. What we just said ensures that QC(X) ⊂ L 0 (Cap). It is readily verified that QC(X) is an algebra.
Let us now discuss a notion of convergence particularly relevant in relation with QC(X): Definition 1.15 (Local quasi-uniform convergence). Let f n : X → R, n ∈ N ∪ {∞} be Borel functions. Then we say that f n locally quasi-uniformly converges to f ∞ as n → ∞ provided for any B ⊂ X bounded and any ε > 0 there exists a set E ⊂ X with Cap(E) < ε such that f n → f ∞ uniformly on B \ E. In this case, we shall write f n QU → f ∞ .
As before, nothing changes if one even requires the set E to be open in the above definition and the notion of local quasi-uniform convergence is invariant under modification of the functions in Cap-null sets. Local quasi-uniform convergence is (almost) the convergence induced by the following distance: where (A k ) k is any sequence as in Definition 1.8. Indeed, it is trivial to verify that d QU is actually a distance (notice that d QU ≤ 1, as one can see by picking E = X in (1.25)), moreover we have: then any subsequence n k has a further subsequence, not relabeled, Proof.
(i) Let ε > 0 and use the definition of local quasi-uniform convergence to find some subsets (E k ) k∈N of X such that Cap(E k ) < ε/2 k and f n → f ∞ uniformly on A k \ E k for any k ∈ N. Choosing the set E := k∈N E k in (1.25) yields (fork ∈ N sufficiently big) and the conclusion follows by the arbitrariness of ε.
Let ε > 0 and B ⊂ X bounded be fixed. Pickk ∈ N such that B ⊂ Ak. Then (1.26) grants the existence ofn ∈ N with n≥n Cap(E n ∩ Ak) < ε, thus E := n≥n E n ∩ Ak satisfies Cap(E) < ε. Therefore we have that This grants that f n QU → f ∞ , as required.
Proposition 1.17. The following properties hold: i) The metric space QC(X), d QU is complete.
ii) It holds that d Cap (f, g) ≤ d QU (f, g) ≤ 2 d Cap (f, g) for every f, g ∈ QC(X). In particular, the canonical embedding of QC(X) in L 0 (Cap) is continuous and has closed image. iii) QC(X) is the closure in L 0 (Cap) of the space of (equivalence classes up to Cap-null sets of ) continuous functions. Proof.
(i) To prove completeness, fix a d QU -Cauchy sequence (f n ) n of quasi-continuous functions. Up to passing to a (not relabeled) subsequence, we can suppose that d QU (f n , f n+1 ) < 2 −n for all n. For any n ∈ N we can pick a set E n ⊂ X such that the function f n is continuous on X \ E n and Now define F k,m := n≥m E n ∩ A k and F k := m∈N F k,m for every k, m ∈ N. Hence one has for any given k, m ∈ N, so that f n n → g m uniformly on the set A k \ F k,m for some continuous function g k,m : Moreover, we know from (1.27) that n∈N Cap(E n ∩ A k ) ≤ 2 k Cap(A k ) ∨ 1 n∈N 2 −n < +∞ holds for every k ∈ N, whence item c) of Proposition 1.1 ensures that Cap(F k ) = 0 for all k ∈ N. By item b) of Proposition 1.1 we see that which shows that the function f ∞ is quasi-continuous. We claim that f n QU → f ∞ , which is enough to conclude by item i) of Proposition 1.16. Given any ε > 0 and any bounded set B ⊂ X, we can pickk,n ∈ N such that B ⊂ Ak and Cap(E) < ε, where we set E := En ∩ Ak. Therefore we have which implies that f n QU → f ∞ , as desired.
(ii) Fix f, g ∈ QC(X) and take (A k ) k as in Definition 1.8. Given any E ⊂ X, it holds that for every k ∈ N. By summing over k ∈ N and then passing to the infimum over E ⊂ X, we On the other hand, let us consider the set E λ := |f − g| ∧ 1 > λ for any λ > 0. Therefore for every k ∈ N one has that Cap(E λ ∩ A k ) ≤ λ −1 A k |f − g| ∧ 1 dCap by item v) of Proposition 1.1 and that sup A k \E λ |f − g| ∧ 1 ≤ λ, thus accordingly By letting λ ↓ d Cap (f, g) we conclude that d QU (f, g) ≤ 2 d Cap (f, g), as required.
(iii) Let f : X → R be a Borel function whose equivalence class up to Cap-null sets belongs to QC(X) and ε > 0. Then by definition there is an open set Ω with Cap(Ω) < ε and f | X\Ω is continuous. By the Tietze extension theorem there is g ∈ C(X) which agrees with f on X \ Ω, and -since this latter condition ensures that d QU (f, g) < ε -the proof is achieved.
We now turn to the relation between quasi-continuity and Sobolev functions, and to do so it is useful to emphasise whether we speak about functions up to Cap-null sets or up to m-null sets. We shall therefore write [f ] Cap (resp. [f ] m ) for the equivalence class of the Borel function f : X → R up to Cap-null (resp. m-null) sets.
We start noticing that -since m is absolutely continuous with respect to Cap, i.e. Cap-null sets are also m-null (recall (i) of Proposition 1.7) -there is a natural projection map Since in general there are m-null sets which are not Cap-null, such projection operator is typically non-injective. This is why the following result is interesting: and the quasi-continuity assumption gives the conclusion.
Proof. For any λ > 0 let Ω λ := |f − g| > λ , so that by definition of d QU we have Notice that Ω λ is an open set by continuity of |f − g|. Moreover,

Main result
2.1. L 0 (Cap)-normed L 0 (Cap)-modules. The language of L 0 (m)-normed L 0 (m)-modules over a metric measure space (X, d, m) has been proposed and investigated by the second author in [9], with the final aim of developing a differential calculus on RCD spaces. In the present paper, we assume the reader to be familiar with such language. We shall use the term L 0 (m)-module in place of L 0 (m)-normed L 0 (m)-module and we will typically denote by M m any such object. We refer to [9,10] for a detailed account about this topic. Here we introduce a new notion of normed module, called L 0 (Cap)-normed L 0 (Cap)-module or, more simply, L 0 (Cap)-module, in which the measure under consideration is the capacity Cap instead of the reference measure m.
Let (X, d, m) be a metric measure space as in (1.8) and (A k ) k a sequence as in Definition 1.8.
is complete and induces the topology τ .
Much like starting from L 0 (Cap) and passing to the quotient up to m-a.e. equality we obtain L 0 (m), in the same way by passing to an appropriate quotient starting from an arbitrary L 0 (Cap)module we obtain a L 0 (m)-module. Let us describe this procedure.
Let  Proof. The latter statement is an obvious consequence of the former, so we concentrate on this one. Let v, w ∈ M be such that v ∼ m w and notice that ≤ Pr |v − w| = 0 holds m-a.e..
Thus T passes to the quotient and defines a map T Pr : M m → N m making the diagram (2.2) commute. It is clear that T Pr is linear and continuous (the latter being a consequence of (2.1) and the definition), thus to conclude it is sufficient to prove L 0 (m)-linearity. By linearity and continuity this will follow if we show that m for any Borel set E ⊂ X; in turn, this will follow if we prove that To show this, notice that from and the conclusion follows.

Remark 2.3.
In analogy with the case of L 0 (m)-modules, one could be tempted to define the dual of a L 0 (Cap)-module M Cap as the space of L 0 (Cap)-linear continuous maps L : M Cap → L 0 (Cap) and to declare that the pointwise norm |L| of any such L is the minimal element of L 0 (Cap) (where minimality is intended in the Cap-a.e. sense) such that the inequality |L| ≥ L(v) holds Cap-a.e. for any v ∈ M Cap that Cap-a.e. satisfies |v| ≤ 1. Technically speaking, for L 0 (m)-modules this can be achieved by using the notion of essential supremum of a family of Borel functions. Nevertheless, it seems that this tool cannot be adapted to the situation in which we want to consider the capacity instead of the reference measure, as suggested by Example 1.2.
Then the operator ·, · is L 0 (Cap)-bilinear and satisfies v, w ≤ |v||w| v, v = |v| 2 Cap-a.e. for every v, w ∈ H . (2.6) 2.2. Tangent L 0 (Cap)-module. Let (X, d, m) be an RCD(K, ∞) space, for some K ∈ R. A fundamental class of Sobolev functions on X is that of test functions, denoted by Test(X) (cf. [9]). We point out that we are in a position to apply Theorem 1.20 above, since Lipschitz functions with bounded support are dense in W 1,2 (X), as proven in [1]. Moreover, a fact that is fundamental for our discussion (see [15]) is the following: ∇f, ∇g ∈ W 1,2 (X) for every f, g ∈ Test(X).
In particular, by taking g = f in (2.7) we get |Df | 2 ∈ W 1,2 (X) for every f ∈ Test(X). Let us use the notation L 0 m (T X) to indicate the tangent L 0 (m)-module over (X, d, m). Recall that TestV(X) ⊆ L 0 m (T X) denotes the class of test vector fields on X, while H 1,2 C (T X) is the closure of TestV(X) in the Sobolev space W 1,2 C (T X). We know from [10,Proposition 2.19] that for any v ∈ H 1,2 C (T X) ∩ L ∞ m (T X) one has that |v| 2 ∈ W 1,2 (X) and (2.8) d|v| 2 (w) = 2 ∇ w v, v m-a.e. for every w ∈ L 0 m (T X), whence in particular D|v| 2 ≤ 2 |∇v| HS |v| holds m-a.e.. This in turn implies the following: C (T X) be fixed. Then |v| ∈ W 1,2 (X) and (2.9) D|v| ≤ |∇v| HS holds m-a.e. in X.
Proof. First of all, we prove the statement for v ∈ TestV(X). Given any ε > 0, let us define the Lipschitz function ϕ ε : [0, +∞) → R as ϕ ε (t) := √ t + ε for any t ≥ 0. Hence by applying the chain rule for minimal weak upper gradients we see that ϕ ε • |v| 2 ∈ S 2 (X) (cf. [2] for the notion of Sobolev class S 2 (X)) and This grants the existence of G ∈ L 2 (m) and a sequence ε j ց 0 such that D(ϕ εj • |v| 2 ) ⇀ G weakly in L 2 (m) as j → ∞ and G ≤ |∇v| HS in the m-a.e. sense. Since ϕ εj • |v| 2 → |v| pointwise m-a.e. as j → ∞, we deduce from the lower semicontinuity of minimal weak upper gradients that |v| ∈ W 1,2 (X) and that D|v| ≤ |∇v| HS holds m-a.e. in X. Now fix v ∈ H 1,2 C (T X). Pick a sequence (v n ) n ⊆ TestV(X) that W 1,2 C (T X)-converges to v. In particular, |v n | → |v| and |∇v n | HS → |∇v| HS in L 2 (m). By the first part of the proof we know that |v n | ∈ W 1,2 (X) and D|v n | ≤ |∇v n | HS for all n ∈ N, thus accordingly (up to a not relabeled subsequence) we have that D|v n | ⇀ H weakly in L 2 (m), for some H ∈ L 2 (m) such that H ≤ |∇v| HS holds m-a.e. in X. Again by lower semicontinuity of minimal weak upper gradients we conclude that |v| ∈ W 1,2 (X) with D|v| ≤ |∇v| HS in the m-a.e. sense, proving the statement.
We now introduce the so-called tangent L 0 (Cap)-module L 0 Cap (T X) over X, which is a L 0 (Cap)module in the sense of Definition 2.1.
Theorem 2.6 (Tangent L 0 (Cap)-module). Let (X, d, m) be an RCD(K, ∞) space. Then there exists a unique couple L 0 Cap (T X),∇ , where L 0 Cap (T X) is a L 0 (Cap)-module over X and the operator∇ : Test(X) → L 0 Cap (T X) is linear, such that the following properties hold: i) For any f ∈ Test(X) we have that the equality |∇f | = QCR |Df | holds Cap-a.e. on X (note that |Df | ∈ W 1,2 (X) as a consequence of Lemma 2.5). ii) The space of n∈N χ En∇ f n , with (f n ) n ⊆ Test(X) and (E n ) n Borel partition of X, is dense in L 0 Cap (T X). Uniqueness is intended up to unique isomorphism: given another couple (M Cap ,∇ ′ ) with the same properties, there exists a unique isomorphism Φ : Cap (T X) is called tangent L 0 (Cap)-module associated to (X, d, m), while its elements are said to be Cap-vector fields on X. Moreover, the operator∇ is called gradient.

Proof.
Uniqueness. Consider any simple vector field v ∈ L 0 Cap (T X), i.e. v = n∈N χ En∇ f n for some (f n ) n ⊆ Test(X) and (E n ) n Borel partition of X. We are thus forced to set Such definition is well-posed, as granted by the Cap-a.e. equalities which also show that Φ preserves the pointwise norm of simple vector fields. In particular, the map Φ is linear and continuous, whence it can be uniquely extended to a linear and continuous operator Φ : L 0 Cap (T X) → M Cap by density of simple vector fields in L 0 Cap (T X). It follows from Proposition 1.12 that Φ preserves the pointwise norm. Moreover, we know from the definition (2.10) that Φ(f v) = f Φ(v) is satisfied for any simple f and v, whence also for all f ∈ L 0 (Cap) and v ∈ L 0 Cap (T X) by Proposition 1.12. To conclude, just notice that the image of Φ is dense in M Cap by density of simple vector fields in M Cap , thus accordingly Φ is surjective (as its image is closed, being Φ an isometry). Therefore we proved that there exists a unique module isomorphism Φ : L 0 Cap (T X) → M Cap such that Φ •∇ =∇ ′ , as required. Existence. We define the 'pre-tangent module' Ptm as the set of all sequences (E n , f n ) n , where (f n ) n ⊆ Test(X) and (E n ) n is a Borel partition of X. We now define an equivalence relation ∼ on Ptm: we declare that (E n , f n ) n ∼ (F m , g m ) m provided QCR |D(f n − g m )| = 0 Cap-a.e. on E n ∩ F m for every n, m ∈ N.
The equivalence class of (E n , f n ) n will be denoted by [E n , f n ] n . Moreover, let us define for every α, β ∈ R and [E n , f n ] n , [F m , g m ] m ∈ Ptm/ ∼, so that Ptm/ ∼ inherits a vector space structure; well-posedness of these operations is granted by the locality property of minimal weak upper gradients and by Theorem 1.20. We define the pointwise norm of any given element [E n , f n ] n ∈ Ptm/ ∼ as (2.11) [E n , f n ] n := n∈N χ En QCR |Df n | ∈ L 0 (Cap).
Then we define L 0 Cap (T X) as the completion of the metric space Ptm/ ∼ , d L 0 Cap (T X) , where for every test function f ∈ Test(X), thus obtaining a linear operator∇ : Test(X) → L 0 Cap (T X). Item i) of the statement is thus clearly satisfied. Observe that [E n , f n ] n = n∈N χ En∇ f n for every [E n , f n ] n ∈ Ptm/ ∼, so that also item ii) is verified, as a consequence of the density of Ptm/ ∼ in L 0 Cap (T X). Now let us define the multiplication operator · : Sf(X) × (Ptm/ ∼) → Ptm/ ∼ as follows: Therefore the maps defined in (2.11) and (2.13) can be uniquely extended by continuity to a pointwise norm operator | · | : L 0 Cap (T X) → L 0 (Cap) and a multiplication by L 0 (Cap)-functions · : L 0 (Cap) × L 0 Cap (T X) → L 0 Cap (T X), respectively. It also turns out that the distance d L 0 is expressed by the formula in (2.12) for any v, w ∈ L 0 Cap (T X), as one can readily deduce from Proposition 1.12. Finally, standard verifications show that L 0 Cap (T X) is a L 0 (Cap)-module over (X, d, m), thus concluding the proof.
Remark 2.7. An analogous construction has been carried out in [9] to define the cotangent L 0 (m)module L 0 m (T * X), while the tangent L 0 (m)-module L 0 m (T X) was obtained from the cotangent one by duality. However, since we cannot consider duals of L 0 (Cap)-modules (as pointed out in Remark 2.3), we opted for a different axiomatisation. We just underline the fact that, since RCD spaces are infinitesimally Hilbertian, the modules L 0 m (T * X) and L 0 m (T X) can be canonically identified via the Riesz isomorphism.
Proposition 2.8. The tangent L 0 (Cap)-module L 0 Cap (T X) is a Hilbert module. Proof. Given any f, g ∈ Test(X), we deduce from item i) of Theorem 2.6 and the last statement of Theorem 1.20 that This grants that the pointwise parallelogram identity (2.4) is satisfied whenever v, w are L 0 (Cap)linear combinations of elements of ∇ f : f ∈ Test(X) , whence also for any v, w ∈ L 0 Cap (T X) by approximation. This proves that L 0 Cap (T X) is a Hilbert module, as required.
We now investigate the relation that subsists between tangent L 0 (Cap)-module and tangent L 0 (m)-module. We start with the following result, which shows the existence of a natural projection operatorPr sending∇f to ∇f : Proposition 2.9. There exists a unique linear continuous operatorPr : L 0 Cap (T X) → L 0 m (T X) that satisfies the following properties: i)Pr(∇f ) = ∇f for every f ∈ Test(X). ii)Pr(gv) = Pr(g)Pr(v) for every g ∈ L 0 (Cap) and v ∈ L 0 Cap (T X). Moreover, the operatorPr satisfies The well-posedness of such definition stems from the following m-a.e. equalities: (2.16) Moreover, we also infer that such mapPr -which is linear by construction -is also continuous, whence it admits a unique linear and continuous extensionPr : L 0 Cap (T X) → L 0 m (T X). Property i) is clearly satisfied by (2.15). From the linearity of ∇ and∇, we deduce that property ii) holds for any simple function g ∈ L 0 (Cap), thus also for any g ∈ L 0 (Cap) by approximation. Finally, again by approximation we see that (2.14) follows from (2.16).
The fact that L 0 Cap (T X) can be thought of as a natural 'refinement' of the already known L 0 m (T X) is now encoded in the following proposition, which shows that L 0 m (T X),Pr is the canonical quotient of L 0 Cap (T X) up to m-a.e. equality (recall Proposition 2.2): The implication in (2.19) grants that this is a good definition, i.e. the value of S(v) depends only on v and not on how it is written as finite sum of the form i [ χ Ei ] m ∇f i . It is clear that S is linear and that, by (2.17) and item i) of Theorem 2.6, it holds (2.20) S(v) ≤ |v| m-a.e. for every v ∈ V.
In particular, S is 1-Lipschitz from V (with the L 0 m (T X)-distance) to N m . Since V is dense in L 0 m (T X), S can be uniquely extended to a continuous map -still denoted by S -from L 0 m (T X) to N m . Clearly such extension is linear and, by (2.20), it also satisfies S [ χ E ] m v = [ χ E ] m S(v) (e.g. by mimicking the arguments used in the proof of Proposition 2.2). These two facts easily imply L 0 (m)-linearity, thus showing existence of the desired map S. For uniqueness simply notice that the value of S on the dense subspace V of L 0 m (T X) is forced by the commutativity of the diagram in (2.18).

2.3.
Quasi-continuity of Sobolev vector fields on RCD spaces. Let (X, d, m) be an RCD(K, ∞) space, for some K ∈ R. The aim of this conclusive subsection is to prove that any element of the space H 1,2 C (T X) admits a quasi-continuous representative, in a suitable sense. We begin with the definition of quasi-continuous vector field on X: Definition 2.11 (Quasi-continuity for vector fields). We define the set TestV(X) ⊆ L 0 Cap (T X) as Then the space QC(T X) of quasi-continuous vector fields on X is defined as the d L 0 Cap (T X) -closure of TestV(X) in L 0 Cap (T X). It clearly holds that QC(T X) is a vector subspace of L 0 Cap (T X).
Proposition 2.12. Let v ∈ QC(T X) be given. Then it holds that |v| ∈ QC(X).
Proof. First of all, if v = n i=0 QCR(g i )∇f i ∈ TestV(X) then For general v ∈ QC(T X) we proceed by approximation: chosen any sequence (v n ) n ⊂ TestV(X) such that d L 0 Cap (T X) (v n , v) → 0, or equivalently d Cap |v n − v|, 0 → 0, we have that |v n | → |v| with respect to d Cap , whence accordingly |v| ∈ QC(X) by Proposition 1.17. Proof. Let v, w ∈ QC(T X) be such thatPr(v) =Pr(w). In other words, we have that Pr |v − w| (2.14) = P r(v − w) = 0 holds m-a.e. in X, whence Proposition 1.18 grants that |v − w| = 0 holds Cap-a.e. in X. This shows that v = w, thus proving the claim.
We are ready to state and prove the main result of the paper: any element of H 1,2 C (T X) admits a quasi-continuous representative in QC(T X). This is a generalisation of Theorem 1.20 to vector fields over an RCD space. Proof. Fix v ∈ H 1,2 C (T X). Pick (v n ) n ⊆ TestV(X) such that v n :=Pr(v n ) → v in W 1,2 C (T X). We know from Lemma 2.5 that |v n − v| ∈ W 1,2 (X) and D|v n − v| ≤ ∇(v n − v) HS m-a.e. for all n ∈ N, thus accordingly |v n − v| → 0 in W 1,2 (X) as n → ∞. Proposition 1.19 grants that -up to a (not relabeled) subsequence -we have that QCR |v n − v| → 0 locally quasi-uniformly as n → ∞, whence QCR |v n − v m | → 0 locally quasi-uniformly as n, m → ∞. Thus Proposition 1.17 yields Cap (T X) (v n ,v m ) = d Cap QCR |v n − v m | , 0 −→ 0 as n, m → ∞. This shows that (v n ) n ⊆ L 0 Cap (T X) is Cauchy, thus it converges to somev ∈ L 0 Cap (T X). Hence one hasPr(v) =Pr lim nvn = lim nP r(v n ) = lim n v n = v, so that we defineQ CR(v) :=v. Proposition 2.13 grants that the mapQ CR : H 1,2 C (T X) → QC(T X) is well-defined and is the unique map such thatPr •Q CR coincides with the inclusion H 1,2 C (T X) ⊂ L 0 m (T X). Finally, the last two statements follow from linearity ofPr, Theorem 1.20 and Proposition 2.13.
Remark 2.15. From the defining property ofQ CR and Propositions 2.9, 2.13 we see that QCR(∇f ) =∇f for every f ∈ Test(X). Then it is easy to see thatQ CR TestV(X) = TestV(X). Remark 2.16 (Alternative notion of quasi-continuous vector field). It is well-known that a vector field v : R n → R n in the Euclidean space is quasi-continuous if and only if R n ∋ x → v(x)−∇f (x) is quasi-continuous for every smooth function f : R n → R. This would suggest an alternative definition of quasi-continuous vector field on the RCD(K, ∞) space X: (2.23) QC(T X) := v ∈ L 0 Cap (T X) |v −∇f | ∈ QC(X) for every f ∈ Test(X) .
The well-posedness of the previous definition follows from the fact that quasi-continuity is preserved under modification on Cap-negligible sets. As we are going to show, it holds that (2.24) QC(T X) ⊂ QC(T X).
In order to prove it, let us fix v ∈ QC(T X). Given any f ∈ Test(X) and (v n ) n ⊂ TestV(X) such that v n → v in L 0 Cap (T X), we see (by arguing as in the proof of Lemma 2.12) that |v n −∇f | ∈ QC(X) holds for every n ∈ N, therefore also |v −∇f | ∈ QC(X) as an immediate consequence of the fact that |v n −∇f | → |v −∇f | in L 0 (Cap). Since f ∈ Test(X) is arbitrary, the claim (2.24) is proven.
Notice that due to the non-linearity of the defining condition (2.23) it is not clear if QC(T X) is a vector space or not. In particular, it is not clear if the inclusion in (2.24) can be strict.
We conclude giving a simple and explicit example to which the definitions and constructions presented in the paper can be applied: Example 2.17 (The case X = [0, 1]). Let us see how the definitions we gave work in the case X is the Euclidean segment [0, 1] equipped with its natural distance and measure. It is well known and easy to check that in this space every singleton has positive capacity. It follows that the space L 0 (Cap) coincides, as a set, with the space of all real valued Borel functions on X and similarly the space QC(X) coincides with the space of continuous functions on X. In particular, the quasi-continuous representative of a Sobolev function is, in fact, its continuous representative.
A direct verification of the definitions then shows that for f ∈ D(∆) ⊂ W 1,2 (X) we have f ′ ∈ W 1,2 (X) as well and, identifying f, f ′ with their continuous representatives, it also holds f ′ (0) = f ′ (1) = 0. In particular, for any f ∈ Test(X) we have that (the continuous representative of) f ′ is continuous and equal to 0 in {0, 1}.
We now claim that L 0 Cap (T X) is (=can be identified with) the space of Borel functions on [0, 1] which are 0 on {0, 1}, the corresponding gradient map∇ being the one which assigns to any f ∈ Test(X) the continuous representative of f ′ , which shall hereafter be denoted by f ′ . The verification of this claim follows from the above discussion and the uniqueness part of Theorem 2.6.
It is then clear that QC(T X) consists of continuous elements in L 0 Cap (T X), i.e. of continuous functions which are zero on {0, 1}, and that QC(T X) coincides with the space QC(T X) introduced in the previous remark.
This simple example shows that: a) The constant dimension property of RCD(K, N ) spaces recently obtained in [5], which is known to carry over to the 'standard' tangent module L 0 m (T X), does not carry over to the module L 0 Cap (T X) introduced in this manuscript: adapting the definitions in [9], one can see that in our example the dimension of L 0 Cap (T X) over {0, 1} is 0 and over (0, 1) is 1. b) The estimates obtained in [14] from which one can deduce that the capacity of the critical set of solutions of elliptic PDEs is 0, do not carry over to the RCD setting, and in fact not even in the setting of non-collapsed RCD spaces. Indeed, in our example the critical set of any function on X whose Laplacian is also a function (and not a measure) contains {0, 1}: the problem seems to be the presence of the 'boundary', see also [11] for further comments about the definition of boundary of a ncRCD space.