S ep 2 02 1 REPRESENTATION THEOREMS FOR NORMED MODULES

REPRESENTATION THEOREMS FOR NORMED MODULES arXiv:2109.03509v1 [math.FA] 8 Sep 2021 SIMONE DI MARINO, DANKA LUČIĆ, AND ENRICO PASQUALETTO Abstract. In this paper we study the structure theory of normed modules, which have been introduced by Gigli. The aim is twofold: to extend von Neumann’s theory of liftings to the framework of normed modules, thus providing a notion of precise representative of their elements; to prove that each separable normed module can be represented as the space of sections of a measurable Banach bundle. By combining our representation result with Gigli’s differential structure, we eventually show that every metric measure space (whose Sobolev space is separable) is associated with a cotangent bundle in a canonical way. Contents 1. Introduction 2. Preliminaries 2.1. Liftings of measurable functions 2.2. Measurable correspondences 2.3. Linear isometric embeddings of separable Banach spaces 2.4. Normed L0 (m)-modules 2.5. Normed L∞ (m)-modules 3. Liftings of normed modules 3.1. Definition of normed L∞ (Σ)-module. 3.2. Liftings of normed L∞ (m)-modules 3.3. Fibers of a normed L∞ (Σ)-module 4. Representation of normed modules via embedding 4.1. Separable Banach bundles 4.2. The section functor 4.3. Representation theorem Appendix A. Representation of normed modules via direct limits A.1. Separable Banach bundles A.2. Representation theorem Appendix B. Cotangent bundle on metric measure spaces Appendix C. The pullback bundle C.1. Pullback and section functors commute 2 5 5 7 9 11 16 17 18 19 22 24 24 28 30 33 34 35 37 39 39 Date: September 9, 2021. 2020 Mathematics Subject Classification. 53C23, 28A51, 46G15, 13C05, 18F15, 30L05. Key words and phrases. Normed module, measurable Banach bundle, lifting theory, Serre–Swan theorem. 1 2 SIMONE DI MARINO, DANKA LUČIĆ, AND ENRICO PASQUALETTO C.2. The projection operator Prϕ References 41 45 1. Introduction In recent years, a great deal of attention has been devoted to the study of weakly differentiable objects on abstract metric measure spaces. In this regard, an important contribution is represented by N. Gigli’s book [10], where he proposed a first-order differential calculus tailored to this framework. A key role was played by the notion of normed L0 -module, which has been subsequently refined in [11]; such notion is a variant of a similar concept introduced by N. Weaver in [27], who was in turn inspired by J.-L. Sauvageot’s papers [22,23]. A strictly related notion is that of ‘randomly normed space’, which was extensively studied in [18]. Let (X, d, m) be a metric measure space. We denote by L0 (m) the commutative ring of realvalued Borel functions on X, quotiented up to m-a.e. equality. Then a normed L0 (m)-module is an algebraic L0 (m)-module M equipped with a pointwise norm operator | · | : M → L0 (m), which induces a complete distance dM ; we refer to Definition 2.11 for the precise definition. Roughly speaking, a normed L0 (m)-module can be thought of as the space of measurable sections (up to m-a.e. equality) of some notion of measurable Banach bundle. Nevertheless, this interpretation might be not entirely correct, since it is currently unknown whether every normed L0 (m)-module actually admits this sort of representation. The main purpose of the present paper is, in fact, to show that all those normed L0 (m)-modules M for which the distance dM is separable can be written as spaces of sections of a separable Banach bundle. The result generalises a previous theorem, obtained by the second and third named authors in [20], for normed L0 (m)-modules that are ‘locally finitely-generated’, in a suitable sense. We now pass to a more detailed description of the contents of this manuscript. We point out that, even though in the rest of this introduction we will just focus (for simplicity) on the case of a metric measure space, most of the results can be formulated and proven in the more general framework of σ-finite measure spaces, as we will see later on in the paper. Liftings of normed modules. A propotypical example of normed L0 (m)-module is the space L0 (m) itself, which is generated by the function constantly equal to 1. Its key feature is that it is possible to ‘take precise representatives’ of its elements, by just considering Borel versions. A similar property is – a priori, at least – not shared by all normed L0 (m)-modules, which are intrinsically defined in the m-a.e. sense. This non-trivial issue needs to be addressed in order to be able to provide a representation of (separable) normed L0 (m)-modules as spaces of sections of measurable Banach bundles, since the latter certainly have this sort of property. The whole §3 is dedicated to achieve such a goal, as we are now going to describe. REPRESENTATION THEOREMS FOR NORMED MODULES 3 We denote by L∞ (Σ) the space of bounded, Σ-measurable, real-valued functions on X, where Σ stands for the completion of the Borel σ-algebra B(X). Then there exist linear continuous mappings L : L∞ (m) → L∞ (Σ), called liftings, which preserve products and select m-a.e. representatives; this is the statement of a highly non-trivial result by von Neumann– Maharam (cf. §2.1 below). A natural question arises: Can we generalise von Neumann–Maharam’s theorem to normed modules? (1.1) For technical reasons – namely, due to the fact that von Neumann–Maharam’s liftings cannot be defined on L0 (m), according to [25] – we need to work with normed modules over L∞ (m) and L∞ (Σ). The former have been introduced in [10], but essentially never used nor studied; we will investigate their properties in §2.5. The latter will be introduced and studied in §3.1. With these tools at our disposal, we will prove (in Theorem 3.5) that, given any normed L∞ (m)-module M , there exist a normed L∞ (Σ)-module M¯ and a lifting map L : M → M¯. (Notice that the space M¯ is not given a priori, but its existence is part of the statement.) Given that the elements of M¯ are ‘everywhere defined’, the above result shows that it is possible to select precise representatives of the elements of M , thus providing a positive answer to the question (1.1); the fact of working with normed L∞ (m)-modules rather than normed L0 (m)-modules is harmless, thanks to Lemma 2.24. Observe that liftings exist on every normed L∞ (m)-module, not just on the ones induced by a separable L0 (m)-module. Moreover, given any x ∈ X, we might consider the fiber M¯x := χ{x} · M¯ of M¯ at x, which turns out to be a Banach space. The properties of these fibers – that we will study in §3.3 – play an important role in §4, in the proof of the representation theorem for separable L0 (m)-modules. Representation theorems for normed modules. First of all, we propose in §4.1 a notion of separable Banach bundle: if B is a given universal separable Banach space – i.e., wherein all separable Banach spaces can be embedded linearly and isometrically – then we define the separable Banach B-bundles over X as those weakly measurable (multi-valued) mappings that associate to any x ∈ X a closed linear subspace of B; see Definition 4.1. Given a separable Banach B-bundle E, there is a natural way to define the space Γ(E) of its L0 (m)-sections; see Definition 4.2 and (4.7). Moreover, it is straightforward to check that the space Γ(E) is a separable normed L0 (m)-module, cf. Remark 4.3 and Lemma 4.9. On the other hand, the difficult task is actually to prove that every separable normed M is isomorphic to the space of measurable sections Γ(E) of some separable Banach B-bundle E over X. Let us briefly outline the strategy that we will adopt: L0 (m)-module • Consider the space R(M ) of all bounded elements of M , which is a normed L∞ (m)module. Fix a sequence (vn )n ⊆ R(M ) generating M and a lifting L : R(M ) → M¯. • For any x ∈ X and n ∈ N, we can evaluate L (vn ) at x, thus obtaining an element  L (vn )x of M¯x . The closure E(x) of L (vn )x : n ∈ N is a Banach subspace of M¯x .  • The resulting family E(x) x∈X is a measurable collection of separable Banach spaces; namely, even if the spaces E(x) live in different fibers M¯x , they depend on x in a measurable way, in a weak sense. This working definition is given in Definition 4.5. 4 SIMONE DI MARINO, DANKA LUČIĆ, AND ENRICO PASQUALETTO • The key step is to embed all the spaces E(x) in the ambient space B in a measurable fashion, thus obtaining a separable Banach B-bundle E. This technical result, achieved in Proposition 4.6 and Corollary 4.7, follows from a careful analysis of Banach–Mazur theorem (cf. §2.3), which states that C([0, 1]) is a universal separable Banach space. • The only remaining fact to check is that Γ(E) is isomorphic to M ; cf. Theorem 4.13. Once we have proven that each separable normed L0 (m)-module is the space of sections of some separable Banach B-bundle, it is natural to investigate the relation between bundles and modules at the level of categories. It turns out that the functor associating to any bundle its module of sections is an equivalence of categories: in analogy with the terminology used in [20], we refer to this result as the Serre–Swan theorem; see Theorem 4.15. In §C, we will introduce the concept of pullback bundle and investigate its relation with the pullback of a normed module, a notion extensively used in the field of nonsmooth differential geometry. The duality bundle-module described above resembles – and somehow extends – the theory of direct integrals of Hilbert spaces [26]. However, it is worth pointing out that our results provide a full correspondence between the ‘horizontal’ notion of normed module and the ‘(purely) vertical’ notion of Banach bundle, in the sense that we do not need to require the existence of a countable dense family of sections in order to obtain the Serre–Swan theorem. At a late stage of development of the present paper, we discovered that a comprehensive theory of measurable Banach bundles has been thoroughly studied by A.E. Gutman (see, e.g., the paper [17]). It would be definitely interesting – but outside the scopes of this manuscript – to investigate the relations between our approach and Gutman’s one. Finally, we will propose in §A an alternative (but less descriptive) method to represent separable normed L0 (m)-modules as spaces of sections of some bundle. More precisely, the idea is the following: since separable normed L0 (m)-modules can be approximated by finitelygenerated ones, a representation theorem can be achieved by using the variant of Serre–Swan theorem for finitely-generated modules that has been proven in [20]. See Theorem A.6 below. A notion of cotangent bundle on metric measure spaces. As already mentioned at the beginning of this introduction, the interest towards the theory of normed modules was mainly motivated by the development of a significant measure-theoretical tensor calculus on metric measure spaces. Although in this paper we just focus on ‘abstract’ normed modules, let us spend a few words about some possible applications of our results. Several (essentially) equivalent concepts of Sobolev space on (X, d, m) were introduced and studied in the last two decades, cf. [3, 5, 8, 24]. A common point of all these approaches is that each Sobolev function f is associated with a minimal object |Df |, which behaves more like the modulus of some weak differential of f than the differential itself. Due to this reason, Gigli proposed the theory of normed modules with the aim of providing a linear structure underlying the Sobolev calculus; in a few words, the idea was to define the differential df rather than the modulus of the differential. In this regard, the key object is the so-called REPRESENTATION THEOREMS FOR NORMED MODULES 5 cotangent module L0 (T∗ X), which can be thought of as the space of measurable 1-forms and contains the differentials of all Sobolev functions. The module L0 (T∗ X) has a rich and flexible functional-analytic structure, whence it plays a central role in many works (e.g., [14, 16]). In §B we propose an alternative viewpoint on the differential structure of a vast class of metric measure spaces (i.e., those having separable Sobolev space, which is a quite mild assumption). By combining our Serre–Swan Theorem 4.15 with the results proven in [10], we show in Theorem B.1 that (X, d, m) is canonically associated with a cotangent bundle T∗ X, which does not need the language of normed modules to be formulated. This way, at m-almost every point x ∈ X we obtain a weak notion of cotangent space T∗x X, which has a Banach space structure. However, in this manuscript we do not investigate further the possible geometric and analytic consequences of this more ‘concrete’ representation of the cotangent module. 2. Preliminaries In this section we collect many preliminary definitions and results, which will be needed in the sequel. The contents of §2.1, §2.2, and §2.3 are classical. The material in §2.4 is mostly taken from [10, 11], apart from a few technical statements. The discussion in §2.5 is new, but strongly inspired by already known results. Throughout the whole paper, we will always tacitly adopt the following convention: given any measurable space (X, Σ), it holds that {x} ∈ Σ for every x ∈ X. (2.1) In fact, this assumption plays a role only when considering fibers of normed L∞ (Σ)-modules (starting from §3.3). Notice that in the setting of metric measure spaces (cf. §B), the condition in (2.1) is always in force, since all singletons are closed and thus Borel measurable. 2.1. Liftings of measurable functions. Let us recall the concept of ‘lifting of a measure space’, in the sense of [9, Definition 341A]: Definition 2.1 (Lifting). Let (X, Σ, m) be a measure space. Then a map ℓ : Σ → Σ is said to be a lifting of the measure m provided the following properties are satisfied: i) ℓ is a Boolean homomorphism, i.e., it holds that ℓ(∅) = ∅, ℓ(X) = X, and ℓ(A∆B) =ℓ(A)∆ℓ(B), ℓ(A ∩ B) =ℓ(A) ∩ ℓ(B), for every A, B ∈ Σ, ii) ℓ(N ) = ∅ for every N ∈ Σ such that m(N ) = 0,
iii) m A∆ℓ(A) = 0 for every A ∈ Σ. Liftings can be proven to exist in high generality. For a proof of the following extremely deep result, we refer to [9, Theorem 341K]. Theorem 2.2 (von Neumann–Maharam). Let (X, Σ, m) be a complete, σ-finite measure space such that m(X) > 0. Then there exists a lifting ℓ : Σ → Σ of the measure m. 6 SIMONE DI MARINO, DANKA LUČIĆ, AND ENRICO PASQUALETTO Remark 2.3. We point out that Theorem 2.2 strongly relies upon the Axiom of Choice. Consequently, all our results concerning liftings of normed modules will rely upon the Axiom of Choice as well. Nevertheless, in the study of separable normed modules, the usage of  liftings (thus, of the Axiom of Choice) can be avoided; see Remark 4.14 for the details. Given any measure space (X, Σ, m), let us denote by L∞ (Σ) the space of all bounded measurable functions f¯: X → R, which is a vector space and a commutative ring with respect to the natural pointwise operations. It turns out that L∞ (Σ) is a Banach space and a topological ring when endowed with the norm kf¯kL∞ (Σ) := supX |f¯|. Consider the equivalence relation on L∞ (Σ) given by m-a.e. equality: for any f¯, ḡ ∈ L∞ (Σ), we set f¯ ∼m ḡ ⇐⇒ f¯(x) = ḡ(x) for m-a.e. x ∈ X. Then we denote by L∞ (m) the quotient space L∞ (Σ)/ ∼m . It holds that L∞ (m) is a Banach space and a topological ring when endowed with the quotient norm kf kL∞ (m) := ess supX |f |. We denote by [ · ]m : L∞ (Σ) → L∞ (m) the projection map, which turns out to be a linear and continuous operator. Let us define the family of simple functions Sf(Σ) ⊆ L∞ (Σ) as  X n ai χAi n ∈ N, (ai )ni=1 ⊆ R, (Ai )ni=1 ⊆ Σ partition of X . Sf(Σ) := i=1 Here, χA stands for the characteristic function of the set A ⊆ X, namely, we set χA (x) := 1 for every x ∈ A and χA (x) := 0 for every x ∈ X \ A. It can be readily proved that Sf(Σ) is a dense vector subspace and subring of L∞ (Σ). We define Sf(m) ⊆ L∞ (m) as the image of Sf(Σ) under the map [ · ]m , namely  X n n n χ := ai [ Ai ]m n ∈ N, (ai )i=1 ⊆ R, (Ai )i=1 ⊆ Σ partition of X . (2.2) Sf(m) i=1 Therefore, Sf(m) is a dense vector subspace and subring of L∞ (m). As we are going to prove, any lifting of a measure space can be promoted to a ‘lifting of measurable functions’. This statement is taken from [9, Exercise 341X(e)]: Theorem 2.4 (Lifting of measurable functions). Let (X, Σ, m) be a measure space and let ℓ : Σ → Σ be a lifting of the measure m. Then there exists a unique linear and continuous operator L : L∞ (m) → L∞ (Σ) such that
L [χA ]m = χℓ(A) for every A ∈ Σ. (2.3) Moreover, the following properties hold: i) L is an isometry, i.e., L(f ) L∞ (Σ) = kf kL∞ (m) for every f ∈ L∞ (m).
ii) L [c ]m = c for every constant c ∈ R.   iii) L is a right inverse of [ · ]m , i.e., L(f ) m = f for every f ∈ L∞ (m). iv) L(f g) = L(f ) L(g) for every f, g ∈ L∞ (m).
v) L(f ) = L |f | for every f ∈ L∞ (m). vi) If f, g ∈ L∞ (m) satisfy f ≥ g in the m-a.e. sense, then L(f ) ≥ L(g). REPRESENTATION THEOREMS FOR NORMED MODULES Proof. First of all, we are forced to define the operator L : Sf(m) → Sf(Σ) as follows: X  n n n X X ai [χAi ]m ∈ Sf(m). ai χℓ(Ai ) for every ai [χAi ]m := L (2.4) i=1 i=1 i=1 7 We need to prove the well-posedness of L. It thus remains to show that n X ai [χAi ]m = m X bj [χBj ]m n X ai χℓ(Ai ) = m X bj χℓ(Bj ) . (2.5) j=1 i=1 j=1 i=1 =⇒ The left-hand side of (2.5) is equivalent to saying that m(Ai ∩ Bj ) = 0 for all i, j with ai 6= bj ; this implies that ℓ(Ai ) ∩ ℓ(Bj ) = ℓ(Ai ∩ Bj ) = ∅ for any such i, j, which is equivalent to the right-hand side, whence (2.5) is proven. Moreover, it can be readily checked that L is linear. P P Given any simple function f = ni=1 ai [χAi ]m ∈ Sf(m), we have that |f | = ni=1 |ai | [χAi ]m Pn m-a.e. and L(f ) = i=1 |ai | χℓ(Ai ) , thus accordingly L(f ) L∞ (Σ) = sup L(f ) = max |ai | = max |ai | = ess sup |f | = kf kL∞ (m) . X i=1,...,n: ℓ(Ai )6=∅ i=1,...,n: m(Ai )>0 X

Then the map L is an isometry from Sf(m), k · kL∞ (m) to Sf(Σ), k · kL∞ (Σ) , so it can be uniquely extended to a linear isometry L : L∞ (m) → L∞ (Σ). This proves existence, uniqueness, and item i). Item ii) immediately follows from the 1-homogeneity of L and (2.3).   To prove item iii), it suffices to observe that L(f ) m = f holds for every f ∈ Sf(m) by construction, thus also for any f ∈ L∞ (m) by continuity of [ · ]m and L. Item iv) can be P P χ proved for f, g ∈ Sf(m) by direct computation: if f = ni=1 ai [χAi ]m and g = m j=1 bj [ Bj ]m , P χ then f g = i,j ai bj [ Ai ∩Bj ]m , so that accordingly   X X m n X bj χℓ(Bj ) = L(f ) L(g). ai χℓ(Ai ) L(f g) = ai bj χℓ(Ai ∩Bj ) = i,j i=1 j=1 Since L∞ (m) and L∞ (Σ) are topological rings, we deduce from the density of Sf(m) in L∞ (m) that L(f g) = L(f ) L(g) holds for every f, g ∈ L∞ (m), thus proving item iv) in full generality. Item v) can be readily checked for f ∈ Sf(m), whence the general case follows from an approximation argument. Finally, let us prove item vi). By linearity of L, it suffices to consider f ∈ L∞ (m) such that f ≥ 0 in the m-a.e. sense. Choose any sequence (fn )n ⊆ Sf(m) such that fn ≥ 0 holds m-a.e. for each n ∈ N and limn kfn − f kL∞ (m) = 0. Given that L is continuous and L(fn ) ≥ 0 holds for all n ∈ N by construction, we conclude that L(f ) ≥ 0, as it is a uniform limit of non-negative functions. The proof of the statement is complete.  2.2. Measurable correspondences. Aim of this subsection is to recall some basic definitions and results concerning measurable correspondences. The whole material we are going to discuss can be found, e.g., in [1]. Let (X, Σ) be a measurable space and let (Y, dY ) be a separable metric space. Then any map ϕ : X → 2Y is said to be a correspondence from X to Y and is denoted by ϕ : X ։ Y. Definition 2.5 (Measurable correspondence). A correspondence ϕ : X ։ Y is said to be: 8 SIMONE DI MARINO, DANKA LUČIĆ, AND ENRICO PASQUALETTO  a) weakly measurable, provided x ∈ X : ϕ(x) ∩ U 6= ∅ ∈ Σ for every U ⊆ Y open,  b) measurable, provided x ∈ X : ϕ(x) ∩ C 6= ∅ ∈ Σ for every C ⊆ Y closed. Let us collect a few important properties of (weakly) measurable correspondences: i) Every measurable correspondence from X to Y is weakly measurable. Conversely, every weakly measurable correspondence from X to Y with compact values is measurable. ii) Let ϕ : X ։ Y be a single-valued correspondence, i.e., for every x ∈ X there exists an  element ϕ̄(x) ∈ Y such that ϕ(x) = ϕ̄(x) . Then ϕ is a measurable correspondence if and only if ϕ̄ : X → Y is a measurable map. iii) Let ϕ, ϕ′ : X ։ Y be two measurable correspondences with compact values. Then the intersection correspondence ϕ ∩ ϕ′ : X ։ Y, defined as (ϕ ∩ ϕ′ )(x) := ϕ(x) ∩ ϕ′ (x) for every x ∈ X, is measurable. iv) A correspondence ϕ : X ։ Y with non-empty values is weakly measurable if and only
if X ∋ x 7→ dY y, ϕ(x) is a measurable function for every y ∈ Y. v) Let f : X × Y → R be a Carathéodory function, i.e., f (·, y) : X → R is measurable for every y ∈ Y, f (x, ·) : Y → R is continuous for every x ∈ X.  Define the correspondence ϕ : X ։ Y as ϕ(x) := y ∈ Y : f (x, y) = 0 for every x ∈ X. If the metric space (Y, dY ) is compact, then ϕ is a measurable correspondence. vi) Let ϕ : X ։ Y be a weakly measurable correspondence with closed values. Then the graph of ϕ is measurable, i.e., it holds that  (x, y) ∈ X × Y y ∈ ϕ(x) ∈ Σ ⊗ B(Y). vii) Kuratowski–Ryll-Nardzewski theorem. Suppose that the metric space (Y, dY ) is complete. Let ϕ : X ։ Y be a weakly measurable correspondence with non-empty closed values. Then ϕ admits a measurable selector s : X → Y, i.e., the map s is measurable and satisfies s(x) ∈ ϕ(x) for every x ∈ X. viii) Let ϕ : X ։ Y be a weakly measurable correspondence. Then its closure correspon
dence clY (ϕ) : X ։ Y, which is defined as clY (ϕ)(x) := clY ϕ(x) for every x ∈ X, is weakly measurable. Furthermore, we will also need the following standard results about preimages and compositions of measurable correspondences: Lemma 2.6. Let (X, Σ) be a measurable space. Let (Y, dY ), (Z, dZ ) be separable metric spaces, with (Y, dY ) compact. Let ϕ : X ։ Z be a measurable correspondence and let ψ : Y → Z be a continuous map. Define the preimage correspondence ψ −1 (ϕ) : X ։ Y as
ψ −1 (ϕ)(x) := ψ −1 ϕ(x) ⊆ Y for every x ∈ X. Then ψ −1 (ϕ) is a measurable correspondence. REPRESENTATION THEOREMS FOR NORMED MODULES 9 Proof. Let C ⊆ Y be a closed set. In particular, C is compact, whence ψ(C) ⊆ Z is compact (and thus closed) by continuity of ψ. Therefore, the measurability of ϕ grants that   x ∈ X ψ −1 (ϕ)(x) ∩ C 6= ∅ = x ∈ X ϕ(x) ∩ ψ(C) 6= ∅ ∈ Σ. By arbitrariness of C, we conclude that ψ −1 (ϕ) is a measurable correspondence.  Lemma 2.7. Let (X, ΣX ), (Y, ΣY ) be measurable spaces and (Z, dZ ) a separable metric space. Let u : X → Y be a measurable map and ϕ : Y ։ Z a weakly measurable correspondence. Consider the correspondence ϕ ◦ u : X ։ Z, given by
(ϕ ◦ u)(x) := ϕ u(x) ⊆ Z for every x ∈ X. Then ϕ ◦ u is a weakly measurable correspondence.  Proof. Let U ⊆ Z be an open set. Since y ∈ Y : ϕ(y) ∩ U 6= ∅ ∈ ΣY and u is measurable, it holds that  
x ∈ X (ϕ ◦ u)(x) ∩ U 6= ∅ = u−1 y ∈ Y ϕ(y) ∩ U 6= ∅ ∈ ΣX . By arbitrariness of U , we conclude that ϕ ◦ u is a weakly measurable correspondence.  2.3. Linear isometric embeddings of separable Banach spaces. In this section we collect some results about linear isometric embeddings of Banach spaces.  Given any Banach space E, we shall denote by BE its closed unit ball v ∈ E : kvkE ≤ 1 . We use the notation E ′ to denote the continuous dual space of E, which is a Banach space. We begin by recalling some classical definitions and results (which can be found, e.g., in [1]): i) Cantor set. The Cantor set is the product ∆ := {0, 2}N , where each factor {0, 2} is endowed with the discrete topology. The topology of ∆ is induced by the distance d∆ (a, b) := ∞ X |an − bn | n=1 3n for every a = (an )n , b = (bn )n ∈ ∆. It holds that the Cantor set ∆ is compact. Moreover, ∆ is homeomorphic to the closed  P∞ P∞ n n subset C := n=1 an /3 : a ∈ ∆ of [0, 1] via the map ∆ ∋ a 7→ n=1 an /3 ∈ C. ii) Let K be a non-empty closed subset of ∆. Given any a ∈ ∆, there exists a unique
element r(a) ∈ K such that d∆ a, r(a) = d∆ (a, K). The resulting map r : ∆ → K is continuous and satisfies r|K = idK . We say that r is a retraction of ∆ onto K. iii) Hilbert cube. The Hilbert cube is the product topological space I ∞ := [−1, 1]N . It is compact and its topology is induced by the distance d I∞ (α, β) := ∞ X |αk − βk | k=1 2k for every α = (αk )k , β = (βk )k ∈ I ∞ . Moreover, there exists a continuous surjective map ψ : ∆ → I ∞ . 10 SIMONE DI MARINO, DANKA LUČIĆ, AND ENRICO PASQUALETTO iv) Let E be a separable Banach space. Let (vk )k∈N be a fixed dense subset of BE .
Then the map ι : BE ′ → I ∞ , defined as ι(ω) := ω[vk ] k for every ω ∈ BE ′ , is a homeomorphism with its image (when the domain BE ′ is endowed with the restricted weak∗ topology). v) Let K be a compact Hausdorff topological space. Then the space C(K) of real-valued continuous functions on K is a separable Banach space if endowed with the norm kgkC(K) := sup g(t) for every g ∈ C(K). t∈K We now recall the classical Banach–Mazur theorem [7], which states that any separable Banach space can be embedded linearly and isometrically into C([0, 1]). We also report the proof of this result, as the explicit construction of such an embedding will be needed in §4. Theorem 2.8 (Banach–Mazur). Let E be a separable Banach space. Then there exists a linear isometric embedding I : E → C([0, 1]). Proof. Let ι : BE ′ → I ∞ be a continuous embedding as in item iv). Let ψ : ∆ → I ∞ be a
continuous surjection as in item iii). Note that ψ −1 ι(BE ′ ) ⊆ ∆ is closed (thus compact) by continuity of ψ and weak∗ compactness of BE ′ (the latter is granted by Banach–Alaoglu
theorem). Then consider a retraction r : ∆ → ψ −1 ι(BE ′ ) as in item ii). We claim that the operator I′ : E → C(∆), which is defined as I′ [v](a) := (ι−1 ◦ ψ ◦ r)(a)[v] for every v ∈ E and a ∈ ∆, is well-posed, linear, and isometric. The fact that I′ [v] ∈ C(∆) for every v ∈ E can be easily checked: if a sequence (ai )i ⊆ ∆ converges to some element a ∈ ∆, then (ι−1 ◦ ψ ◦ r)(ai ) converges to (ι−1 ◦ ψ ◦ r)(a) in the weak∗ topology, and accordingly I′ [v](a) = limi I′ [v](ai ). Linearity of I′ immediately follows from the definition. Moreover, it holds that I′ [v] C(∆) = sup I′ [v](a) = sup ω[v] = kvkE a∈∆ for every v ∈ E, ω∈BE ′ where the last equality follows from the fact that the canonical embedding of the space E into its bidual E ′′ is an isometric operator. This shows that the map I′ is an isometry. Finally, denote by h : C → ∆ the homeomorphism described in item i) and write [0, 1] \ C F as a disjoint union i∈N (ti , si ). We define the map e : C(∆) → C([0, 1]) in the following way: given any g ∈ C(∆), we set ( (g ◦ h)(t) if t ∈ C, e(g)(t) := t−ti (g ◦ h)(ti ) + si −ti (g ◦ h)(si ) if t ∈ (ti , si ) for some i ∈ N. It holds that e is linear and isometric. Then the map I : E → C([0, 1]), given by I := e ◦ I′ , is a linear isometric embedding as well. Therefore, the statement is achieved.  The statement of Theorem 2.8 can be reformulated by using the following definition: REPRESENTATION THEOREMS FOR NORMED MODULES 11 Definition 2.9 (Universal separable Banach space). A separable Banach space B is said to be a universal separable Banach space (up to linear isometry) provided for any separable Banach space E there exists a linear isometric embedding I : E → B. Therefore, Theorem 2.8 reads as follows: C([0, 1]) is a universal separable Banach space. Remark 2.10. As a byproduct of the proof of Theorem 2.8, we see that C(∆) is a universal separable Banach space.  2.4. Normed L0 (m)-modules. Let (X, Σ, m) be a given σ-finite measure space. We shall denote by L0 (m) the space of all equivalence classes (up to m-a.e. equality) of measurable functions from X to R, which is a vector space and a commutative ring with respect to the natural pointwise operations. Given a probability measure m′ on (X, Σ) satisfying m ≪ m′ ≪ m – for instance, pick any S sequence (An )n ⊆ Σ such that 0 < m(An ) < +∞ for every n ∈ N and X = n∈N An , and P m|An on X – we define the complete distance dL0 (m) as consider the measure m′ := n∈N 2n m(A n) ˆ dL0 (m) (f, g) := |f − g| ∧ 1 dm′ for every f, g ∈ L0 (m). It turns out that L0 (m) is a topological vector space and a topological ring when endowed with the distance dL0 (m) . The distance dL0 (m) depends on the chosen measure m′ , but its induced topology does not. Observe also that the space L∞ (m) is dL0 (m) -dense in L0 (m). We recall the notion of normed L0 (m)-module, which has been introduced by Gigli in [10]: Definition 2.11 (Normed L0 (m)-module). Let (X, Σ, m) be a σ-finite measure space. Then
a normed L0 (m)-module is a couple M , | · | with the following properties: i) M is an algebraic L0 (m)-module. ii) The map | · | : M → L0 (m), which is called a pointwise norm on M , satisfies |v| ≥ 0 |v + w| ≤ |v| + |w| |f · v| = |f ||v| for every v ∈ M , with equality if and only if v = 0, for every v, w ∈ M , (2.6) for every f ∈ L0 (m) and v ∈ M , where all (in)equalities are intended in the m-a.e. sense. iii) The distance dM on M associated with | · |, which is defined as
for every v, w ∈ M , dM (v, w) := dL0 (m) |v − w|, 0 is complete. Remark 2.12 (Locality/glueing property). Let (X, Σ, m) be a σ-finite measure space. Then any normed L0 (m)-module M has the following properties: • Locality. If (An )n∈N ⊆ Σ is a partition of X and v, w ∈ M are two elements such that [χAn ]m · v = [χAn ]m · w holds for every n ∈ N, then v = w. 12 SIMONE DI MARINO, DANKA LUČIĆ, AND ENRICO PASQUALETTO • Glueing. If (An )n∈N ⊆ Σ is a partition of X and (vn )n∈N is any sequence in M , then there exists v ∈ M such that [χAn ]m · v = [χAn ]m · vn for every n ∈ N. The element v – which is uniquely determined by the locality property – will be denoted
PN P P χ χ by n∈N [χAn ]m · vn . It holds dM n=1 [ An ]m · vn , n∈N [ An ]m · vn → 0 as N → ∞. Both the locality property and the glueing property have been proved in [10].  Let us fix some useful notation about normed L0 (m)-modules: a) Morphism. A map Φ : M → N between two normed L0 (m)-modules M , N is said to be a morphism provided it is an L0 (m)-linear contraction, i.e., Φ(f · v + g · w) = f · Φ(v) + g · Φ(w) Φ(v) ≤ |v| m-a.e. for all f, g ∈ L0 (m) and v, w ∈ M , for all v ∈ M . This allows us to speak about the category NMod(X, Σ, m) of normed L0 (m)-modules. b) Dual. The dual of M is defined as the space M ∗ of all L0 (m)-linear continuous maps from M to L0 (m). It holds that M ∗ is a normed L0 (m)-module if endowed with the natural operations and the pointwise norm  |ω| := ess sup ω(v) v ∈ M , |v| ≤ 1 m-a.e. ∈ L0 (m) for all ω ∈ M ∗ . (2.7) c) Generators. We say that a set S ⊆ M generates M on some set A ∈ Σ provided the smallest (algebraic) L0 (m)-module containing [χA ]m · S is dense in [χA ]m · M . d) Linear independence. Some elements v1 , . . . , vn ∈ M are said to be independent P on a set A ∈ Σ provided for any f1 , . . . , fn ∈ L0 (m) one has ni=1 [χA ]m fi · vi = 0 if and only if f1 , . . . , fn = 0 holds m-a.e. on A. e) Local basis. We say that v1 , . . . , vn ∈ M constitute a local basis for M on A provided they are independent on A and {v1 , . . . , vn } generates M on A. f) Local dimension. The module M has local dimension equal to n ∈ N on A if it admits a local basis on A made of exactly n elements. g) Dimensional decomposition. It holds that the normed L0 (m)-module M admits a unique dimensional decomposition {Dn }n∈N∪{∞} ⊆ Σ, i.e., M has local dimension n on Dn for every n ∈ N and is not finitely-generated on any measurable subset of D∞ having positive m-measure. Uniqueness here is intended up to m-negligible sets. h) Proper module. A normed L0 (m)-module M , whose dimensional decomposition is denoted by {Dn }n∈N∪{∞} , is said to be proper provided m(D∞ ) = 0. We refer to [10, 11] for a thorough discussion about normed L0 (m)-modules. Definition 2.13 (Countably-generated module). Let (X, Σ, m) be any σ-finite measure space. Let M be a normed L0 (m)-module. Then we say that M is countably-generated provided there exists a countable family C ⊆ M that generates M (on X). We call NModcg (X, Σ, m) the category of countably-generated normed L0 (m)-modules. Another class of modules we are interested in is that of separable normed L0 (m)-modules, i.e., those normed L0 (m)-modules M for which (M , dM ) is a separable metric space. We denote REPRESENTATION THEOREMS FOR NORMED MODULES 13 by NMods (X, Σ, m) the category of separable normed L0 (m)-modules. In the forthcoming discussion, we investigate the relation between countably-generated and separable modules. Let (X, Σ, m) be a σ-finite measure space. Given any A, B ∈ Σ, we declare that A ∼m B if and only if m(A∆B) = 0. This way, we obtain an equivalence relation ∼m on Σ. Given any finite measure m′ on (X, Σ) such that m ≪ m′ ≪ m (thus in particular ∼m′ and ∼m coincide), we define the distance dm′ on the quotient set Σ/ ∼m as
dm′ [A]∼m , [B]∼m := m′ (A∆B) for every [A]∼m , [B]∼m ∈ Σ/ ∼m . Then we say that the measure space (X, Σ, m) is separable provided (Σ/ ∼m , dm′ ) is a separable metric space for some finite measure m′ on (X, Σ) such that m ≪ m′ ≪ m. Lemma 2.14. Let (X, Σ, m) be a given σ-finite measure space. Then (X, Σ, m) is a separable
measure space if and only if L0 (m), dL0 (m) is separable. Proof. Necessity. Suppose (X, Σ, m) is separable. Choose any probability measure m′ on (X, Σ) such that m ≪ m′ ≪ m and dm′ is separable. Observe that L0 (m′ ) and L0 (m) coincide. Fix a countable dm′ -dense subset C of Σ/ ∼m . Now consider f ∈ L0 (m) and ε > 0. As already observed, we can find a simple function g ∈ Sf(m) – see (2.2) – such that dL0 (m) (f, g) < ε/2. P Say that g = ni=1 ai [χAi ]m . Without loss of generality, we can assume that a1 , . . . , an ∈ Q.
For any i = 1, . . . , n, pick a set Bi ∈ Σ such that [Bi ]∼m ∈ C and m′ (Bi ∆Ai ) ≤ ε/ 2 |ai | n . Therefore, it holds that  X  X ˆ n n n X ε ′ χ χ χ |ai | m′ (Ai ∆Bi ) ≤ . [ Ai ]m − [ Bi ]m dm = |ai | ai [ Bi ]m ≤ dL0 (m) g, 2 i=1 i=1 i=1 Pn This means that the function h := i=1 ai [χBi ]m satisfies dL0 (m) (f, h) < ε. Given that  X n n n ai [χBi ]m n ∈ N, (ai )i=1 ⊆ Q, (Bi )i=1 ⊆ C i=1 is a countable family, we conclude that L0 (m) is separable, as desired. Sufficiency. Suppose L0 (m) is separable. Observe that the map
(Σ/ ∼m , dm′ ) ∋ [A]∼m 7−→ [χA ]m ∈ L0 (m), dL0 (m) is an isometry. Consequently, we conclude that (Σ/ ∼m , dm′ ) is separable, as required.  Observe that, trivially, any separable normed L0 (m)-module M is countably-generated. The following result aims at determining in which cases the converse implication is satisfied. Proposition 2.15. Let (X, Σ, m) be any σ-finite measure space. Then a countably-generated normed L0 (m)-module M is separable if and only if (X, Σ, m|X\D0 ) is a separable measure space, where {Dn }n∈N∪{∞} ⊆ Σ stands for the dimensional decomposition of the module M . In particular, if (X, Σ, m) is a separable measure space, then any countably-generated normed L0 (m)-module is separable. 14 SIMONE DI MARINO, DANKA LUČIĆ, AND ENRICO PASQUALETTO Proof. Suppose (X, Σ, m|X\D0 ) is separable. Let C be a countable set generating M , i.e., X n fi · vi n ∈ N, (fi )ni=1 ⊆ L0 (m|X\D0 ), (vi )ni=1 ⊆ C i=1  is dense in M . (2.8) Lemma 2.14 ensures that the space L0 (m|X\D0 ) is separable, thus we can pick a countable dense subset D of L0 (m|X\D0 ). Since the multiplication map L0 (m) × M ∋ (f, v) 7→ f · v ∈ M is continuous, we immediately conclude from (2.8) that the countable family X n fi · vi n ∈ N, (fi )ni=1 ⊆ D, (vi )ni=1 ⊆ C i=1  is dense in M . This proves that the metric space (M , dM ) is separable, as desired. Conversely, suppose M is separable. It can be readily checked that there exists v ∈ M such that |v| = 1 holds m-a.e. on X\D0 . (Notice that [χD0 ]m ·v = 0 by definition of D0 .) Therefore, it holds that the module generated by {v} can be identified with L0 (m|X\D0 ) (considered as a normed L0 (m)-module), whence the space L0 (m|X\D0 ) is separable. We can conclude that (X, Σ, m|X\D0 ) is separable by Lemma 2.14, thus completing the proof.  In §A we shall need the concept of direct limit of normed L0 (m)-modules. The following result – which is taken from [21, Theorem 2.1] – states that direct limits always exist in the category of normed L0 (m)-modules. (Actually, the statement below was proven in the case in which X is a Polish space, Σ is the Borel σ-algebra of X, and m is a Radon measure on X; however, the very same proof works in the more general case of a σ-finite measure space.) Theorem 2.16 (Direct limits of normed L0 -modules). Let (X, Σ, m) be any σ-finite measure
space. Let {Mi }i∈I , {ϕij }i≤j be a direct system of normed L0 (m)-modules, i.e., i) (I, ≤) is a directed set, ii) {Mi : i ∈ I} is a family of normed L0 (m)-modules, iii) {ϕij : i, j ∈ I, i ≤ j} is a family of normed L0 (m)-module morphisms ϕij : Mi → Mj such that ϕii = idMi for all i ∈ I and ϕik = ϕjk ◦ ϕij for all i, j, k ∈ I with i ≤ j ≤ k.
Then there exists a unique couple lim M⋆ , {ϕi }i∈I , where lim M⋆ is a normed L0 (m)-module −→ −→ and each map ϕi : Mi → lim M⋆ is a normed L0 (m)-module morphism, such that: −→

a) lim M⋆ , {ϕi }i∈I is a target for {Mi }i∈I , {ϕij }i≤j , i.e., −→ Mi ϕij ϕi Mj ϕj lim M⋆ −→ is a commutative diagram for every i, j ∈ I with i ≤ j. REPRESENTATION THEOREMS FOR NORMED MODULES 15

b) Given any other target N , {ψi }i∈I for {Mi }i∈I , {ϕij }i≤j , there exists a unique normed L0 (m)-module morphism Φ : lim M⋆ → N such that −→ Mi ϕi lim M⋆ −→ Φ ψi N is a commutative diagram for every i ∈ I. A simple example of direct limit of normed L0 (m)-modules is given by the ensuing result: Lemma 2.17. Let (X, Σ, m) be a σ-finite measure space. Let M be a normed L0 (m)-module. S Let (Mn )n be an increasing sequence of normed L0 (m)-submodules of M such that n∈N Mn is dense in M . Call ιnm : Mn ֒→ Mm the inclusion map for every n, m ∈ N with n ≤ m.
Then {Mn }n∈N , {ιnm }n≤m is a direct system of normed L0 (m)-modules and lim M⋆ ∼ = M. −→
Proof. Calling ιn : Mn ֒→ M the inclusion map for any n ∈ N, we see that M , {ιn }n∈N is

a target for {Mn }n∈N , {ιnm }n≤m . Moreover, fix any other target N , {ψn }n∈N . Since the S vector space n∈N ιn (Mn ) is dM -dense in M by assumption, there clearly exists a unique linear continuous map Φ : M → N such that Φ(v) = ψn (v) for every n ∈ N and v ∈ Mn . Finally, this map can be readily proven to be a morphism of normed L0 (m)-modules.  We conclude the subsection by reminding the key notion of pullback of a normed L0 -module: Theorem 2.18 (Pullback of normed L0 -modules). Let (X, ΣX , mX ) and (Y, ΣY , mY ) be σfinite measure spaces. Let ϕ : X → Y be a measurable map satisfying ϕ∗ mX ≪ mY . Let M be a normed L0 (mY )-module. Then there exists a unique couple (ϕ∗ M , ϕ∗ ) such that ϕ∗ M is a normed L0 (mX )-module and ϕ∗ : M → ϕ∗ M is a linear map with the following properties: i) It holds that |ϕ∗ v| = |v| ◦ ϕ in the mX -a.e. sense for every v ∈ M . ii) The family {ϕ∗ v : v ∈ M } generates ϕ∗ M . Uniqueness has to be intended up to unique isomorphism: given any other couple (N , T ) satisfying the same properties, there exists a unique isomorphism Φ : ϕ∗ M → N of normed L0 (mX )-modules such that M ϕ∗ T ϕ∗ M Φ N is a commutative diagram. The pullback of a normed module has been introduced in [10], but the variant for normed L0 -modules presented above has been considered in [15] and [6]. 16 SIMONE DI MARINO, DANKA LUČIĆ, AND ENRICO PASQUALETTO 2.5. Normed L∞ (m)-modules. We recall the notion of normed L∞ (m)-module, which has been introduced in [10]: Definition 2.19 (Normed L∞ (m)-module). Let (X, Σ, m) be any σ-finite measure space.
Then a normed L∞ (m)-module is a couple M , | · | having the following properties: i) M is an algebraic L∞ (m)-module. ii) The map | · | : M → L∞ (m), which is called a pointwise norm on M , satisfies |v| ≥ 0 |v + w| ≤ |v| + |w| |f · v| = |f ||v| for every v ∈ M , with equality if and only if v = 0, for every v, w ∈ M , (2.9) ∞ for every f ∈ L (m) and v ∈ M , where all (in)equalities are intended in the m-a.e. sense. iii) The space M has the glueing property, i.e., given a partition (An )n∈N ⊆ Σ of X and a sequence (vn )n∈N ⊆ M with supn ess supAn |vn | < +∞, there is (a unique) v ∈ M such P that [χAn ]m ·v = [χAn ]m ·vn for all n ∈ N. The element v is denoted by n∈N [χAn ]m ·vn . iv) The norm k · kM on M associated with | · |, which is defined as kvkM := |v| L∞ (m) for every v ∈ M , is complete. We point out that in iii) we do not require limN PN χ n=1 [ An ]m ·vn − P χ n∈N [ An ]m ·vn M = 0. Remark 2.20. The above notion of normed L∞ (m)-module is equivalent to the concept of L∞ (m)-normed L∞ (m)-module introduced in [10]. Here we do not specify – for the sake of brevity – that the pointwise norm operator takes values into the space L∞ (m), the reason being that in the present manuscript this is the only type of pointwise norm we are going to consider over L∞ (m)-modules. However, in [10, 11] also Lp (m)-normed L∞ (m)-modules, for any exponent p ∈ [1, ∞), are studied.  Remark 2.21 (Locality property). It can be readily checked that normed L∞ (m)-modules have the locality property: if (An )n∈N ⊆ Σ is a partition of X and v, w ∈ M are two elements such that [χAn ]m · v = [χAn ]m · w for every n ∈ N, then v = w. This ensures that in item iii) P of Definition 2.19 the element n∈N [χAn ]m · vn is uniquely determined.  Remark 2.22 (Lack of glueing). We point out that item iii) of Definition 2.19 is not granted P by i), ii), and iv). For instance, let us consider the Radon measure m := n∈N δn on N and the space c0 of all real-valued sequences that converge to 0, which is an algebraic module over
the ring ℓ∞ ∼ = L∞ (m). We define the pointwise norm | · | : c0 → ℓ∞ as (an )n := |an | n for every (an )n ∈ c0 , which clearly satisfies items ii) and iv). Nevertheless, the glueing property fails: the elements en := (δin )i∈N with n ∈ N belong to c0 , but by ‘glueing’ them we would obtain the sequence constantly equal to 1, which is not an element of c0 .  We now aim to investigate the relation between normed L∞ (m)-modules and normed For the sake of clarity, we denote by M ∞ the former, by M 0 the latter. L0 (m)-modules. REPRESENTATION THEOREMS FOR NORMED MODULES 17 Definition 2.23 (Completion/restriction). Let (X, Σ, m) be a σ-finite measure space. i) Completion. Let M ∞ be a normed L∞ (m)-module. Then its completion C(M ∞ )
is defined as the metric completion of M ∞ , dC(M ∞ ) , where the distance dC(M ∞ ) is
given by dC(M ∞ ) (v, w) := dL0 (m) |v − w|, 0 for every v, w ∈ M ∞ . ii) Restriction. Let M 0 be a normed L0 (m)-module. Then its restriction R(M 0 ) is  defined as R(M 0 ) := v ∈ M 0 : |v| ∈ L∞ (m) . It can be readily checked – by arguing as in [11, Theorem/Definition 2.7] – that C(M ∞ ) inherits a normed L0 (m)-module structure. Moreover, R(M 0 ) is a normed L∞ (m)-module. The following result says that ‘the completion map is the inverse of the restriction map’: Lemma 2.24 (‘C = R−1 ’). Let (X, Σ, m) be a σ-finite measure space. Then it holds that:
i) C R(M 0 ) ∼ = M 0 for every normed L0 (m)-module M 0 ,
ii) R C(M ∞ ) ∼ = M ∞ for every normed L∞ (m)-module M ∞ . Proof.
i) Let M 0 be a normed L0 (m)-module. Observe that C R(M 0 ) can be identified with the dM 0 -closure of R(M 0 ) in M 0 , thus to conclude it suffices to show that R(M 0 ) is dM 0 -dense  in M 0 . To this aim, fix v ∈ M 0 and call An := |v| ≤ n for every n ∈ N. Hence, it clearly
holds that [χAn ]m · v n ⊆ R(M 0 ) and limn dM 0 [χAn ]m · v, v) = 0, as required. ii) Let M ∞ be a normed L∞ (m)-module. We call ι the L∞ (m)-linear isometric embedding
of (M ∞ , dC(M ∞ ) ) into C(M ∞ ). Notice that ι(M ∞ ) ⊆ R C(M ∞ ) by definition of R. To

conclude, it is enough to prove that actually ι(M ∞ ) = R C(M ∞ ) . Fix any w ∈ R C(M ∞ )
and ε > 0. Pick a sequence (vn )n ⊆ M ∞ with limn dC(M ∞ ) ι(vn ), w = 0. By using Egorov theorem, we can find a partition (Ai )i∈N ⊆ Σ of X and a sequence (ni )i∈N ⊆ N such that ess sup ι(vni ) − w ≤ ε for every i ∈ N. Ai Notice that supi ess supAi |vni | ≤ ε + ess supX |w| < +∞, thus the glueing property of M ∞ P P grants the existence of v := i∈N [χAi ]m · vni . It holds that ki=1 [χAi ]m · vni → v as k → ∞ with respect to the distance dC(M ∞ ) (but not with respect to k · kM ∞ , in general). This
P Pk χ ensures that ki=1 [χAi ]m · ι(vni ) = ι i=1 [ Ai ]m · vni → ι(v) as k → ∞, thus accordingly
the inequality ι(v) − w ≤ ε holds m-a.e.. We conclude that ι(M ∞ ) = R C(M ∞ ) .  Let us mention that both the correspondences M ∞ 7→ C(M ∞ ) and M 0 7→ R(M 0 ) can be made into functors, which turn out to be equivalences of categories – one the inverse of the other. We omit the details, referring to [20, Appendix B] for a similar discussion. 3. Liftings of normed modules Aim of this section is to generalise the theory of liftings to the setting of normed modules. In §3.1 we introduce and study a notion of normed module over L∞ (Σ), whose elements are ‘everywhere defined’. In §3.2 we prove that every normed L∞ (m)-module can be lifted to a 18 SIMONE DI MARINO, DANKA LUČIĆ, AND ENRICO PASQUALETTO normed L∞ (Σ)-module. In §3.3 we focus our attention on the functional-analytic properties of the fibers of a normed L∞ (Σ)-module. 3.1. Definition of normed L∞ (Σ)-module. We propose a notion of normed module over the commutative ring L∞ (Σ): Definition 3.1 (Normed L∞ (Σ)-module). Let (X, Σ, m) be a σ-finite measure space. Then
a normed L∞ (Σ)-module is a couple M¯, | · | that satisfies the following properties: i) M¯ is an algebraic L∞ (Σ)-module. ii) The map | · | : M¯ → L∞ (Σ), which is called a pointwise norm on M¯, satisfies |v̄| ≥ 0 |v̄ + w̄| ≤ |v̄| + |w̄| |f¯ · v̄| = |f¯||v̄| for every v̄ ∈ M¯, with equality if and only if v̄ = 0, for every v̄, w̄ ∈ M¯, (3.1) for every f¯ ∈ L∞ (Σ) and v̄ ∈ M¯. iii) M¯ satisfies the glueing property, i.e., given any partition (An )n∈N ⊆ Σ of X and a sequence (v̄n )n∈N ⊆ M¯ such that supn supAn |v̄n | < +∞, there is (a unique) v̄ ∈ M¯ P such that χAn · v̄ = χAn · v̄n for all n ∈ N. The element v̄ is denoted by n∈N χAn · v̄n . iv) The norm k · kM¯ on M¯ associated with | · |, which is defined as kv̄kM¯ := |v̄| L∞ (Σ) for every v̄ ∈ M¯, is complete. Remark 3.2 (Locality property). It can be readily checked that normed L∞ (Σ)-modules have the locality property: if (An )n∈N ⊆ Σ is a partition of X and v̄, w̄ ∈ M¯ are two elements such that χAn · v̄ = χAn · w̄ for every n ∈ N, then v̄ = w̄. This ensures that in item iii) of P  Definition 3.1 the element n∈N χAn · v̄n is uniquely determined. Remark 3.3 (Lack of glueing). The glueing property is not granted by items i), ii), iv) of  Definition 3.1, as shown by the same counterexample we provided in Remark 2.22. Given any normed L∞ (Σ)-module M¯, we can introduce the following equivalence relation: two elements v̄, w̄ ∈ M¯ are equivalent – shortly, v̄ ∼ w̄ – provided |v̄ − w̄| = 0 holds m-a.e.. Then we define the space Πm (M¯) as follows: Πm (M¯) := M¯/ ∼ . (3.2) Given any element v̄ ∈ M¯, we will denote by [v̄]∼ ∈ Πm (M¯) its equivalence class modulo ∼. The canonical projection map v̄ 7→ [v̄]∼ will be denoted by πm : M¯ → Πm (M¯). Lemma 3.4. Let (X, Σ, m) be a σ-finite measure space. Let M¯ be a normed L∞ (Σ)-module. Then M := Πm (M¯) is a normed L∞ (m)-module and πm : M¯ → M is linear and continuous. Proof. Given any v, w ∈ M and f ∈ L∞ (m) – say v = [v̄]∼ , w = [w̄]∼ , and f = [f¯]m – we set v + w := [v̄ + w̄]∼ ∈ M , f · v := [f¯ · v̄]∼ ∈ M ,   |v| := |v̄| m ∈ L∞ (m). REPRESENTATION THEOREMS FOR NORMED MODULES 19 It can be readily checked that the above operations are well-posed, meaning that they do not depend on the specific choice of the representatives v̄, w̄, and f¯. Moreover, we have that the operator | · | : M → L∞ (m) satisfies (2.6) as an immediate consequence of (3.1). To prove the glueing property of M , fix a partition (An )n∈N ⊆ Σ of X and a sequence (vn )n∈N ⊆ M such that supn ess supAn |vn | < +∞. Choose a representative v̄n of vn for each n ∈ N, so that there exists an m-negligible set N ∈ Σ such that supn supAn \N |v̄n | < +∞. Since M¯ has the glueing property, there is v̄ ∈ M¯ such that χAn \N · v̄ = χAn \N · v̄n holds for all n ∈ N. Therefore, [χAn ]m · πm (v̄) = πm (χAn \N · v̄) = πm (χAn \N · v̄n ) = [χAn ]m · vn holds for every n ∈ N, which shows that M has the glueing property. Now let us define kvkM := ess supX |v| for
every element v ∈ M . We aim to prove that M , k · kM is a Banach space, so fix a Cauchy sequence (vn )n ⊆ M , say vn = [v̄n ]∼ for all n. Hence, there exists N ∈ Σ with m(N ) = 0 such that supX\N |v̄n − v̄m | → 0 as n, m → ∞. This means that the sequence (χX\N · v̄n )n∈N is Cauchy in M¯, thus kχX\N · v̄n − v̄kM¯ → 0 for some v̄ ∈ M¯. Then we have that vn − πm (v̄) M = ess sup vn − πm (v̄) ≤ sup χX\N · v̄n − v̄ −→ 0 X as n → ∞, X as required. Finally, linearity and continuity of πm : M¯ → M can be trivially verified.  3.2. Liftings of normed L∞ (m)-modules. The following result shows that any lifting of measurable functions can be made into a ‘lifting of normed modules’, much like in Theorem 2.4 we ‘raised’ a lifting of a measure space to a lifting of measurable functions. Theorem 3.5 (Lifting of normed L∞ (m)-modules). Let (X, Σ, m) be a σ-finite measure space. Let M be a normed L∞ (m)-module. Let ℓ be a lifting of m and call L : L∞ (m) → L∞ (Σ) its associated operator (as in Theorem 2.4). Then there exists a unique couple (M¯, L ), called the L-lifting of M , where M¯ is a normed L∞ (Σ)-module and L : M → M¯ is a linear map that satisfies the following properties:
i) It holds that L (v) = L |v| for every v ∈ M . P ii) The linear subspace V of all elements v̄ ∈ M¯ of the form v̄ = n∈N χAn · L (vn ), where (An )n∈N ⊆ Σ is a partition of X and (vn )n∈N ⊆ M , is dense in M¯. Uniqueness is intended up to unique isomorphism: given any other couple (N¯, L ′ ) with the same properties, there exists a unique L∞ (Σ)-module isomorphism Ψ : M¯ → N¯ preserving the pointwise norm such that L M L′ M¯ Ψ N¯ is a commutative diagram. (3.3) 20 SIMONE DI MARINO, DANKA LUČIĆ, AND ENRICO PASQUALETTO Proof. Existence. First of all, let us define the pre-module Pm as    (An )n∈N ⊆ Σ is a partition of X, (An , vn ) n∈N Pm := . (vn )n ⊆ M , supn ess supAn |vn | < ∞  
We declare that (An , vn ) n ∼ (Bm , wm ) m provided χAn ∩Bm · L |vn − wm | = 0 holds for  every n, m ∈ N. We denote by [An , vn ]n the equivalence class of the sequence (An , vn ) n with respect to the equivalence relation ∼. We endow Pm/ ∼ with the following operations: [An , vn ]n + [Bm , wm ]m := [An ∩ Bm , vn + wm ]n,m , X  k ci χCi · [An , vn ]n := [An ∩ Ci , ci vn ]n,i , i=1 [An , vn ]n := X n∈N [An , vn ]n M¯
χAn L |vn | , (3.4) := sup [An , vn ]n , X P for every [An , vn ]n , [Bm , wm ]m ∈ Pm/ ∼ and ki=1 ci χCi ∈ Sf(Σ). Then we define M¯ as the
completion of the normed space Pm/ ∼, k · kM¯ . It can be readily checked – since the space Sf(Σ) is dense in L∞ (Σ) – that the operations in (3.4) can be uniquely extended to M¯, which has a natural structure of L∞ (Σ)-module endowed with a pointwise norm | · | : M¯ → L∞ (Σ) satisfying items ii) and iv) of Definition 3.1. Define L : M → M¯ as L (v) := [X, v] ∈ Pm/ ∼ for every v ∈ M . It is clear that L is a linear operator that satisfies i). To show that M¯ has the glueing property, fix a sequence (w̄n )n∈N ⊆ M¯ and a partition (An )n∈N ⊆ Σ of X. Given k ) k any n, k ∈ N, there exist a sequence (vn,i i∈N ⊆ M and a partition (An,i )i∈N ⊆ Σ of An such k ] −χ k k −k k ¯ that [Akn,i , vn,i i An · w̄n M¯ ≤ 2 . Call z̄ := [An,i , vn,i ]n,i ∈ M for every k ∈ N. Given that supAn |z̄ k+1 − z̄ k | ≤ 2−k+1 for all n, k ∈ N, we have z̄ k+1 − z̄ k M¯ ≤ 2−k+1 for all k ∈ N. Then (z̄ k )k∈N is a Cauchy sequence, whence it converges to some element w̄ ∈ M¯. Since for any n ∈ N it holds that χAn · w̄ = limk χAn · z̄ k = χAn · w̄n , the glueing property is proved. Finally, it only remains to show ii). Notice that it is enough to prove that V = Pm/ ∼. To this aim, fix a partition (An )n∈N ⊆ Σ of X and a sequence (vn )n ⊆ M . We denote by v̄ the P element n∈N χAn · L (vn ) ∈ M¯. Then it holds that v̄ = [An , vn ]n : for any m ∈ N, we have χAm · v̄ = χAm · L (vm ) = χAm · [X, vm ] = [Am ∩ An , vn ]n = χAm · [An , vn ]n . This proves that V = Pm/ ∼ and accordingly that ii) is verified. Uniqueness. First of all, observe that we are forced to set  X X X χAn · L (vn ) ∈ V. χAn · L ′ (vn ) for every χAn · L (vn ) := Ψ n∈N n∈N n∈N P Its well-posedness stems from the fact that n∈N χAn · L (vn ) and n∈N χAn · L ′ (vn ) have the same pointwise norm by i). Hence, by ii) the map Ψ can be uniquely extended to an L∞ (Σ)-linear operator Ψ : M¯ → N¯ preserving the pointwise norm. Finally, since Ψ(V) is dense in N¯ again by ii), we conclude that Ψ is a normed L∞ (Σ)-module isomorphism.  P REPRESENTATION THEOREMS FOR NORMED MODULES 21 Remark 3.6. We highlight an important byproduct of the proof of Theorem 3.5: for every v ∈ M and f ∈ L∞ (m). (3.5) P n By density of Sf(m) in L∞ (m), it is enough to prove it for f = i=1 ai [χAi ]m ∈ Sf(m). Since



χℓ(Ai ) · L |ai v − f · v| = L [χAi ]m L |ai v − f · v| = L [χAi ]m |ai v − f · v| = 0   holds for every i = 1, . . . , n, we see that ℓ(Ai ), ai v i = [X, f · v]. Therefore, we deduce from the second line of (3.4) that n X    L(f ) · L (v) = ai χℓ(Ai ) · [X, v] = ℓ(Ai ), ai v i = [X, f · v] = L (f · v), L (f · v) = L(f ) · L (v) i=1 as required.  A natural question arises: given a normed L∞ (m)-module M and calling M¯ its L-lifting, do Πm (M¯) and M coincide? The following result shows that the answer is positive. Lemma 3.7. Let (X, Σ, m) be a σ-finite measure space. Let M be a normed L∞ (m)-module. Let ℓ be any lifting of m, with associated operator L : L∞ (m) → L∞ (Σ). Denote by (M¯, L ) the L-lifting of M . Then it holds that Πm (M¯) ∼ = M. Proof. To prove the statement, it suffices to show that the map T := πm ◦ L : M → Πm (M¯) is an isomorphism of normed L∞ (m)-modules. We know that T is L∞ (m)-linear: it is linear as composition of linear operators, while for any f ∈ L∞ (m) and v ∈ M it holds that
 
(3.5)   T (f · v) = πm L (f · v) = πm L(f ) · L (v) = L(f ) · L (v) ∼ = f · L (v) ∼ = f · T (v). Furthermore, for every v ∈ M we have that   
   T (v) = L (v) ∼ = L (v) m = L |v| m = |v| holds m-a.e. on X, in other words, the map T preserves the pointwise norm. In order to conclude, it suffices to prove that T is surjective. Let [w̄]∼ ∈ Πm (M¯) be fixed. Then for any n ∈ N we can pick an P element w̄n = i∈N χAni · L (vin ) ∈ M¯ in such a way that limn kw̄n − w̄kM¯ = 0. Now let us P set vn := i∈N [χAni ]m · vin ∈ M for all n ∈ N, which is well-defined as sup ess sup |vin | ≤ sup sup L (vin ) = kw̄n kM¯ < +∞ i∈N An i i∈N An i for every n ∈ N. Since for any n, m ∈ N it holds that h i X X   n m χAn ∩Am ]m L (vin ) − L (vjm ) |vn − vm | = [χAni ∩Am ] |v − v | = [ = |w̄n − w̄m | m , m i j j i j i,j∈N i,j∈N m we see that the sequence (vn )n∈N ⊆ M is Cauchy, thus it converges to some element v ∈ M .   In general L (v) and w̄ may be different, but for sure one has that L (v) − w̄ m = 0, indeed h i h i h i     L (v) − w̄ ≤ L (v − vn ) + L (vn ) − w̄n + |w̄n − w̄| m = |v − vn | + |w̄n − w̄| m m m m   is satisfied m-a.e. and |v − vn | + |w̄n − w̄| m → 0 with respect to the L∞ (m)-norm. This means that T (v) = [w̄]∼ , whence the operator T is surjective, as desired.  22 SIMONE DI MARINO, DANKA LUČIĆ, AND ENRICO PASQUALETTO 3.3. Fibers of a normed L∞ (Σ)-module. Let (X, Σ, m) be a σ-finite measure space and let M¯ be a normed L∞ (Σ)-module. Given any x ∈ X, we define the submodule M¯x of M¯ as M¯x := χ{x} · M¯. Since the ideal (χ{x} ) ⊆ L∞ (Σ) generated by χ{x} can be identified with the real field R, we deduce that M¯x inherits a vector space structure. Moreover, let us define kv̄kx := |v̄|(x) for every v̄ ∈ M¯x .
Therefore, M¯x , k · kx is a Banach space. We call it the fiber of M¯ over the point x. Remark 3.8. Given any element v̄ ∈ M¯, we shall make use of the shorthand notation v̄x := χ{x} · v̄ ∈ M¯x for every x ∈ X. Then v̄ ∈ M¯ can be thought of as a map assigning to any x ∈ X a vector v̄x ∈ M¯x .  Fix a normed L0 (m)-module M and consider the duality pairing h·, ·i : M ∗ × M −→ L0 (m), hω, vi := ω(v) for every ω ∈ M ∗ and v ∈ M . (3.6) Choose any lifting ℓ of m. Denote by L the operator associated with ℓ as in Theorem 2.4. Call (M¯, L ) and (N¯, L ∗ ) the L-liftings of the normed L∞ (m)-modules R(M ) and R(M ∗ ), respectively. Notice that the duality pairing in (3.6) restricts to an L∞ (m)-bilinear and continuous map h·, ·i : R(M ∗ ) × R(M ) → L∞ (m). We now want to lift it to a duality pairing between M¯ and N¯. Given any two sequences (vn )n∈N ⊆ R(M ), (ωm )m∈N ⊆ R(M ∗ ) and any two partitions (An )n∈N , (Bm )m∈N ⊆ Σ of X, let us define  X X X
χAn ∩Bm L hωm , vn i ∈ L∞ (Σ). χAn · L (vn ) := χBm · L ∗ (ωm ), (3.7) n,m∈N n∈N m∈N The well-posedness of the previous definition stems from the following inequality: X X

χAn ∩Bm L hωm , vn i χAn ∩Bm L hωm , vn i = n,m∈N n,m∈N ≤ X n,m∈N = X n,m∈N =
χAn ∩Bm L |ωm ||vn |

χAn ∩Bm L |ωm | L |vn | X χAn ∩Bm L ∗ (ωm ) L (vn ) X χBm · L ∗ (ωm ) n,m∈N = m∈N X χAn · L (vn ) . n∈N The same inequality grants that the map in (3.7) can be uniquely extended to a pairing h·, ·i : N¯ × M¯ −→ L∞ (Σ), REPRESENTATION THEOREMS FOR NORMED MODULES 23 i.e., to an L∞ (Σ)-bilinear and continuous map satisfying the inequality hω̄, v̄i ≤ |ω̄||v̄| everywhere on X (3.8) for every ω̄ ∈ N¯ and v̄ ∈ M¯. Remark 3.9. We underline that
L ∗ (ω), L (v) = L ω(v) for every v ∈ R(M ) and ω ∈ R(M ∗ ), (3.9) by the very definition (3.7) of h·, ·i : N¯ × M¯ → L∞ (Σ).  Now fix a point x ∈ X. Then the pairing h·, ·i : N¯ × M¯ → L∞ (Σ) naturally induces a pairing h·, ·ix : N¯x × M¯x → R, which is the bilinear and continuous function given by for every ω̄ ∈ N¯x and v̄ ∈ M¯x . hω̄, v̄ix := hω̄, v̄i(x) Observe that M¯x ∋ v̄ 7→ hω̄, v̄ix ∈ R is a linear and continuous function for every ω̄ ∈ N¯x . Therefore, it makes sense to define the map Rx : N¯x → (M¯x )′ as Rx (ω̄) := hω̄, ·ix for every ω̄ ∈ N¯x . (3.10) It is clear that Rx is a linear operator. Proposition 3.10. Under the above assumptions, it holds that the map Rx : N¯x → (M¯x )′ is an isometric embedding. Proof. First of all, given any ω̄ ∈ N¯x it holds that hω̄, v̄ix ≤ kω̄kx kv̄kx for every v̄ ∈ M¯x as a consequence of (3.8), whence accordingly Rx (ω̄) (M¯x )′ ≤ kω̄kx for every ω̄ ∈ N¯x . (3.11) In particular, the operator Rx is continuous. Then to prove the statement it suffices to show that Rx is an isometry when restricted to any dense subset of N¯x .  Observe that D := L ∗ (ω)x : ω ∈ R(M ∗ ) is a dense subspace of N¯x by the very definition of L-lifting. Fix any ω ∈ R(M ∗ ). Then it holds that
Rx L ∗ (ω)x = sup L ∗ (ω)x , v̄ x ≥ sup L ∗ (ω), L (v) (x) ¯ ′ (Mx ) v̄∈M¯x : kv̄kx ≤1 (3.9) = sup v∈R(M ): |v|≤1 m-a.e. v∈R(M ): |v|≤1 m-a.e. (3.12)
L ω(v) (x). Given any ε > 0, there exists v ∈ R(M ) such that |v| ≤ 1 and ω(v) ≥ |ω| − ε are satisfied


m-a.e. by (2.7). Thus L ω(v) ≥ L |ω| − ε and accordingly L ω(v) (x) ≥ L ∗ (ω)x x − ε. This fact – if read in conjunction with (3.11) and (3.12) – ensures that the map Rx is an isometry when restricted to D, thus proving the statement.  Problem 3.11. Are the maps Rx : N¯x → (M¯x )′ defined in (3.10) isomorphisms? 24 SIMONE DI MARINO, DANKA LUČIĆ, AND ENRICO PASQUALETTO 4. Representation of normed modules via embedding In this section we study separable Banach B-bundles and their strong connections with separable normed L0 (m)-modules. Is §4.1 we introduce our notion of separable Banach bundle and we describe a constructive procedure to obtain one such bundle. In §4.2 we show that the space of sections of a given bundle is a normed L0 (m)-module; also, we extend the operation of ‘taking sections’ to the level of categories, thus obtaining a well-behaved section functor. In §4.3 we prove that the section functor is actually an equivalence of categories; this means, roughly speaking, that there is a full correspondence between separable Banach bundles and separable normed L0 (m)-modules. 4.1. Separable Banach bundles. Given a Banach space B, we denote by Gr(B) the family of its closed linear subspaces. Definition 4.1 (Separable Banach bundle). Let (X, Σ) be a measurable space and B a universal separable Banach space. Then a map E : X → Gr(B) is said to be a separable Banach B-bundle over X provided E : X ։ B is a weakly measurable correspondence. Given a separable Banach B-bundle E over X, let us define [ TE := {x} × E(x) ⊆ X × B. (4.1) x∈X Observe that TE ∈ Σ ⊗ B(B) by item vi) of §2.2. Any bundle is naturally associated with the space of its (measurable) sections: Definition 4.2 (Sections of a separable Banach bundle). Let (X, Σ) be a measurable space and B a universal separable Banach space. Let E be a separable Banach B-bundle over X. Then by section of E we mean a measurable selector of E, namely, a measurable map s̄ : X → B such that s̄(x) ∈ E(x) for every x ∈ X. The vector space of sections of E is denoted by Γ̄(E). Given any s̄ ∈ Γ̄(E), we define the function |s̄| : X → [0, +∞) as |s̄|(x) := s̄(x) for every x ∈ X. (4.2)  Moreover, we define the linear subspace Γ̄b (E) ⊆ Γ̄(E) as Γ̄b (E) := s̄ ∈ Γ̄(E) : |s̄| ∈ L∞ (Σ) . B Remark 4.3. It is straightforward to check that Γ̄b (E) is a normed L∞ (Σ)-module when endowed with the natural pointwise operations and the pointwise norm | · | : Γ̄b (E) → L∞ (Σ) that has been defined in (4.2).  Proposition 4.4. Let (X, Σ) be a measurable space and B a universal separable Banach space. i) Let (s̄n )n be any sequence of measurable maps s̄n : X → B. Define E : X ։ B as    for every x ∈ X. E(x) := clB span s̄n (x) n ∈ N Then E is a separable Banach B-bundle over X. ii) Let E be a separable Banach B-bundle over X. Then there exists a countable Q-linear  subspace C of Γ̄(E) such that E(x) = clB s̄(x) : s̄ ∈ C for every x ∈ X. REPRESENTATION THEOREMS FOR NORMED MODULES 25 Proof. i) Given any open set U ⊆ B, it holds that  x ∈ X E(x) ∩ U 6= ∅ = [ [ n∈N q1 ,...,qn ∈Q  x∈X n X i=1 qi s̄i (x) ∈ U  ∈ Σ. By arbitrariness of U , we conclude that E : X → Gr(B) is a separable Banach B-bundle. ii) Let (vn )n ⊆ B be a fixed dense sequence. Given any n, k ∈ N, let us define the correspondence ϕnk : X ։ B as (
if E(x) ∩ B1/k (vn ) 6= ∅, clB E(x) ∩ B1/k (vn ) ϕnk (x) := {0B } otherwise. Notice that ϕnk is weakly measurable as a consequence of item viii) of §2.2. Then by applying Kuratowski–Ryll-Nardzewski theorem (item vii) of §2.2) we obtain a measurable selector s̄nk of the correspondence ϕnk . Given any x ∈ X, v ∈ E(x), and ε > 0, we can find n, k ∈ N such that 1/k < ε/2 and kv − vn kB < 1/k. Therefore, it holds that v − s̄nk (x) B < ε. This shows  that s̄nk (x) : n, k ∈ N is a dense subset of E(x) for every x ∈ X. The claim follows by taking as C the Q-linear subspace of Γ̄(E) generated by {s̄nk : n, k ∈ N}.  In the sequel, we will need the following working definition of a measurable collection of Banach spaces. Roughly speaking, it is an intermediate construction that will be used to cook up the separable Banach bundle underlying a given separable normed L0 (m)-module. Definition 4.5 (Measurable collection of separable Banach spaces). Let (X, Σ) be a given  measurable space. Then a family E(x) x∈X of separable Banach spaces is said to be a measurable collection of separable Banach spaces provided there exist elements vn (x) ∈ E(x) and ωn (x) ∈ E(x)′ , with n ∈ N and x ∈ X, such that the following properties hold:
a) vn (x) n is a dense subset of E(x) for every x ∈ X,   b) ωn (x) E(x)′ = 1 and ωn (x) vn (x) = vn (x) E(x) whenever n ∈ N and x ∈ X are such that vn (x) 6= 0E(x) ,   c) X ∋ x 7→ ωn (x) vk (x) ∈ R is a measurable function for every n, k ∈ N. In particular, the function X ∋ x 7→ vn (x) E(x) ∈ R is measurable for every n ∈ N. In the next two technical results we explain how to get a separable Banach bundle out of a measurable collection of separable Banach spaces. Here, the explicit construction in the proof of Banach–Mazur Theorem 2.8 plays a role. Theorem 4.6 (Measurable family of embeddings). Let (X, Σ) be a measurable space. Let B  be a universal separable Banach space and E(x) x∈X a measurable collection of separable
Banach spaces. Choose elements vn (x) n∈N ⊆ E(x) for x ∈ X as in Definition 4.5. Then there exists a family {Ix }x∈X of linear isometric embeddings Ix : E(x) → B such that   (4.3) X ∋ x 7−→ Ix vn (x) ∈ B is a measurable map for every n ∈ N. We say that {Ix }x∈X is a measurable family of (linear isometric) embeddings. 26 SIMONE DI MARINO, DANKA LUČIĆ, AND ENRICO PASQUALETTO Proof. First of all, let us define the objects we will need throughout the proof: i) Given any k ∈ N, we denote by πk : I ∞ → [−1, 1] the projection on the k th component of the Hilbert cube I ∞ , namely, we define πk (α) := αk for every α = (αk′ )k′ ∈ I ∞ . Each πk is continuous, as I ∞ is endowed with the product topology (item iii) of §2.3). ii) Given any x ∈ X, we define the map ιx : BE(x)′ → I ∞ as     ω vk (x) ∈ I ∞ for every ω ∈ BE(x)′ . (4.4) ιx (ω) := vk (x) E(x) ∨ 1 k iii) iv) v) vi) vii) Then it holds that ιx is a homeomorphism with its image (when the domain BE(x)′ is endowed with the restricted weak∗ topology). Recall item iv) of §2.3.
Let us define the correspondence K ′ : X ։ I ∞ as K ′ (x) := ιx BE(x)′ for every x ∈ X. Observe that K ′ has compact values by virtue of Banach–Alaoglu theorem. Fix a continuous surjective map ψ : ∆ → I ∞ (recall item iii) of §2.3). Denote by K : X ։ ∆ the preimage correspondence ψ −1 (K ′ ), defined as in Lemma 2.6. Since ψ is continuous and ∆ is compact, it holds that K has compact values. Given any x ∈ X, we define the retraction rx : ∆ → K(x) as in item ii) of §2.3. Namely, for any point a ∈ ∆ we have that rx (a) is the unique element of K(x) satisfying the

identity d∆ a, rx (a) = d∆ a, K(x) . Given any x ∈ X, we define the operator I′x : E(x) → C(∆) as I′x [v](a) := (ι−1 x ◦ ψ ◦ rx )(a)[v] for every v ∈ E(x) and a ∈ ∆. (4.5) Then each map I′x is a linear isometric embedding (recall the proof of Theorem 2.8). viii) Fix a linear isometric map I : C(∆) → B. Given any x ∈ X, we define the linear isometric embedding Ix : E(x) → B as Ix [v] := (I ◦ I′x )[v] for every v ∈ E(x).   It remains to prove that X ∋ x 7→ Ix vk (x) ∈ C([0, 1]) is a measurable map for any k ∈ N.
Fix ωn (x) n∈N ⊆ BE(x)′ for x ∈ X as in Definition 4.5. We set the family of indexes Q as   M X Q := q = (qn )n ∈ Q qn ≥ 0 for every n ∈ N and qn = 1 , n∈N N where N Q stands for the set of all sequences q = (qn )n ∈ QN such that qn = 0 for all but finitely many n ∈ N. Let us define X ω q (x) := qn ωn (x) ∈ BE(x)′ for every q ∈ Q and x ∈ X. L n∈N  Observe that ω q (x) q∈Q is weak∗ dense in BE(x)′ for every x ∈ X, whence it follows that 
ιx ω q (x) q∈Q is dI ∞ -dense in K ′ (x). Moreover, given any α ∈ I ∞ and λ > 0, we have that o n o [n


x ∈ X dI ∞ α, K ′ (x) < λ = x ∈ X dI ∞ α, ιx ω q (x) < λ) q∈Q [ = x∈X (4.4) q∈Q    ∞ X qn ωn (x) vk (x) X 1 α − < λ ∈ Σ, k 2k v (x) ∨ 1 k E(x) n∈N k=1 REPRESENTATION THEOREMS FOR NORMED MODULES 27   as a consequence of the measurability of each function X ∋ x 7→ ωn (x) vk (x) . Therefore, the
function X ∋ x 7→ dI ∞ α, K ′ (x) is measurable for every α ∈ I ∞ , thus accordingly K ′ is a weakly measurable correspondence by item iv) of §2.2. Thanks to item i) of §2.2, we deduce that K ′ is a measurable correspondence, whence K is a measurable correspondence as well by Lemma 2.6. For any a ∈ ∆, let us consider the correspondence Za : X ։ ∆ given by n
o for every x ∈ X. Za (x) := b ∈ ∆ d∆ (a, b) = d∆ a, K(x)
It holds that X × ∆ ∋ (x, b) 7→ d∆ (a, b) − d∆ a, K(x) ∈ R is a Carathéodory function, since
∆ ∋ b 7→ d∆ (a, b) is continuous and X ∋ x 7→ d∆ a, K(x) is measurable (the latter follows from the measurability of K, by taking items i) and iv) of §2.2 into account). Therefore, we have that Za : X ։ ∆ is a measurable correspondence by item v) of §2.2, whence the intersection correspondence Za ∩ K : X ։ ∆ is measurable as well by item iii) of §2.2. Since  it holds that Za (x) ∩ K(x) = rx (a) for every x ∈ X, we deduce from item ii) of §2.2 that X ∋ x 7−→ rx (a) ∈ ∆ is a measurable map for every a ∈ ∆. (4.6) Let us fix k ∈ N, a dense subset (ai )i∈N of ∆, and an element g ∈ C(∆). Observe that     (4.4)
(4.5) i I′x vk (x) (ai ) = (ι−1 = (πk ◦ ψ) rx (ai ) x ◦ ψ ◦ rx )(a ) vk (x) for all x ∈ X and i ∈ N.   Being πk ◦ ψ : ∆ → [−1, 1] continuous, we deduce from (4.6) that X ∋ x 7→ I′x vk (x) (ai ) ∈ R is measurable for every i ∈ N. Since we have that   g − I′x vk (x) (4.5) C(∆)   = sup g(ai ) − I′x vk (x) (ai ) for every x ∈ X, i∈N   we deduce that X ∋ x 7→ g − I′x vk (x) C(∆) ∈ R is measurable for every element g ∈ C(∆).   Therefore, it holds that X ∋ x 7→ I′x vk (x) ∈ C(∆) is a measurable map for all k ∈ N. Recalling that Ix = I ◦ I′x for every x ∈ X and that I is continuous, we can finally conclude    that X ∋ x 7→ Ix vk (x) ∈ B is a measurable map for all k ∈ N, as required. Corollary 4.7. Let (X, Σ) be a measurable space and B a universal separable Banach space.  Let E(x) x∈X be a measurable collection of separable Banach spaces. Consider the associated measurable family {Ix }x∈X of linear isometric embeddings Ix : E(x) → B as in Theorem 4.6.
Then the map X ∋ x 7→ E(x) := Ix E(x) ∈ Gr(B) is a separable Banach B-bundle over X.
Proof. Choose elements vn (x) n∈N ⊆ E(x) for x ∈ X as in Definition 4.5. Let us define   s̄n (x) := Ix vn (x) for every n ∈ N and x ∈ X. Theorem 4.6 guarantees that each map s̄n : X → B is measurable, thus accordingly the correspondence E : X ։ B, which is given by   
for every x ∈ X, E(x) = Ix E(x) = clB span s̄n (x) n ∈ N is a separable Banach B-bundle over X by item i) of Proposition 4.4.  28 SIMONE DI MARINO, DANKA LUČIĆ, AND ENRICO PASQUALETTO 4.2. The section functor. Let (X, Σ, m) be a σ-finite measure space and B a universal separable Banach space. Let E be a separable Banach B-bundle over X. Then we define

(4.7) Γb (E) := Πm Γ̄b (E) , Γ(E) := C Γb (E) , where the operations Πm and C are defined as in (3.2) and Definition 2.23, respectively. Remark 4.8. Observe that Γ(E) can be identified with the quotient space Γ̄(E)/ ∼, where the equivalence relation ∼ on Γ̄(E) is defined in the following way: given any s̄, t̄ ∈ Γ̄(E), we 
declare that s̄ ∼ t̄ provided m x ∈ X : s̄(x) 6= t̄(x) = 0.  Lemma 4.9. Let (X, Σ, m) be a σ-finite measure space and B a universal separable Banach space. Let E be a separable Banach B-bundle over X. Then the normed L0 (m)-module Γ(E) is countably-generated. In particular, if (X, Σ, m) is separable, then Γ(E) is separable. Proof. Thanks to item ii) of Proposition 4.4, we can find a sequence (s̄n )n ⊆ Γ̄(E) such that  the identity E(x) = clB s̄n (x) : n ∈ N is satisfied for every x ∈ X. Given any n, k ∈ N, we  define Bnk := x ∈ X : |s̄n |(x) ≤ k and snk := πm (χBnk · s̄n ), where πm : Γ̄b (E) → Γb (E) stands for the canonical projection map. We claim that (snk )n,k generates Γ(E). To prove it, fix t ∈ Γ(E) and ε > 0. Then there exist k ∈ N and A ∈ Σ with |t| < k − ε/2 m-a.e. on A and dΓ(E) (t0 , t) < ε/2, where we set t0 := [χA ]m · t ∈ Γb (E). Choose any element t̄0 ∈ Γ̄b (E) such that t0 = πm (t̄0 ). Pick a partition (An )n ⊆ Σ of A such that |s̄n − t̄0 | ≤ ε/2 on An for P every n ∈ N. Therefore, we have that [χ ] · s − t0 ≤ ε/2 holds m-a.e. on X. This
n∈N An m nk P χ implies that dΓ(E) n∈N [ An ]m · snk , t < ε, thus proving the first claim. The second one is  then an immediate consequence of Proposition 2.15. Definition 4.10 (Morphism of separable Banach bundles). Let (X, Σ, m) be a σ-finite measure space and B a universal separable Banach space. Let E, F be two separable Banach B-bundles over X. Then a pre-morphism ϕ̄ from E to F is a measurable map ϕ̄ : TE → TF such that
ϕ̄ {x} × E(x) ⊆ {x} × F(x) ∼ = F(x) for every x ∈ X and ϕ̄(x, ·) : E(x) → F(x) is a linear contraction for every x ∈ X. We declare two pre-morphisms ϕ̄1 , ϕ̄2 from E to F to be equivalent if there exists a set N ∈ Σ with m(N ) = 0 such that ϕ̄1 (x, ·) = ϕ̄2 (x, ·) for every x ∈ X \ N . This defines an equivalence relation, whose equivalence classes are called morphisms and usually denoted by ϕ : E → F. We denote by SBBB (X, Σ, m) the category having the separable Banach B-bundles over the space X as objects and the morphisms of separable Banach B-bundles as arrows. Let us consider two separable Banach B-bundles E, F over X and a morphism ϕ : E → F. Fix any pre-morphism ϕ̄ : TE → TF that is a representative of ϕ. Then we define the morphism of normed L0 (m)-modules Γ(ϕ) : Γ(E) → Γ(F) as follows: given any s ∈ Γ(E), we define Γ(ϕ)(s) as the equivalence class (under the relation ∼ introduced in Remark 4.8) of
X ∋ x 7−→ ϕ̄ x, s̄(x) ∈ F(x), REPRESENTATION THEOREMS FOR NORMED MODULES 29 where s̄ ∈ Γ̄(E) is any representative of s. It can be readily checked that this way we obtain a covariant functor Γ : SBBB (X, Σ, m) → NModcg (X, Σ, m), which we call the section functor. (For brevity, in our notation the dependence of Γ on the space B is omitted.) Lemma 4.11 (Γ is full). Let (X, Σ, m) be a σ-finite measure space and B a universal separable Banach space. Let E, F be two separable Banach B-bundles and Φ : Γ(E) → Γ(F) a morphism of normed L0 (m)-modules. Then there exists a morphism ϕ : E → F of separable Banach B-bundles such that Γ(ϕ) = Φ. Proof. Thanks to item ii) of Proposition 4.4, there is a countable Q-linear subspace C of Γ̄(E)  such that the Q-linear space C(x) := s̄(x) : s̄ ∈ C is dense in E(x) for every x ∈ X. Given
any s̄ ∈ C, choose a representative Φ̄(s̄) ∈ Γ̄(F) of Φ [s̄]∼ ∈ Γ(F). Then there exists N ∈ Σ with m(N ) = 0 such that the following properties hold: Φ̄(s̄ + t̄ )(x) = Φ̄(s̄)(x) + Φ̄(t̄ )(x), for every x ∈ X \ N, s̄, t̄ ∈ C, and q ∈ Q. Φ̄(q s̄)(x) = q Φ̄(s̄)(x), Φ̄(s̄)(x) F(x) ≤ s̄(x) E(x) (4.8) , Given any x ∈ X, we define the map ϕ̄x : C(x) → F(x) as (
if x ∈ X \ N, Φ̄(s̄)(x) ϕ̄x s̄(x) := if x ∈ N. 0F(x) The properties in (4.8) grant that each map ϕ̄x is a Q-linear contraction, thus it can be uniquely extended to an R-linear contraction ϕ̄x : E(x) → F(x). Then we define ϕ̄ : TE → TF
as ϕ̄(x, v) := x, ϕ̄x (v) for every x ∈ X and v ∈ E(x). In order to prove that ϕ̄ is a premorphism, it is sufficient to check its measurability. To this aim, we just have to show that
ϕ̄−1 A × B̄r (w) ∈ Σ ⊗ B(B) for any A ∈ Σ, w ∈ B, and r > 0. This follows from the identity n o [ \

ϕ̄−1 A × B̄r (w) = S ∪ (x, v) ∈ (A \ N ) × B ∩ TE ϕ̄x (v) ∈ B̄r (w) = S ∪ An,s̄ , n∈N s̄∈C
where we set S := (A ∩ N ) × B ∩ TE ∈ Σ if kwkB ≤ r, while S := ∅ if kwkB > r, and  
1 1 := ∈ Σ. An,s̄ (x, v) ∈ (A \ N ) × B ∩ TE v − s̄(x) B < , w − Φ̄(s̄)(x) B < r + k k Therefore, ϕ̄ is a pre-morphism from E to F. We denote by ϕ : E → F its equivalence class.

Observe that Γ(ϕ) [s̄]∼ = Φ [s̄]∼ for every s̄ ∈ C by construction. Finally, by arguing  exactly as in the proof of Lemma 4.9, we deduce that [s̄]∼ : s̄ ∈ C generates Γ(E), whence we can conclude that Γ(ϕ) = Φ. Consequently, the statement is achieved.  Lemma 4.12 (Γ is faithful). Let (X, Σ, m) be a σ-finite measure space and B a universal separable Banach space. Let E, F be separable Banach B-bundles. Let ϕ, ψ : E → F be two morphisms of separable Banach B-bundles such that ϕ 6= ψ. Then Γ(ϕ) 6= Γ(ψ). 30 SIMONE DI MARINO, DANKA LUČIĆ, AND ENRICO PASQUALETTO Proof. Choose representatives ϕ̄, ψ̄ of ϕ, ψ, respectively. Pick a set P ′ ∈ Σ such that m(P ′ ) > 0 and ϕ̄(x, ·) 6= ψ̄(x, ·) for every x ∈ P ′ . By item ii) of Proposition 4.4, there exists (s̄n )n ⊆ Γ̄(E)
such that s̄n (x) n is dense in E(x) for all x ∈ X. Therefore, there exist n ∈ N and P ∈ Σ

such that P ⊆ P ′ , m(P ) > 0, and ϕ̄ x, s̄n (x) 6= ψ̄ x, s̄n (x) for every x ∈ P . This ensures

that Γ(ϕ) [s̄n ]∼ 6= Γ(ψ) [s̄n ]∼ and thus accordingly Γ(ϕ) 6= Γ(ψ), as required.  4.3. Representation theorem. We are finally in a position – by combining the whole machinery developed so far – to prove that every separable normed L0 (m)-module is the space of sections of a separable Banach bundle. Theorem 4.13 (Representation theorem). Let (X, Σ, m) be a complete, σ-finite measure space. Let M be a countably-generated normed L0 (m)-module. Let B be a universal separable Banach space. Then there exists a separable Banach B-bundle E over X such that Γ(E) ∼ = M. Proof. Step 1. First, fix a countable Q-linear subspace (vn )n of R(M ) that generates M . Choose a sequence (ωn )n ⊆ R(M ∗ ) such that the identities |ωn | = 1 and ωn (vn ) = |vn | hold m-a.e. for every n ∈ N. Let ℓ be any lifting of m, whose existence is granted by Theorem 2.2. Consider the operator L : L∞ (m) → L∞ (Σ) associated with ℓ as in Theorem 2.4. Denote by (M¯, L ) and (N¯, L ∗ ) the L-liftings of R(M ) and R(M ∗ ), respectively; recall Theorem 3.5. Therefore, L ∗ (ωn ) (x) = 1, L ∗ (ωn ), L (vn ) (x) = L (vn ) (x), for every n ∈ N and x ∈ X. (Recall the discussion about the duality pairing h·, ·i between M¯ and N¯ in §3.3.) Given any point x ∈ X, we define the separable Banach subspace E(x) of M¯x as  E(x) := clM¯x L (vn )x n ∈ N . (4.9) (4.10) Consider the isometric embedding Rx : N¯x → (M¯x )′ , which has been introduced in (3.10) and studied in Proposition 3.10. Let us define v̄n (x) := L (vn )x ∈ E(x),
ω̄n (x) := Rx L ∗ (ωn )x |E(x) ∈ E(x)′ , It follows from the second line in (4.9) that   ω̄n (x) v̄n (x) = v̄n (x) E(x) for every n ∈ N and x ∈ X. for every n ∈ N and x ∈ X. Moreover, observe that for any n ∈ N and x ∈ X one has that
= L ∗ (ωn )x ω̄n (x) E(x)′ ≤ Rx L ∗ (ωn )x ¯ ′ (Mx ) (4.11) (4.9) N¯x = 1. Hence, if n ∈ N and x ∈ X satisfy v̄n (x) 6= 0E(x) , then (4.11) forces ω̄n (x) E(x)′ = 1. Finally,   given any n, k ∈ N, we have that the function X ∋ x 7→ ω̄n (x) v̄k (x) = L ∗ (ωn ), L (vk ) (x) is  measurable. All in all, we have proven that E(x) x∈X is a measurable collection of separable Banach spaces (in the sense of Definition 4.5) when equipped with (v̄n )n , (ω̄n )n . Therefore, let us consider a measurable family {Ix }x∈X of linear isometric embeddings Ix : E(x) → B, REPRESENTATION THEOREMS FOR NORMED MODULES 31 whose existence is granted by Theorem 4.6. We thus denote by E : X → Gr(B) the map
X ∋ x 7→ Ix E(x) , which is a separable Banach B-bundle over X thanks to Corollary 4.7. Step 2. Let v ∈ R(M ) be fixed. We claim that L (v)x ∈ E(x) for m-a.e. x ∈ X. (4.12) P k k k k mk ∞ Indeed, we can find a sequence (uk )k – where uk = m i=1 fi · ui for some (fi )i=1 ⊆ L (m) m k and (uki )i=1 ⊆ {vn }n – such that limk dM (uk , v) = 0. Then (up to taking a not relabelled subsequence) we have that L (uk ) − L (v) (x) → 0 for m-a.e. point x ∈ X, or equivalently that limk L (uk )x − L (v)x x = 0 for m-a.e. x ∈ X. Since L (uk )x = mk X L(fik )(x) L (uki )x ∈ E(x) for every k ∈ N and x ∈ X, i=1 we obtain (4.12). Now let us define the map Ī(v) : X → B as (   Ix L (v)x if L (v)x ∈ E(x), Ī(v)(x) := otherwise. 0B (4.13) Choose any set N ∈ Σ such that m(N ) = 0 and L (v)x = limk L (uk )x for every x ∈ X \ N . Hence, we have that mk X     L(fik )(x) Ix L (uki )x Ī(v)(x) = lim Ix L (uk )x = lim k→∞ k→∞ for every x ∈ X \ N. i=1 By recalling Theorem 4.6 and the fact that the measure space (X, Σ, m) is complete, we deduce that Ī(v) is a measurable map from X to B. In other words, it holds that Ī(v) ∈ Γ̄(E). Then   let us denote by I : R(M ) → Γ(E) the map given by I(v) := Ī(v) ∼ for every v ∈ R(M ). Step 3. We aim to prove that I maps R(M ) to Γb (E) and that I : R(M ) → Γb (E) is an isomorphism of normed L∞ (m)-modules. This is sufficient to conclude that the spaces M and Γ(E) are isomorphic as normed L0 (m)-modules by item i) of Lemma 2.24. We first check the L∞ (m)-linearity of I: if v, w ∈ R(M ) and f, g ∈ L∞ (m), then for m-a.e. x ∈ X we have     Ī(f · v + g · w)(x) = Ix L (f · v + g · w)x = Ix L(f )(x) · L (v)x + L(g)(x) · L (w)x = L(f )(x) · Ī(v)(x) + L(g)(x) · Ī(w)(x), thus accordingly I(f · v + g · w) = f · I(v) + g · I(w). Moreover, given any v ∈ R(M ) one has
Ī(v)(x) B = L (v)x x = L (v) (x) = L |v| (x) for m-a.e. x ∈ X, whence I(v) = |v| holds in the m-a.e. sense. This ensures that I maps R(M ) to Γb (E) and that I : R(M ) → Γb (E) is a morphism of normed L∞ (m)-modules preserving the pointwise norm. Finally, to prove that the map I is surjective, it is enough to show that its image is dense in Γb (E). Fix s ∈ Γb (E) and ε > 0. Choose any representative s̄ ∈ Γ̄b (E) of s. Given
that the sequence Ī(vn )(x) n is dense in E(x) for every x ∈ X by (4.10) and (4.13), we can find a partition (An )n ⊆ Σ of X such that s̄(x) − Ī(vn )(x) ≤ ε for every n ∈ N and x ∈ An . P This implies that the inequality s̄ − n∈N χAn · Ī(vn ) ≤ ε holds everywhere on X. 32 SIMONE DI MARINO, DANKA LUČIĆ, AND ENRICO PASQUALETTO P Now let us define v := n∈N [χAn ]m · vn ∈ R(M ). Clearly s − I(v) ≤ ε holds m-a.e. by construction, so that s − I(v) Γ (E) ≤ ε. This yields surjectivity of I, which is consequently b an isomorphism of normed L∞ (m)-modules. Hence, the statement is finally achieved.  Remark 4.14 (Proof of the representation theorem without the Axiom of Choice). In the above proof of Theorem 4.13, we made use of the theory of liftings of normed modules that we developed in §3. Nevertheless, it is possible to provide an alternative proof which does not rely upon the Axiom of Choice (and, thus, without appealing to von Neumann’s theory of lifting). Some weaker form of the Axiom of Choice is needed anyway, e.g., in the proof of Banach–Mazur Theorem 2.8, where Banach–Alaoglu theorem is used. We now sketch the argument of the alternative proof of Theorem 4.13, leaving its verification to the reader. Fix a countable Q-linear subspace (vn )n of M that generates M . Choose (ωn )n ⊆ M ∗ such that |ωn | = 1 and ωn (vn ) = |vn | hold m-a.e. for every n ∈ N. Given any n, k ∈ N, pick a measurable representative ωn (vk ) of ωn (vk ). Then we can find a m-null set N ⊆ X such that ωn (vk + vk′ )(x) = ωn (vk )(x) + ωn (vk′ )(x), ωn (qvk )(x) = q ωn (vk )(x), ωn (vk )(x) ≤ ωk (vk )(x) for every n, k, k′ ∈ N, q ∈ Q, and x ∈ X \ N . Observe that for any x ∈ X \ N the family n o
Vx := ωn (vk )(x) n∈N k ∈ N ⊆ ℓ∞ is a Q-linear subpace of ℓ∞ . Hence, calling E(x) := {0ℓ∞ } for all x ∈ N and E(x) := clℓ∞ (Vx )  for all x ∈ X \ N , we have that E(x) x∈X is a family of separable Banach subspaces of ℓ∞ . Moreover, for any x ∈ X and n, k ∈ N, we define ṽk (x) ∈ E(x) and ω̃n (x) ∈ E(x)′ as follows: trivially, ṽk (x) := 0E(x) and ω̃n (x) := 0E(x)′ if x ∈ N ; if x ∈ / N , then we set
ṽk (x) := ωn′ (vk )(x) n′ ∈N ∈ E(x), while we denote by ω̃n (x) : E(x) → R the unique linear and continuous operator satisfying    ω̃n (x) ṽk′ (x) = ωn (vk′ )(x) for every k′ ∈ N. Then it holds that E(x) x∈X is a measurable collection of separable Banach spaces – together with ṽk (x) and ω̃n (x). Finally, one can also prove that the associated separable Banach B-bundle E satisfies Γ(E) ∼ = M , as desired.  In analogy with [20], we have a Serre–Swan theorem for separable normed L0 (m)-modules: Theorem 4.15 (Serre–Swan theorem). Let (X, Σ, m) be a complete, σ-finite measure space. Let B be a universal separable Banach space. Then the section functor Γ : SBBB (X, Σ, m) −→ NModcg (X, Σ, m) is an equivalence of categories. In particular, if (X, Σ, m) is a separable measure space, then Γ : SBBB (X, Σ, m) −→ NMods (X, Σ, m) is an equivalence of categories. REPRESENTATION THEOREMS FOR NORMED MODULES 33 Proof. The first claim follows from Lemma 4.11, Lemma 4.12, and Theorem 4.13. The second  claim follows from the first one by taking Proposition 2.15 into account. Problem 4.16. Does there exist some notion of measurable Banach bundle that is sufficient to describe also the duals of separable normed L0 (m)-modules? In this regard, it is well-known that all duals of separable Banach spaces can be embedded linearly and isometrically into ℓ∞ , thus the space ℓ∞ would be a good candidate for the ‘ambient space’ of the fibers of the bundle. Finally, we now extend Banach–Mazur Theorem 2.8 to the setting of separable normed i.e., as we are going to see, we prove the existence of universal such modules. L0 (m)-modules; Definition 4.17 (Universal separable normed L0 -module). Let (X, Σ, m) be a σ-finite measure space. Let M be a separable normed L0 (m)-module. Then we say that M is a universal separable normed L0 (m)-module if for any separable normed L0 (m)-module N there exists a normed L0 (m)-module morphism I : N → M that preserves the pointwise norm. Given a measurable space (X, Σ) and a universal separable Banach space B, we shall denote by Γ(B) the space of sections of the separable Banach B-bundle X ∋ x 7→ B ∈ Gr(B). Theorem 4.18 (Existence of universal modules). Let (X, Σ, m) be a complete, σ-finite, separable measure space. Let B be a universal separable Banach space. Then Γ(B) is a universal separable normed L0 (m)-module. Proof. Let M be any given separable normed L0 (m)-module. Theorem 4.15 grants the existence of a separable Banach B-bundle E over X such that Γ(E) ∼ = M . With a slight abuse of notation, we use the symbol B to denote the separable Banach B-bundle X ∋ x 7→ B. Consider the pre-morphism ϕ̄ from E to B defined as follows: given any x ∈ X, we declare that ϕ̄(x, ·) : E(x) → B is the inclusion map. Call ϕ : E → B the equivalence class of ϕ̄. Therefore, it holds that Γ(ϕ) : Γ(E) → Γ(B) is a morphism of normed L0 (m)-modules that preserves the pointwise norm. The statement is achieved.  Appendix A. Representation of normed modules via direct limits In this section we provide an alternative description of separable normed L0 (m)-modules, which builds upon the representation results for proper modules that have been proven in [20]. Roughly speaking, the strategy we will adopt is the following: any separable normed L0 (m)module M can be obtained as direct limit of finite-dimensional normed L0 (m)-modules (Mk )k ; each module Mk is the space of sections of some finite-dimensional Banach bundle Ek , thus by ‘patching together’ the bundles Ek we obtain some notion of separable Banach bundle E, whose space of sections can be eventually identified with M . Even though this approach is much more ‘implicit’ than the one proposed in §4, it has the advantage of clarifying how to approximate separable normed L0 (m)-modules by proper ones. A word on notation: for simplicity, we will use some terminology that has been already used in §4 (such as ‘separable Banach bundle” and so on), but with a different meaning. Since this section is independent of §4, we believe that this will not cause any ambiguity. 34 SIMONE DI MARINO, DANKA LUČIĆ, AND ENRICO PASQUALETTO Let (X, Σ, m) be a fixed σ-finite measure space. For the sake of simplicity, we shall write mA := m|A for every A ∈ Σ such that m(A) > 0. As observed in [14, Section 2], any normed L0 (mA )-module N can be canonically viewed as a normed L0 (m)-module; we shall denote it by ExtA (N ) and call it the extension of N . On the other hand, a normed L0 (m)-module M can be ‘localised’ on A as follows: we define M |A as the pullback of M under the identity map from (X, Σ, mA ) and (X, Σ, m) (cf. [10, Section 1.6]), so that M |A is a normed L0 (mA )-module. It is clear that ExtA (M |A )  is isomorphic to the normed L0 (m)-submodule [χA ]m · M = [χA ]m · v : v ∈ M of M . A.1. Separable Banach bundles. We propose an alternative notion of separable Banach bundle over X, which extends the one that has been introduced in [20]. The language we adopt here is slightly different from that of [20], but it can be readily checked that the two resulting theories are fully consistent. We fix the notation N̄ := N ∪ {∞}. Given any n ∈ N̄, we define the vector space Vn as ( if n < ∞, Rn Vn := if n = ∞, c00 where c00 stands for the space of all sequences in R having only finitely many non-zero terms. Calling (en )n∈N the canonical basis of V∞ (i.e., en := (δin )i∈N for all n ∈ N), we shall always implicitly identify Vn with the subspace of V∞ spanned by e1 , . . . , en . The topology we shall consider on the space V∞ is the one induced by the ℓ∞ -norm. It is straightforward to check that a set S ⊆ c00 belongs to the Borel σ-algebra associated to such topology if and only if S ∩ Vn is a Borel subset of (Rn , dEucl ) for every n ∈ N. Definition A.1 (Banach bundle of dimension n). Let n ∈ N̄ be given. Then we say that a couple E = (A, n) is a Banach bundle of dimension n over a given set A ∈ Σ provided the function n : A × Vn → [0, +∞) is measurable and satisfies the following property: n(x, ·) is a norm on Vn for every x ∈ A. Notice that (A, n) is a Banach bundle of dimension n ∈ N̄ if and only if (A, n|A×Vk ) is a Banach bundle of dimension k for every k ∈ N satisfying k ≤ n. Remark A.2. We observe that if n = ∞, then the norms n(x, ·) on V∞ cannot be complete,  as the vector space c00 does not support any complete norm; cf. for instance [1]. Let us consider a Banach bundle E = (A, n) of dimension n ∈ N. Then the space ΓA (E) of sections of E is defined as the family of all measurable maps s : A → Rn , considered up to mA -a.e. equality. As shown in [20], it turns out that ΓA (E) is a normed L0 (mA )-module when endowed with the natural pointwise operations and the following pointwise norm:
|s|(x) := n x, s(x) for mA -a.e. x ∈ A for every s ∈ ΓA (E). More precisely, ΓA (E) is a free L0 (mA )-module of rank n. REPRESENTATION THEOREMS FOR NORMED MODULES 35 We now define the space of sections of a Banach bundle E = (A, n) of any dimension n ∈ N̄, possibly n = ∞. Call I the set of all k ∈ N with k ≤ n. We set Ek := (A, n|A×Vk ) for all k ∈ I. Given any j, k ∈ I with j ≤ k, we have a canonical inclusion map ιjk : ΓA (Ej ) ֒→ ΓA (Ek ). Namely, ιjk is the map sending (the equivalence class of) any section s = (s1 , . . . , sj ) of Ej to (the equivalence class of) the section (s1 , . . . , sj , 0 . . . , 0) of Ek . It can be readily checked
 that ΓA (Ek ) k∈I , {ιjk }j≤k is a direct system in the category of normed L0 (mA )-modules. Then we define ΓA (E) := lim ΓA (E⋆ ), −→ whose existence is granted by Theorem 2.16. Notice that such definition of ΓA (E) is consistent with the previous one when E is a Banach bundle of finite dimension; cf. [21, Lemma 2.10].  Definition A.3 (Banach bundle). We say that E = (An , En ) n∈N is a separable Banach bundle over X provided {An }n∈N ⊆ Σ is a partition of X and each En = (An , nn ) is a Banach bundle of dimension n. Moreover, we say that E is proper provided m(A∞ ) = 0.  Let E = (An , En ) n∈N be a separable Banach bundle over X. Then we define the space of its sections as Y
ExtAn ΓAn (En ) . Γ(E) := n∈N The direct product Γ(E) inherits an (algebraic) L0 (m)-module structure. Moreover, the fact that the sets An are pairwise disjoint grants that the following definition is meaningful: X [χAn ]m |sn | for every s = {sn }n∈N ∈ Γ(E). |s| := (A.1) n∈N It is straightforward to verify that (A.1) actually defines a pointwise norm on Γ(E), whose associated distance dΓ(E) is complete. Therefore, Γ(E) is a normed L0 (m)-module. A.2. Representation theorem. The purpose of this subsection is to show that any separable normed L0 (m)-module M is isomorphic to the space of sections Γ(E) of some separable Banach bundle E over X. In the sequel, we will need the following consequence of [20, Theorem 3], which we rephrase in the current language. Actually, the result was obtained for modules on metric measure spaces, but the same proof can be repeated verbatim in the case of σ-finite measure spaces. Theorem A.4 (Representation theorem for proper modules). Let (X, Σ, m) be a σ-finite measure space. Let M be a proper normed L0 (m)-module. Then there exists a proper Banach bundle E over X such that M is isomorphic to Γ(E) as a normed L0 (m)-module. Remark A.5. Let M be a separable normed L0 (m)-module that is not finitely-generated on any measurable subset of X having positive m-measure. We claim that there exists an increasing sequence (Nn )n∈N of normed L0 (m)-submodules of M with the following properties: i) Each Nn has local dimension n on X. S ii) The set n∈N Nn is dense in M . 36 SIMONE DI MARINO, DANKA LUČIĆ, AND ENRICO PASQUALETTO We construct the desired modules Nn is a recursive way. Fix any dense subset (vn )n∈N of M . We aim to build a sequence (Nn )n∈N of normed L0 (m)-modules satisfying i) and such that each Nn contains the elements v1 , . . . , vn . This would clearly imply ii). First, we define N1 as the normed L0 (m)-module generated by the element X  χ{|vk |>0}\S {|vj |>0} · vk ∈ M . m j<k k∈N Then N1 has dimension 1 (as M is not finitely-generated on any measurable set) and v1 ∈ N1 . Now suppose to have already defined Nn for some n ∈ N. We want to define Nn+1 . Fix a local basis w1 , . . . , wn of Nn . For any k ≥ n + 1 we call Bk′ the set where w1 , . . . , wn , vk are Sk−1 Bj′ ; then we define Nn+1 as the normed L0 (m)-module independent and Bk := Bk′ \ j=n+1 P generated by Nn ∪ {wn+1 }, where we put wn+1 := k≥n+1 [χBk ]m · vk . Hence, Nn+1 has local dimension equal to n + 1 on X (as M is not finitely-generated on any measurable set) and contains the elements v1 , . . . , vn+1 by construction.  By building on top of Theorem A.4, we can eventually prove the following result: Theorem A.6 (Representation theorem). Let (X, Σ, m) be any σ-finite measure space. Let M be a separable normed L0 (m)-module. Then there exists a separable Banach bundle E over the space X such that Γ(E) ∼ = M. Proof. Let us call {An }n∈N the dimensional decomposition of the module M . Consider the normed L0 (mA∞ )-module N := M |A∞ . As shown in Remark A.5, one can build an increasing sequence (Nk )k∈N of normed L0 (mA∞ )-submodules of N such that each Nk has local S dimension equal to k on the set A∞ and k∈N Nk is dense in N . Let us pick any sequence (vk )k∈N ⊆ N such that v1 , . . . , vk is a local basis for Nk on A∞ for every k ∈ N. (A.2) Given k ∈ N, we can find (by Theorem A.4) a Banach bundle Fk′ = (A∞ , n′k ) of dimension k such that ΓA∞ (Fk′ ) ∼ = Nk . Under such isomorphism, the elements v1 , . . . , vk ∈ Nk correspond to some sections sk1 , . . . , skk ∈ Γ(Fk′ ), respectively. Pick representatives s̄k1 , . . . , s̄kk of them. Then we know from (A.2) that there is a measurable set N ⊆ A∞ with m(N ) = 0 such that s̄k1 (x), . . . , s̄kk (x) is a basis of Rk for every k ∈ N and x ∈ A∞ \ N. Therefore, for any k ∈ N we define a new Banach bundle Fk = (A∞ , nk ) of dimension k as (

if x ∈ A∞ \ N, n′k x, λ1 s̄k1 (x) + . . . + λk s̄kk (x) nk x, (λ1 , . . . , λk ) := 1/2 2 2 (λ1 + . . . + λk ) if x ∈ N. ∼ Nk and nk+1 |A ×V = nk for every k ∈ N. Hence, we can consider It follows that ΓA∞ (Fk ) = ∞ k the Banach bundle E∞ = (A∞ , n∞ ) of dimension ∞, where n∞ : A∞ × V∞ → [0, +∞) is defined as the unique function satisfying n∞ |A∞ ×Vk = nk for all k ∈ N. As granted by Lemma 2.17, the direct limit of ΓA∞ (Fk ) ∼ = Nk is isomorphic to N , thus ΓA∞ (E∞ ) ∼ = M |A∞ . To 0 conclude, notice that M |X\A∞ is a proper normed L (mX\A∞ )-module, whence accordingly  M |X\A∞ ∼ = ΓX\A∞ (E ′ ) for some proper Banach bundle E ′ = (An , En ) n∈N REPRESENTATION THEOREMS FOR NORMED MODULES  as a consequence of Theorem A.4. Hence, the separable Banach bundle E := (An , En ) satisfies Γ(E) ∼ = M , as required. 37 n∈N  Remark A.7 (‘Serre–Swan theorem’). Given a σ-finite measure space (X, Σ, m) and two   separable Banach bundles E = (An , En ) n∈N̄ and F = (Bm , Fm ) m∈N̄ over X, we can define a pre-morphism between E and F as a family ϕ = {ϕnm }n,m∈N̄ of measurable maps ϕnm : (An ∩ Bm ) × Vn −→ (An ∩ Bm ) × Vm such that ϕnm (x, Vn ) ⊆ {x} × Vm and

F ϕnm (x, ·) : Vn , nE n (x, ·) −→ Vm , nm (x, ·) is a linear contraction F for every n, m ∈ N̄ and x ∈ An ∩ Bm , where we call En = (An , nE n ) and Fm = (Bm , nm ). We declare two pre-morphisms {ϕnm }n,m∈N̄ and {ψnm n,m∈N̄ between E and F to be equivalent provided there exists a set N ∈ Σ with m(N ) = 0 such that ϕnm (x, ·) = ψnm (x, ·) for every x ∈ X \ N and n, m ∈ N̄. Therefore, it makes sense to consider the category of separable Banach bundles over X having the equivalence classes of pre-morphisms as arrows. We point out that – in analogy with what done in §4.2 – it is possible to promote the correspondence E 7→ Γ(E) to a functor (called the section functor ) from the category of separable Banach bundles over X to that of separable normed L0 (m)-modules. Furthermore, the section functor can be shown to be an equivalence of categories (the so-called ‘Serre–Swan theorem’). We omit the details.  Appendix B. Cotangent bundle on metric measure spaces As mentioned in the introduction, in the study of the differential structure of metric measure spaces a key role is played by the so-called cotangent module L0 (T∗ X), which has been introduced by Gigli in [10]. We now propose an alternative axiomatisation, at least in the case in which W 1,2 (X) is separable. In the approach we are going to present, we directly introduce a notion of cotangent bundle T∗ X over (X, d, m), which does not require the theory of normed modules to be formulated (cf. Remark B.2). Even though we just consider p = 2 for simplicity, a similar construction could be performed for any p ∈ (1, ∞). By metric measure space (X, d, m) we mean a complete, separable metric space (X, d), endowed with a non-negative Radon measure m. Calling Σ the completion of the Borel σalgebra B(X) and m̄ the completion of the measure m, it holds that (X, Σ, m̄) is a complete, σ-finite measure space, thus in particular the results of §4.3 can be applied. Observe that normed L0 (m)-modules and normed L0 (m̄)-modules can be identified in a canonical way. We denote by LIP(X) the space of all real-valued Lipschitz functions on X. Given any function f ∈ LIP(X), its slope lip(f ) : X → [0, +∞) is defined as lip(f )(x) := lim y→x f (x) − f (y) d(x, y) if x ∈ X is an accumulation point, 38 SIMONE DI MARINO, DANKA LUČIĆ, AND ENRICO PASQUALETTO and lip(f )(x) := 0 otherwise. We define the Cheeger energy Ch : L2 (m) → [0, +∞] on X as   ˆ 2 2 2 Ch(f ) := inf lim lip (fn ) dm (fn )n ⊆ LIP(X), [fn ]m ∈ L (m), [fn ]m → f in L (m) n→∞ for every f ∈ L2 (m). Then (following [8]) we define the Sobolev space on (X, d, m) as o n W 1,2 (X) := f ∈ L2 (m) Ch(f ) < +∞ . Given any f ∈ W 1,2 (X), there exists a distinguished function |Df | ∈ L2 (m) – called the ´ minimal relaxed slope of f – which provides the integral representation Ch(f ) = |Df |2 dm of the Cheeger energy. It turns out that W 1,2 (X) is a Banach space if endowed with the norm ˆ 1/2 for every f ∈ W 1,2 (X). |f |2 dm + Ch(f ) kf kW 1,2 (X) := In the case in which the measure m is boundedly finite, some equivalent notions of Sobolev spaces have been introduced in [3, 5, 24]. It is worth pointing out that in most cases the Sobolev space W 1,2 (X) is separable. Indeed, as proven in [2], the separability is granted by the reflexivity of the Sobolev space, which is in turn known to hold on a vast class of metric measure spaces; for instance, whenever the underlying metric space (X, d) is doubling (cf. [2]) or carries a ‘nice’ geometric structure (such as Euclidean spaces, Carnot groups, Finsler manifolds, subRiemannian manifolds, and locally CAT(κ) spaces; cf. the introduction of [19]). To the best of our knowledge, the only known examples of a non-separable Sobolev space are described in [2]. With the terminology introduced above at our disposal, we can now state and prove our existence and uniqueness result about the cotangent bundle of a metric measure space: Theorem B.1 (Cotangent bundle). Let (X, d, m) be a metric measure space such that the Sobolev space W 1,2 (X) is separable. Let B be a universal separable Banach space. Then there exists a unique couple (T∗ X, d), where T∗ X is a separable Banach B-bundle over X (in the sense of Definition 4.1) called the cotangent bundle of (X, d, m) and d : W 1,2 (X) → Γ(T∗ X) is a linear operator called the differential, such that the following properties are satisfied: i) It holds that |df | = |Df | in the m-a.e. sense for every f ∈ W 1,2 (X). ii) Given any dense sequence (fn )n in W 1,2 (X), it holds that  dfn (x) n ∈ N is dense in T∗x X := T∗ X(x) for m-a.e. x ∈ X. Uniqueness is intended up to unique isomorphism: given any other couple (E, d′ ) with the same properties, there exists a unique isomorphism ϕ : T∗ X → E such that d W 1,2 (X) d′ Γ(T∗ X) Γ(ϕ) Γ(E) is a commutative diagram. REPRESENTATION THEOREMS FOR NORMED MODULES 39 Proof. First of all, recall the notion of cotangent module introduced in [10, Definition 2.2.1] (cf. [13, Proposition 4.1.8] for the formulation we will present, via normed L0 (m)-modules):
there exists a unique couple L0 (T∗ X), d , where L0 (T∗ X) is a separable normed L0 (m)module and d : W 1,2 (X) → L0 (T∗ X) is a linear operator, such that: i′ ) |df | = |Df | holds m-a.e. for every f ∈ W 1,2 (X). ii′ ) Given any (fn )n dense in W 1,2 (X), the family {dfn : n ∈ N} generates L0 (T∗ X). (The separability of L0 (T∗ X) is granted, e.g., by [13, Lemma 3.1.17].) Uniqueness is intended in the following sense: given another couple (M , d′ ) having the same properties, there exists a unique isomorphism Φ : L0 (T∗ X) → M of normed L0 (m)-modules such that Φ ◦ d = d′ . By applying Lemma 4.11, we find a separable Banach B-bundle T∗ X over the space X such that Γ(T∗ X) ∼ = L0 (T∗ X). It is clear that the map d : W 1,2 (X) → Γ(T∗ X) satisfies the item i) as a consequence of i′ ). Moreover, the validity of the item ii) can be deduced from ii′ ) by suitably adapting the arguments in the proof of Proposition 4.4. Finally, the uniqueness of the couple (T∗ X, d) up to unique isomorphism can be obtained by combining the analogous property
of L0 (T∗ X), d with the fact (Theorem 4.15) that the section functor is an equivalence of categories. Therefore, the statement is achieved.  Remark B.2. The theory of normed modules is not really used in Theorem B.1, in the sense that the spaces Γ(T∗ X) and Γ(E) can be just considered as vector spaces, without looking at their module structure. The same observation applies to the morphism Γ(ϕ) as well.  Appendix C. The pullback bundle In the field of nonsmooth differential geometry, a prominent role is played by the notion of pullback of a normed module, which we recalled in Theorem 2.18. The primary purpose of this section is to show that separable Banach bundles come with a natural notion of pullback (see Definition C.1) which is consistent with that of normed modules (see Theorem C.3). For technical reasons (namely, because we will need the Disintegration Theorem C.2), we will mostly work with metric measure spaces (instead of general σ-finite measure spaces). Furthermore, we study the projection operator Prϕ associated with a normed L0 -module M (see Theorem C.9), which is a distinguished left inverse of the pullback map ϕ∗ : M → ϕ∗ M . In the setting of separable normed modules, we also provide a fiberwise description of the operator Prϕ ; see Proposition C.11 for the details. C.1. Pullback and section functors commute. Let us begin with the definition of pullback of a separable Banach bundle. Definition C.1 (Pullback bundle). Let (X, ΣX ), (Y, ΣY ) be measurable spaces and B a universal separable Banach space. Let ϕ : X → Y be a measurable map. Let E be a separable Banach B-bundle over Y. Then we define the map ϕ∗ E : X → Gr(B) as
ϕ∗ E(x) := E ϕ(x) for every x ∈ X. 40 SIMONE DI MARINO, DANKA LUČIĆ, AND ENRICO PASQUALETTO It follows from Lemma 2.7 that ϕ∗ E is a separable Banach B-bundle over X. We call it the pullback bundle of E under the map ϕ. Let (X, dX ), (Y, dY ) be separable metric spaces. By P(X) we mean the family of all Borel probability measures µ on X, i.e., with µ(X) = 1. Then a family {µy }y∈Y ⊆ P(X) is said to be a Borel family of measures provided for any Borel function f : X → [0, +∞] it holds that ˆ Y ∋ y 7−→ f dµy ∈ [0, +∞] is Borel measurable. We need the following classical result, whose proof can be found, e.g., in [4, Theorem 5.3.1]. Theorem C.2 (Disintegration theorem). Let (X, dX , mX ), (Y, dY , mY ) be metric measure spaces, with mX , mY finite. Let ϕ : X → Y be a Borel map satisfying ϕ∗ mX = mY . Then there exists a mY -a.e. uniquely determined Borel family of measures {myX }y∈Y ⊆ P(X) such that
myX X \ ϕ−1 (y) = 0 for mY -a.e. y ∈ Y, (C.1a)  ˆ ˆ ˆ (C.1b) f dmX = f dmyX dmY (y) for every f : X → [0, +∞] Borel. We abbreviate the conditions (C.1a) and (C.1b) to the single expression mX = ´ myX dmY (y). We are now ready to prove that the sections of the pullback bundle can be identified with the elements of the pullback of the space of sections of the bundle itself. Theorem C.3 (Pullback and section functors commute). Let (X, dX , mX ), (Y, dY , mY ) be metric measure spaces, with mX , mY finite. Let B be a universal separable Banach space and let ϕ : X → Y be a Borel map satisfying ϕ∗ mX = mY . Let E be a separable Banach B-bundle over Y. Then it holds that ϕ∗ Γ(E) ∼ = Γ(ϕ∗ E),
the pullback map ϕ∗ : Γ(E) → Γ(ϕ∗ E) being given by (ϕ∗ s)(x) := s ϕ(x) for mX -a.e. x ∈ X.
Proof. First of all, observe that |ϕ∗ s|(x) = (s◦ϕ)(x) B = |s|◦ϕ (x) holds for every s ∈ Γ(E) and mX -a.e. x ∈ X. Consequently, in order to prove the statement, it suffices to show that the  family ϕ∗ s : s ∈ Γ(E) generates Γ(ϕ∗ E) on X. To this aim, let t ∈ Γ(ϕ∗ E) and ε > 0 be fixed. Thanks to the separability of B, we can find a Borel partition (An )n∈N of X such that for every n ∈ N it holds that t(x) − t(x′ ) B ≤ ε for mX -a.e. x, x′ ∈ An . By using Theorem ´ C.2, we can disintegrate the measure mX along ϕ as myX dmY (y). Then let us define X t̃ := [χAn ]mX · ϕ∗ sn ∈ Γ(ϕ∗ E), (C.2) n∈N where for every n ∈ N the section sn ∈ Γ(E) is given by sn (y) := An t(x) dmyX (x) ∈ E(y) for mY -a.e. y ∈ Y, (C.3) ffl with the convention that An t dmyX := 0E(y) if myX (An ) = 0. Some verifications are in order: in view of (C.1a), we know that for mY -a.e. y ∈ Y it holds t(x) ∈ E(y) for myX -a.e. x ∈ X. REPRESENTATION THEOREMS FOR NORMED MODULES 41 Being the map t Borel (thus strongly Borel, as B is separable) and bounded on the set An , ffl we have that the Bochner integral An t dmyX ∈ E(y) exists for mY -a.e. y ∈ Y, whence the well-posedness of (C.2) follows. Moreover, given any element ω ∈ B′ , it holds that ´   ω[t(x)] dmyX (x) ´ ω sn (y) = χ{y′ ∈Y : my′ (A )>0} (y) for mY -a.e. y ∈ Y. χAn dmyX n X   Given that {myX }y∈Y is a Borel family of measures, we deduce that Y ∋ y 7→ ω sn (y) ∈ R is a Borel function, thus accordingly the map sn : Y → B is weakly Borel. Being B separable, we conclude that sn is Borel. Therefore, we have that (sn )n∈N ⊆ Γ(E), so that the definition in (C.2) is meaningful. Given any n ∈ N and mX -a.e. x ∈ An , we may estimate t̃(x) − t(x) B = t(x) − (ϕ∗ sn )(x) = An B ϕ(x) = t(x) − An
ϕ(x) t(x) − t(x′ ) dmX (x′ ) t(x′ ) dmX B t(x) − t(x′ ) ≤ B (x′ ) An ϕ(x) B dmX (x′ ) ≤ ε, whence dΓ(ϕ∗ E) (t̃, t) ≤ ε. This shows that ϕ∗ s : s ∈ Γ(E) generates Γ(ϕ∗ E) on X.   Remark C.4. Some of the assumptions of Theorem C.3 might be dropped: i) The result holds whenever mX and mY are σ-finite. Indeed, by arguing as we did at the beginning of §2.4, one can find m′X ∈ P(X) such that mX ≪ m′X ≪ mX . Observe that mY ≪ ϕ∗ m′X ≪ mY . Given that L0 (m′X ) = L0 (mX ) and L0 (ϕ∗ m′X ) = L0 (mY ), the claim immediately follows from Theorem C.3. ii) In addition, the assumption ϕ∗ mX = mY can be relaxed to ϕ∗ mX ≪ mY (requiring that the measure ϕ∗ mX is σ-finite when mX (X) = +∞). To show it, consider the following equivalence relation ∼ on a given normed L0 (mY )-module M : for any v, w ∈ M , we declare that v ∼ w if and only if |v − w| = 0 holds ϕ∗ mX -a.e. on Y. Then the quotient Mϕ∗ mX := M / ∼ inherits a natural structure of normed L0 (ϕ∗ mX )-module. Moreover, it is easy to prove that ϕ∗ Mϕ∗ mX ∼ = ϕ∗ M . Hence, given a separable Banach B-bundle E over Y, we can deduce from Theorem C.3 that ϕ∗ Γ(E) ∼ = ϕ∗ Γ(E)ϕ∗ mX ∼ = Γ(ϕ∗ E). We omit the details and will not insist further on these observations.  C.2. The projection operator Prϕ . Let us begin by recalling a few basic notions and results about the projection operator Prϕ for bounded functions. Cf. [12] for a related discussion. Definition C.5 (The operator Prϕ for functions). Let (X, ΣX , mX ), (Y, ΣY , mY ) be σ-finite measure spaces. Let ϕ : X → Y be a measurable map satisfying ϕ∗ mX = mY . Then we define the projection operator Prϕ : L∞ (mX ) → L∞ (mY ) as the linear, 1-Lipschitz mapping Prϕ (f ) := ± dϕ∗ (f + mX ) dϕ∗ (f − mX ) − ∈ L∞ (mY ) dmY dmY for every f ∈ L∞ (mX ), (f mX ) where dϕ∗dm stands for the Radon–Nikodým derivative of ϕ∗ (f ± mX ) with respect to mY , Y + while f := f ∨0 and f − := −f ∧0 are the positive part and the negative part of f , respectively. 42 SIMONE DI MARINO, DANKA LUČIĆ, AND ENRICO PASQUALETTO Let us briefly comment on the well-posedness of Prϕ : given that
ϕ∗ (f ± mX ) ≤ ϕ∗ kf kL∞ (mX ) mX = kf kL∞ (mX ) ϕ∗ mX = kf kL∞ (mX ) mY , we know that ϕ∗ (f ± mX ) is σ-finite and absolutely continuous with respect to mY , so that the (f ± mX ) Radon–Nikodým derivative dϕ∗dm exists. Moreover, the same estimate shows that Y dϕ∗ (f + mX ) dϕ∗ (f − mX ) dϕ∗ (f + mX ) dϕ∗ (f − mX ) ≤ − + dmY dmY dmY dmY
dϕ∗ (f + + f − ) mX dϕ∗ (|f | mX ) = ≤ kf kL∞ (mX ) = dmY dmY Prϕ (f ) = holds mY -a.e. on Y for any given function f ∈ L∞ (mX ), thus Prϕ (f ) L∞ (m ) ≤ kf kL∞ (mX ) . Y Finally, the linearity of Prϕ can be directly checked from its definition. Remark C.6. It is straightforward to check the validity of the following properties: Prϕ (c) = c Prϕ (f ) ≤ Prϕ (g) Prϕ (g ◦ ϕ f ) = g Prϕ (f ) for every c ∈ R, (C.4a) ∞ for every f, g ∈ L (mX ) with f ≤ g, ∞ ∞ for every f ∈ L (mX ) and g ∈ L (mY ), where all equalities and inequalities are intended in the almost everywhere sense. (C.4b) (C.4c)  Example C.7. In general, the projection operator Prϕ : L∞ (mX ) → L∞ (mY ) cannot be extended to a continuous map from L0 (mX ) to L0 (mY ), as shown by the following example. P Let us consider X := N and Y := {0}, endowed with the Borel measures mX := i∈N 2−i δi and mY := δ0 , respectively. The unique map ϕ : X → Y (sending all elements to 0) is Borel and satisfies ϕ∗ mX = mY . We argue by contradiction: suppose there exists a continuous P extension T : L0 (mX ) → L0 (mY ) of Prϕ . Define fn := ni=1 2i χi ∈ L∞ (mX ) for every n ∈ N P and f := i∈N 2i χi ∈ L0 (mX ). Notice that fn → f in L0 (mX ) as n → ∞ and L0 (mY ) ∼ = R. Given that Prϕ (fn ) = n for every n ∈ N, we deduce that T (f ) = lim T (fn ) = lim Prϕ (fn ) = +∞, n→∞ n→∞ which leads to a contradiction. Therefore, Prϕ cannot be extended to such a map T .  In the metric measure space setting, the operator Prϕ : L∞ (mX ) → L∞ (mY ) can be alternatively described (thanks to the disintegration theorem) in the following way: Proposition C.8 (Characterisation of Prϕ for functions). Let (X, dX , mX ), (Y, dY , mY ) be metric measure spaces, with mX , mY finite. Let ϕ : X → Y be a Borel map with ϕ∗ mX = mY . Then for every function f ∈ L∞ (mX ) it holds that ˆ Prϕ (f )(y) = f dmyX for mY -a.e. y ∈ Y, (C.5) where mX = ´ myX dmY (y). REPRESENTATION THEOREMS FOR NORMED MODULES 43 (f mX ) . Therefore, in order to prove Proof. Given that mX is finite, it holds that Prϕ (f ) = dϕ∗dm Y the statement it is sufficient to show that  ˆ ˆ y f dmX dmY (y) for every A ⊆ Y Borel. ϕ∗ (f mX )(A) = (C.6) A Since for mY -a.e. y ∈ Y it holds ϕ(x) = y for myX -a.e. x ∈ X by (C.1a), we have that  ˆ ˆ ˆ
(C.1b) y χ χ ϕ∗ (f mX )(A) = A ϕ(x) f (x) dmX (x) dmY (y) A ◦ ϕ f dmX =  ˆ ˆ = f dmyX dmY (y), A thus proving (C.6). Consequently, the proof is complete.  We are now in a position to generalise the object Prϕ to the framework of normed modules. This construction has been first obtained in [10] and later studied in [12]. Here we work with normed L∞ -modules equipped with a pointwise norm taking values in L∞ , thus we need to provide a slightly different proof, but the ideas are essentially borrowed from [10, 12]. Theorem C.9 (The operator Prϕ for modules). Let (X, ΣX , mX ), (Y, ΣY , mY ) be σ-finite measure spaces. Let ϕ : X → Y be a measurable map satisfying ϕ∗ mX = mY . Let M ∞ be a normed L∞ (mY )-module. Denote M 0 := C(M ∞ ). Then there exists a unique linear and continuous operator Prϕ : R(ϕ∗ M 0 ) → M ∞ – called the projection operator – such that Prϕ (f · ϕ∗ v) = Prϕ (f ) · v for every f ∈ L∞ (mX ) and v ∈ M ∞ . (C.7) Moreover, it holds that
Prϕ (w) ≤ Prϕ |w| mY -a.e. for every w ∈ R(ϕ∗ M 0 ). (C.8) Proof. Denote by V the family of all those elements w ∈ R(ϕ∗ M 0 ) that can be written as P w = n∈N [χAn ]mX · ϕ∗ vn , for a partition (An )n∈N ⊆ ΣX of X and a sequence (vn )n∈N ⊆ M ∞ . Note that V is a dense linear subspace of R(ϕ∗ M 0 ). We are forced to set Prϕ : V → M ∞ as X X
Prϕ (w) := Prϕ [χAn ]mX · vn ∈ M ∞ for every w = [χAn ]mX · ϕ∗ vn ∈ V, n∈N n∈N where the sum is intended with respect to the distance dM 0 . Let us check that such sum is actually well-defined: given any k ∈ N, we have the mY -a.e. inequality k X n=1 k k X


(C.4c) X Prϕ [χAn ]mX |vn | ◦ ϕ Prϕ [χAn ]mX |vn | = Prϕ [χAn ]mX · vn = n=1 n=1 = Prϕ  X k [χAn ]mX n=1  (C.4b)
∗ · ϕ vn ≤ Prϕ |w| ,

P whence it follows that n∈N Prϕ [χAn ]mX · vn ≤ Prϕ |w| holds mY -a.e. on Y. This grants
P that the sum n∈N Prϕ [χAn ]mX · vn exists in M 0 and defines an element Prϕ (w) ∈ M ∞ . Moreover, the same estimates show that (C.8) holds for every w ∈ V, thus Prϕ can be uniquely extended to a linear and continuous map Prϕ : R(ϕ∗ M 0 ) → M ∞ , which also satisfies (C.8). 44 SIMONE DI MARINO, DANKA LUČIĆ, AND ENRICO PASQUALETTO By construction, the resulting map Prϕ is the unique linear and continuous operator satisfying (C.7) for functions f ∈ L∞ (mX ) of the form f = [χA ]mX , where A ∈ ΣX . Finally, since simple  functions are dense in L∞ (mX ), one can easily deduce that (C.7) is verified. Remark C.10. Observe that L∞ (mX ) and L0 (mX ) have a natural structure of normed
L∞ (mX )-module and normed L0 (mX )-module, respectively, and C L∞ (mX ) = L0 (mX ); the same holds for L∞ (mY ) and L0 (mY ). Moreover, the pullback map ϕ∗ : L0 (mY ) → L0 (mX ) is given by ϕ∗ g = g ◦ ϕ for every g ∈ L0 (mY ). Therefore, by using (C.4c) we deduce that the operator Prϕ : L∞ (mX ) → L∞ (mY ) actually coincides with the projection operator between normed modules, thus accordingly no ambiguity may arise.  We conclude by showing that the projection operator Prϕ (associated with a separable normed module) can be characterised in a fiberwise way, thus generalising Proposition C.8. Proposition C.11 (Characterisation of Prϕ for modules). Let (X, dX , mX ), (Y, dY , mY ) be metric measure spaces, with mX , mY finite. Let ϕ : X → Y be a Borel map with ϕ∗ mX = mY . Let B be a universal separable Banach space. Let E be a separable Banach B-bundle over Y. Then the projection operator Prϕ : Γb (ϕ∗ E) → Γb (E) is given by ˆ Prϕ (t)(y) = t(x) dmyX (x) ∈ E(y) for every t ∈ Γb (ϕ∗ E) and mY -a.e. y ∈ Y, where mX = ´ myX dmY (y).

Proof. First of all, let us observe that Theorem C.3 gives Γb (ϕ∗ E) = R Γ(ϕ∗ E) ∼ = R ϕ∗ Γ(E) , thus the projection operator Prϕ associated with Γ(E) can be seen as a mapping from Γb (ϕ∗ E) ´ to Γb (E). Given any t ∈ Γb (ϕ∗ E), we define Φ(t)(y) := t dmyX ∈ E(y) for mY -a.e. y ∈ Y. By arguing as in the proof of Theorem C.3, one can see that Φ(t) ∈ Γb (E). The resulting map Φ : Γb (ϕ∗ E) → Γb (E) is linear and 1-Lipschitz. For any f ∈ L∞ (mX ) and s ∈ Γb (E) we have ˆ  ˆ

(C.5) y y ∗ Φ(f · ϕ s)(y) = f (x) s ϕ(x) dmX (x) = f dmX s(y) = Prϕ (f ) · s (y) for mY -a.e. y ∈ Y, thus accordingly Φ = Prϕ by Theorem C.9, which yields the statement.  Acknowledgements. The authors would like to thank Nicola Gigli for having suggested the study of liftings in the setting of normed modules. They would also like to thank Tapio Rajala for his useful comments on §4, as well as Milica Lučić, Ivana Vojnović, and Daniele Semola for the careful reading of a preliminary version of the manuscript. The second and third named authors acknowledge the support by the Academy of Finland, project number 314789. The second named author also acknowledges the support by the project 2017TEXA3H “Gradient flows, Optimal Transport and Metric Measure Structures”, funded by the Italian Ministry of Research and University. The third named author also acknowledges the support by the Balzan project led by Luigi Ambrosio. REPRESENTATION THEOREMS FOR NORMED MODULES 45 References [1] C. 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Pap, ed.), Elsevier, Amsterdam, 2002. [26] M. Takesaki, Theory of operator algebras. I, Springer-Verlag, New York-Heidelberg, 1979. [27] N. Weaver, Lipschitz algebras and derivations. II. Exterior differentiation, J. Funct. Anal., 178 (2000), pp. 64–112. Università di Genova (DIMA), MaLGa, Via Dodecaneso 35, 16146 Genova, Italy Email address: [email protected] Università di Pisa, Dipartimento di Matematica, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy Email address: [email protected] Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy Email address: [email protected]