Toward an optimal theory of integration for functions taking values in quasi-Banach spaces

We present a new approach to define a suitable integral for functions with values in quasi-Banach spaces. The integrals of Bochner and Riemann have deficiencies in the non-locally convex setting. The study of an integral for p-Banach spaces initiated by Vogt is neither totally satisfactory, since there are quasi-Banach spaces which are p-convex for all 0<p<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<p<1$$\end{document}, so it is not always possible to choose an optimal p to develop the integration. Our method puts the emphasis on the galb of the space, which permits a precise definition of its convexity. The integration works for all spaces of galbs known in the literature. We finish with a fundamental theorem of calculus for our integral.


Introduction
If X is a non-locally convex space, it is easy to construct a sequence of simple functions as n goes to infinity, where μ denotes the Lebesgue measure (cf. [31,pp. 121-123]). Therefore, Bochner-Lebesgue integration cannot be extended to non-locally convex spaces. On the other hand, the definition of the Riemann integral extends verbatim for functions defined on an interval [a, b] with values in an F-space X. However, it has some problems in the non-locally convex setting. For example, Mazur and Orlicz [28] proved that the F-space X is non-locally convex if and only if there is a continuous function f : [0, 1] → X which is not Riemann integrable. But the main drawback is that the Riemann integral operator I R , acting from the set of X-valued simple functions S([a, b], X) to X by is not continuous when X is not locally convex (see [1,Theorem 2.3]). An important attempt (somehow missed in the literature) to develop a theory of integration based on operators for functions with values in a quasi-Banach-space (i.e. a locally bounded F-space) was initiated by Vogt [40]. A remarkable theorem of Aoki and Rolewicz [5,30] says that any quasi-normed space is p-convex for some 0 < p ≤ 1. The idea of Vogt was the following. Given a quasi-Banach space X, let 0 < p ≤ 1 be such that X is p-convex. For this fixed p, he developed a theory of integration based on an identification of tensor spaces with function spaces (see [40,Satz 4]). Among the papers that approach integration of quasi-Banach-valued functions from Vogt's point of view we highlight [27].
The main advantage of Vogt's integration with respect other approaches to integration in the non locally convex setting is that it provides a bounded operator from the space of integrable functions into the target quasi-Banach space. Regarding the limitations, its main drawback is that it depends heavily on the convexity parameter p chosen, and for some spaces there is no optimal choice of p. Take, for instance, the weak Lorentz space L 1,∞ = L 1,∞ (R). This classical space, despite not being locally convex, is p-convex for any 0 < p < 1 (see [18,(2.3) and (2.6)]).
The concept that permits a precise definition of the convexity of a space was introduced and developed by Turpin in a series of papers (cf. [36,37]) and a monograph ( [38]) in the early 1970's. Given an F-space X, its galb, denoted by G (X), is the vector space of all sequences (a n ) ∞ n=1 of scalars such that whenever (x n ) ∞ n=1 is a sequence in X with lim x n = 0, the series ∞ n=1 a n x n converges in X. We say that a sequence space Y galbs X if Y ⊆ G (X). With this terminology, X is p-convex if and only if p ⊆ G (X).
The galb of certain classical spaces is known. Turpin [36] computed the galb of locally bounded, non-locally convex Orlicz function spaces L ϕ (μ), where μ is either a nonatomic measure or the counting measure, and showed that the result is an Orlicz sequence space φ modeled after a different Orlicz function φ. Hernández [13][14][15] continued the study initiated by Turpin and computed, in particular, the galb of certain vector-valued Orlicz spaces. The study of the convexity of Lorentz spaces took a different route. Before Turpin invented the notion of galb, Stein and Weiss [35] proved that the Orlicz sequence space log galbs L 1,∞ , and used this result to achieve a Fourier multiplier theorem for L 1,∞ . Sjögren [33] concluded the study by (implicitely) proving that G (L 1,∞ ) = log . Later on, the convexity type of Lorentz spaces L 1,q for 0 < q < ∞ was estudied (see [10,34]). In [9], general weighted Lorentz spaces were considered.
The geometry of spaces of galbs is quite unknown, however. Probably, the most significant advance in this direction since seminal Turpin work was made in [19]. Solving a question raised in [38], Kalton proved that if X is p-convex and is not q-convex for any q > p, then G (X) = p .
In this paper, we use galbs to develop a theory of integration for functions taking values in quasi-Banach spaces in the spirit of Vogt that fits as well as possible the convexity of the target space. Our construction is closely related to tensor products, and to carry out it we construct topological tensor products adapted to our neeeds. More precisely, given a quasi-Banach space X and a σ -finite measure space (Ω, Σ, μ), for an appropriate function quasi-norm λ over N (see Definition 1) we define the tensor product space X ⊗ λ L 1 (μ) so that there are bounded linear canonical maps If I factors through J , that is, there is a map I (defined on the range of J ) such that the diagram X ⊗ λ L 1 (μ) J I L λ 1 (μ, X) := J (X ⊗ λ L 1 (μ)) I X commutes, then I defines a suitable integral for functions in L λ 1 (X). Thus, we say that (λ, X) is amenable if λ galbs X (i.e., (a n ) ∞ n=1 ∈ G (X) whenever λ((a n ) ∞ n=1 ) < ∞) and I factors through J .
There is a tight connection between the existence of the integral I and the injectivity of J . In fact, we will prove that if (λ, X) is amenable, then J is one-to-one (see Theorem 5). This connection leads us to study the injectivity of J . More generally, we consider the map J : X ⊗ λ L ρ → L ρ (X) associated with the quasi-Banach space X, the function quasi-norm λ and a function quasinorm ρ over (Ω, Σ, μ), and we obtain results that generalize those previously obtained for Lebesgue spaces L q (μ) and tensor quasi-norms in the sense of p , 0 < p ≤ q ≤ ∞ (see [40,Satz 4]).
With the terminology of this paper, Vogt proved that if X is a p-Banach space (see the precise definition in the next section), 0 < p ≤ 1, then ( p , X) is amenable. So, in order to exhibit the applicability of the theory of integration developed within this paper, we must exhibit new examples of amenable pairs. Since the space of galbs of the quasi-Banach space X arises from a function quasi-norm on N, say λ X , the question of whether the pair (λ X , X) is amenable arises. For answering it, one first need to know whether the space of galbs G (X) is always 1-concave as a quasi-Banach lattice or not. See Questions 2 and 4. As long as there is no general answer to these questions, we focus on the spaces of galbs that have appeared in the literature. In Theorem 6, we prove that for all of them Question 4 has a positive answer.
Once the theory is built, the first goal should be the study of its integration properties. By construction, our integral behaves linearly and has suitable convergence properties. Hence, we finish with a fundamental theorem of calculus for our integral (see Theorem 7).
The paper is organized as follows. In Sect. 2, we introduce the terminology and notation that will be employed. The theory of function norms (i.e., the locally convex setting) has been deeply developed (cf. [6,26]). However, even though some generalizations of results from harmonic analysis to quasi-Banach function spaces are known (see, e.g., [9,16,17] for recent work within this area), a systematic study in the non-locally convex setting is missing. For that reason, in Sect. 3, we do a brief survey on function quasi-norms covering the most relevant aspects, and all the results that we need. Section 4 is devoted to galbs. In Sect. 5, we briefly collect some results on tensor products. In Sect. 6, we present our main results on integration for functions taking values in quasi-Banach spaces. Finally, in Sect. 7, we give a fundamental theorem of calculus that improves [1,Theorem 5.2].

Terminology
We use standard terminology and notation in Banach space theory as can be found, e.g., in [3]. The unfamiliar reader will find general information about quasi-Banach spaces in [23]. We next gather the notation on quasi-Banach spaces that we will use.
A quasi-normed space will be a vector space over the real or complex field F endowed with a quasi-norm, i.e., a map · : X → [0, ∞) satisfying (Q.1) x = 0 if and only if x = 0; (Q.2) t x = |t| x for t ∈ F and x ∈ X; and (Q.3) there is a constant κ ≥ 1 so that for all x and y in X we have The smallest number κ in (Q.3) will be called the modulus of concavity of the quasi-norm. If it is possible to take κ = 1 we obtain a norm. A quasi-norm clearly defines a metrizable vector topology on X whose base of neighborhoods of zero is given by sets of the form {x ∈ X : x < 1/n}, n ∈ N. Given 0 < p ≤ 1, a quasi-normed space is said to be p-convex if it has an absolutely p-convex neighborhood of the origin. A quasi-normed space X is p-convex if and only if there is a constant C such that If, besides (Q.1) and (Q.2), (1) holds with C = 1 we say that · is a p-norm. Any p-norm is a quasi-norm with modulus of concavity at most 2 1/ p−1 . A p-normed space is a quasi-normed space endowed with a p-norm. By the Aoki-Rolewicz theorem [5,30] any quasi-normed space is p-convex for some 0 < p ≤ 1. In turn, any p-convex quasi-normed space can be equipped with an equivalent p-norm. Hence, any quasi-normed space becomes, for some 0 < p ≤ 1, a p-normed space under suitable renorming.
A p-Banach (resp. quasi-Banach) space is a complete p-normed (resp. quasi-normed) space. It is known that a p-convex quasi-normed space is complete if and only if for every sequence (x n ) ∞ n=1 in X such that ∞ n=1 x n p < ∞ the series ∞ n=1 x n converges.
A semi-quasi-norm on a vector space X is a map · : X → [0, ∞) satisfying (Q.2) and (Q.3). A standard procedure, to which we refer as the completion method allow us to manufacture a quasi-Banach from a semi-quasi-norm (see e.g. [2,Sect. 2.2]).
As the Hahn-Banach Theorem depends heavily on convexity, it does not pass through general quasi-Banach spaces. In fact, there are quasi-Banach spaces as L p ([0, 1]) for 0 < p < 1 whose dual space is null (see [11]). Following [23], we say that the quasi-Banach space X has point separation property if for every Given a σ -finite measure space (Ω, Σ, μ) and a quasi-Banach space X, we denote by L + 0 (μ) the set consisting of all measurable functions from Ω into [0, ∞], and by L 0 (μ, X) the vector space consisting of all measurable functions from Ω into X. As usual, we identify almost everywhere (a.e. for short) coincident functions. We set L 0 (μ) = L 0 (μ, F) and We denote by S(μ, X) the vector space consisting of all integrable X-valued simple functions. That is, An order ideal in L 0 (μ) will be a (linear) subspace L of L 0 (μ) such that the conjugate function f ∈ L whenever f ∈ L, and max{ f , g} ∈ L whenever f and g are real-valued functions in L. A cone in L + 0 (μ) will be a subset C of L + 0 (μ) such that for all f , g ∈ C and all α, β ≥ 0 we have f < ∞ a.e., α f + βg ∈ C, and max{ f , g} ∈ C. It is immediate that if L is an order ideal in L 0 (μ), then L + := L ∩ L + 0 (μ) is a cone in L + 0 (μ); and reciprocally, if C is a cone in L + 0 (μ), there is a unique order ideal L with L + = C. Namely, Given a quasi-Banach space X, we say that a quasi-Banach space U is complemented in X via a map S : U → X if there is a map P : X → U such that P • S = Id U . The unit vector system is the sequence (e k ) ∞ k=1 in F N defined by e k = (δ k,n ) ∞ n=1 , where δ k,n = 1 if k = n and δ k,n = 0 otherwise. A block basis sequence with respect to the unit vector system is a sequence ( f k ) ∞ k=1 such that f k = n k n=1+n k−1 a n e n , k ∈ N for some sequence (a n ) ∞ n=1 in F N and some increasing sequence (n k ) ∞ k=0 of non-negative scalars with n 0 = 0.
we do not impose them to satisfy a Fatou property (something that Bennet and Sharpley [6] do for function norms). We devote a subsection to the study of this property. Then we study the properties of absolute continuity and domination for function quasi-norms, as well as Minkowski-type inequalities. We also discuss the use of conditional expectation (via the notion of leveling function quasi-norms), which will be relevant for the proof of Theorem 5. We conclude the section with some comments on function quasi-norms over N endowed with the counting measure, a specially important particular case.
Definition 2 A function norm is a function quasi-norm with modulus of concavity 1. More generally, given 0 < p ≤ 1, a function p-norm is a function ρ : The inequality a p + b p ≤ 2 1− p (a + b) p for all a, b ∈ [0, ∞] and p ∈ (0, 1] yields that any function p-norm is a function quasi-norm with modulus of concavity at most 2 1/ p−1 . This generalization of the notion of a function norm follows ideas from [6,26]. Asides (F.5), the main differences between our definition and that adopted by Luxemburg and Zaanen in [26] lie in restricting ourselves to σ -finite spaces, and in imposing condition (F.3), which, on the one hand, prevents from existing non null sets E on which ρ is trivial (in the sense that if f ∈ L + 0 (μ) is null outside E then ρ( f ) is either 0 or ∞) and, on the other hand, guarantees the existence of enough functions with finite quasi-norm. Regarding the approach in [6], we point out that Bennet and Sharpley imposed a function norm to satisfy The most natural examples of functions quasi-norms are L p -quasi-norms, 0 < p < ∞, defined by To avoid introducing cumbrous notations, sometimes the symbol L p (μ) will mean the function quasi-norm defining the space L p (μ) instead of the space itself, and the same convention will be used for Lorentz and Orlicz spaces. Since, if μ is not purely atomic and 0 < p < 1, L p (μ) does not satisfy (F.7), imposing this condition to all function quasi-norms is somewhat nonsense in the non-locally convex setting. Thus we impose its natural substitute (F. 4) instead. Also, unlike Bennet and Sharpley, we do not a priori impose ρ to satisfy Fatou property (see Sect. 3.1). Given f ∈ L + 0 (μ) and a function quasi-norm ρ over (Ω, Σ, μ), we set ). If ρ is the function quasi-norm associated with L 1 (μ), then μ f := ρ f is the distribution function of f . We say f has a finite distribution function if μ f (s) < ∞ for all s > 0.

Definition 3
We say that a function quasi-norm ρ is rearrangement invariant if every function f ∈ L + 0 (μ) with ρ( f ) < ∞ has a finite distribution function, and ρ( f ) = ρ(g) whenever The proof of the following lemma is based on the elementary inequality Lemma 1 Let ρ be a function quasi-norm over a σ -finite measure space (Ω, Σ, μ).
Definition 4 A function quasi-norm ρ is said to be p-convex if there is a constant C such that Proof It is clear that any function p-norm is p-convex, and p-convexity is inherited by passing to an equivalent function quasi norm. Reciprocally, if ρ is a p-convex function quasi-norm over a σ -finite measure space (Ω, Σ, μ), then it is immediate that the map λ : L + 0 (μ) → [0, ∞] given by is a function p-norm equivalent to ρ.
Corollary 1 Any function quasi-norm is equivalent to a function p-norm for some 0 < p ≤ 1.
Proof It follows from Proposition 1 and Lemma 2.
In light of Corollary 1, it is natural, and convenient in some situations, to restrict ourselves to function quasi-norms that are function p-norms for some p. However, we emphasize that some p-convex spaces arising naturally in Mathematical Analysis are given by a function quasi-norm that is not a p-norm. Take, for instance the 1-convex (i.e., locally convex) function space L r ,∞ , r > 1. So, when working in the general framework of non-locally convex spaces, it is convenient to know whether a given property pass to equivalent function quasi-norms.

Definition 6
Let X be a quasi-Banach space, and let ρ be a function quasi-norm over a σ -finite measure space (Ω, Σ, μ). The space endowed with the gauge · ρ will be called the vector-valued Köthe space associated with ρ and X. The space L ρ = L ρ (F) will be called the Köthe space associated with ρ.
Note that we do not impose the functions in L ρ (X) to be strongly measurable. If ρ is the function quasi-norm associated to the Lebesgue space L p (μ), 0 < p < ∞, we set L p (μ, X) := L ρ (X). If A ∈ Σ, we set L p (A, μ, X) := L p (μ| A , X), where μ| A is the restriction of μ to Σ ∩ P( A). In general, if ρ| A is the function quasi-norm defined by It is clear that L ρ is an order ideal in L 0 (μ). By Lemma 1 (ii), its cone is given by

Lemma 3
Let ρ be a function quasi-norm over a σ -finite measure space (Ω, Σ, μ) and X be a quasi-Banach space.
(iii) If we endow L 0 (μ, X) with the vector topology of the local convergence in measure, then L ρ (X) ⊆ L 0 (μ, X) continuously.
Proof Statements (i), (ii), and (iii) are straightforward from the very definition of function quasi-norm and Lemma 1. Now let K be a closed subset of X, and let x be a function in L ρ (X) \ L ρ (K) (assuming that this set is non-empty). There is ε > 0 and A ⊆ Σ with μ(A) > 0 such that x(a) − k ≥ ε for all a ∈ A and all k ∈ K. Therefore x − y ρ ≥ ερ(χ A ) > 0 for all y ∈ L ρ (K), and we obtain (iv).

Lemma 4
Let ρ be a function quasi-norm, and let X be a Banach space. If a sequence (x n ) ∞ n=1 converges to x in L ρ (X), then ( x n ) ∞ n=1 converges to x in L ρ .
Proof It follows from the inequality | x n − x | ≤ x n − x for all n ∈ N.
Proposition 2 Let ρ be a function quasi-norm over a σ -finite measure space (Ω, Σ, μ), let X be a quasi-Banach space, and let (x n ) ∞ n=1 be a sequence in L 0 (μ, X) such that lim n x n − x ρ = 0 for some x ∈ L 0 (μ, X). Then, there is a subsequence (y n ) ∞ n=1 of (x n ) ∞ n=1 such that lim n y n = x a.e.

The Fatou property
Definition 7 Suppose that ρ is a function quasi-norm over a σ -finite measure space (Ω, Σ, μ). We say that ρ has the rough Fatou property if there is a constant C such that If the above holds with C = 1 we say that ρ has the Fatou property. We say that ρ has the weak Fatou property if ρ(lim n f n ) < ∞ whenever the non-decreasing sequence Note that Fatou property says that if f n f then ρ( f n ) ρ( f ), so it does not pass to equivalent function quasi-norms. In contrast, both rough and weak Fatou property are preserved. In fact, these two notions are equivalent.

Proposition 3 (cf. [4,Lemma]) If ρ is a function quasi-norm with the weak Fatou property, then it also has the rough Fatou property.
Proof Let ρ be a function quasi-norm over a σ -finite measure space (Ω, Σ, μ). By Corollary 1, we can assume without loss of generality that it is a function p-norm for some 0 < p ≤ 1. Suppose that ρ does not have the rough Fatou property. Then, for each is non-decreasing, and we have 2 −k/ p f k,n ≤ g := lim n g n , k ≤ n.
On the other hand, since ρ is a function p-norm, ρ p (g n ) ≤ n k=1 2 −k ≤ 1 for all n ∈ N. Therefore ρ does not have the weak Fatou property.
Using that ρ has the rough Fatou property (due to Proposition 3) and Proposition 1, we obtain that ρ( f ) < ∞. Since Definition 8 Let 0 < p ≤ 1 and let ρ be a function quasi-norm over a σ -finite measure space (Ω, Σ, μ). We say that ρ has the Riesz-Fischer p-property if for every sequence Lemma 5 (cf. [4,Theorem]) Let ρ be a p-convex function quasi-norm with the weak Fatou property. Then ρ has the Riesz-Fischer p-property. Proof Hence lim m ρ( m n=1 f n ) < ∞, and therefore ρ( ∞ n=1 f n ) < ∞ (since ρ has the weak Fatou property). That is, ρ has the Riesz-Fischer p-property.

Proposition 5 Let ρ be a function quasi-norm over a σ -finite measure space (Ω, Σ, μ).
Given 0 < p ≤ 1, the following statements are equivalent: Moreover, the optimal constant in (ii) is the p-convexity constant of L ρ . In particular, L ρ is a p-Banach space if and only if (ii) holds with C = 1.
Also, we have that any of the previous statements hold for some 0 < p ≤ 1 if and only if (iv) L ρ (X) is a quasi-Banach space for any (resp. some) nonzero quasi-Banach space X.

Proof
Let us see first that if (i) holds for some 0 < p ≤ 1, then (ii) also holds for the same p. We use an argument by contradiction. Fix 0 < p ≤ 1 and suppose that there is no Hence ρ does not have the Riesz-Fischer p-property, as we wanted to prove. Now suppose that (ii) holds for some 0 < p ≤ 1, and let X be any nonzero quasi-Banach space. Let us see that L ρ (X) is a quasi-Banach space. By Lemma 3 (i), we already know that L ρ (X) is a quasi-normed space. To obtain the completeness of the space, notice that it suffices to prove that the series ∞ n=1 f n converges in where κ is the modulus of concavity of X. Using (ii) and Lemma 1 (ii), we obtain that ∞ n=1 κ n f n converges a.e. in Ω; say it converges in Ω \ N where μ(N ) = 0. Set g n := f n χ Ω\N . Obviously g n ≤ f n , so ρ( g n ) ≤ ρ( f n ) for all n ∈ N. Then (2) is also true if we put g n instead of f n .
Therefore, L ρ (X) is a quasi-Banach space, as wanted. Now suppose that L ρ (X) is a quasi-Banach space for some nonzero quasi-Banach space X. Take a nonzero vector x in X. Since obviously F is isomorphic to {t x : t ∈ F}, which is a closed subset of X, it follows that L ρ is a quasi-Banach space using Lemma 3 (iv). By the Aoki-Rolewicz theorem, L ρ is p-convex for some 0 < p ≤ 1. Hence (iv) implies (iii).
Thus ρ has the Riesz-Fischer p-property, and we have proved that (iii) implies (i).
This completes the proof of the theorem.

Definition 9
Suppose that ρ is a function quasi-norm over a σ -finite measure space If the above holds only in the case when lim n f n = 0, we say that f is dominating. We denote by L a ρ (resp. L d ρ ) the set consisting of all f ∈ L 0 (μ) such that | f | is absolutely continuous (resp. dominating). We say that ρ is absolutely continuous for every E ∈ Σ(μ), we say that ρ is locally absolutely continuous (resp. locally dominating).
Notice that domination is preserved under equivalence of function quasi-norms, but absolute continuity is not. Propostion 6 below yields that if the function quasi-norm is continuous (in the sense that lim n x n ρ = x ρ whenever (x n ) ∞ n=1 and x in L ρ satisfy lim n x n − x ρ = 0), then both concepts are equivalent. Notice that any function p-norm, 0 < p ≤ 1, is continuous. So, the existence of non-continuous function quasi-norms is a 'pathology' which only occurs in the non-locally convex setting. We must point out that, since it is by no means clear whether absolutely continuous norms are continuous, the terminology could be somewhat confusing. Notwithstanding, we prefer to use terminology similar to that it is customary within framework of function norms.

Proposition 6
Let ρ be a function quasi-norm over a σ -finite measure space (Ω, Σ, μ). Suppose that f ∈ L + ρ is dominating. Then lim n x n = x in L ρ (X) for every quasi-Banach space X and every sequence (x n ) ∞ n=1 in L 0 (μ, X) with lim n x n = x a.e. and x n ≤ f a.e. for all n ∈ N.
Proof Let N be a null set such that sup n x n (ω) ≤ f (ω) < ∞ and lim n x n (ω) = x(ω) for all ω ∈ Ω \ N . Then x(ω) ≤ κ f (ω) for all ω ∈ Ω \ N , where κ is the modulus of concavity of the quasi-norm · . Set Let κ be the modulus of concavity of ρ, and fix ε > 0.
Given a function quasi-norm ρ and a set E ∈ Σ we define Notice that the function Φ[E, ρ] is non-negative and nondecreasing. In particular, there exists the limit of Φ[E, ρ](t) when t → 0 + .

Corollary 2 A function quasi-norm ρ is locally dominating if and only if
, using Proposition 7 we obtain that ρ is locally dominating. Now assume that s : decreases to a null set and ρ(χ B n ) ≥ s/2 for all n ∈ N, so χ B 1 is not dominating. Hence ρ is not locally dominating.

Definition 10
Let ρ be a function quasi-norm over a σ -finite measure space (Ω, Σ, μ). We say say L ⊆ L ρ is an order ideal with respect to ρ if it is an order ideal and it is closed in L ρ . Lemma 6 (cf. [6,Theorem 3.8]) Let ρ be a function quasi-norm over a σ -finite measure space (Ω, Σ, μ). Then L d ρ is an order ideal with respect to ρ.
Hence lim n ρ(g n ) = 0. So | f | is dominating, as we wanted to prove.
Proof It is obvious that L b ρ is an order ideal in L 0 (μ), and it is closed in L ρ by definition. Hence L b ρ is an order ideal with respect to ρ. Let f be a function in C, and set E : Then This means that C is contained in L b,+ ρ . Therefore, the closure of C in L ρ is also contained in L b,+ ρ . On the other hand, it is obvious that every non-negative simple function which is finite a.e. belongs to C. So the second part of the statement follows.
Proof It is enough to prove that L d,+ be an increasing sequence in Σ(μ) whose union is { f > 0} ⊆ Ω. Pick an increasing sequence ( f j ) ∞ j=1 of measurable positive simple functions with lim n f n = f . We have lim n ρ( f − f χ A n ) = 0 and lim j ρ( f χ A n − f j χ A n ) = 0 for each n ∈ N. Since f j χ A n ∈ L b,+ ρ for all j, n ∈ N, we infer that f ∈ L b,+ ρ .

Corollary 3 A function quasi-norm ρ is locally dominating if and only if
Proof It is a straightforward consequence of Proposition 8 Since we could need to deal with non-continuous function quasi-norms, we give some results pointing to ensure that lim n x n ρ = x ρ under the assumption that (x n ) ∞ n=1 converges to x. ρ be a function quasi-norm over a σ -finite measure space (Ω, Σ, μ) with the Fatou property, and let ( f n ) ∞ n=1 be a sequence in L + 0 (μ). Then

Lemma 8 Let
Proof Just apply Fatou property to inf k≥n f k , n ∈ N.

Lemma 9
Let ρ be a function quasi-norm with the Fatou property and X be a Banach space.
n=1 be a subsequence of (x n ) ∞ n=1 such that lim n ρ( y n ) = lim inf n ρ( x n ). Since lim n ρ( x − y n ) = 0, by Lemma 4 we have lim n ρ( x − y n ) = 0. Then Proposition 2 guarantees the existence of a subsequence (z n ) ∞ n=1 of (y n ) ∞ n=1 such that lim n z n = x . Using Lemma 8 we obtain as we wanted to prove.

Lemma 10
Let ρ be a function quasi-norm over a σ -finite measure space (Ω, Σ, μ) with the Fatou property, and let X be a Banach space. If x ∈ L 0 (μ, X) and (x n ) ∞ n=1 ⊆ L 0 (μ, X) satisfy lim n x n = x a.e., and sup n x n ≤ g for some g ∈ L a,+ ρ , then lim n ρ( x n ) = ρ( x ).
Proof Note that since lim n x n = x a.e. and X is a Banach space, we have lim n x n = x a.e.
Consider two particular cases. First, suppose that x n ≤ x for all n ∈ N. Obviously, lim sup n ρ( x n ) ≤ ρ( x ). Then, by Lemma 8, ρ( x ) ≤ lim inf n ρ( x n ). Second, suppose that x n ≥ x for all n ∈ N. Obviously, lim inf n ρ( x n ) ≥ ρ( x ). Set g n = sup k≥n x k . Then g ≥ g 1 and (g n ) ∞ n=1 is non-increasing with lim n g n = x a.e. Using the absolute continuity of g, we have lim sup n ρ( x n ) ≤ lim ρ(g n ) = ρ( x ). In the general case, set g n = min{ x n , x } and h n = max{ x n , x }. Then both (ρ(g n )) ∞ n=1 and (ρ(h n )) ∞ n=1 converge to ρ( x ). Since g n ≤ x n ≤ h n , the statement follows.

Proposition 9
Let ρ be a function quasi-norm over a σ -finite measure space (Ω, Σ, μ) with the Fatou property, and let X be a Banach space. If x ∈ L 0 (μ, X) and (x n ) ∞ n=1 ⊆ L 0 (μ, X) satisfy lim n x − x n ρ = 0, and sup n x n ≤ g for some g ∈ L a,+ ρ , then lim n ρ( x n ) = ρ( x ).
Proof It suffices to prove that any subsequence of (x n ) ∞ n=1 has a further subsequence (y n ) ∞ n=1 with lim n y n ρ = x ρ . But this follows combining Proposition 2 with Lemma 10.

The role of lattice convexity and Minkowski-type inequalities
Function spaces built from function quasi-norms have a lattice structure. Let ρ be a function quasi-norm over a σ -finite measure space (Ω, Σ, μ). Given 0 < p ≤ ∞, we say that ρ is lattice p-convex (resp. concave) if L ρ is. Equivalently, ρ is lattice p-convex (resp. concave) if and only if there is a constant C such that G ≤ C H (resp. H ≤ CG) for every n ∈ N and ρ p ( f j )) 1/ p . If the above holds for disjointly supported families, we say that ρ satisfies an upper (resp. lower) p-estimate.
If ρ is lattice p-convex, then it is p-convex, where p = min{1, p}. The notions of 1convexity and lattice 1-convexity are equivalent. This identification does not extend to p < 1 since there are function quasi-norms over N which are lattice p-convex for no p > 0 (see [22]). Kalton [21] characterized quasi-Banach lattices (in particular, function quasi-norms) that are p-convex for some p as those that are L-convex. We say that a function quasi-norm It is straightforward to check that ρ (r ) is a function quasi-norm. If ρ has the Fatou (resp. weak Fatou) property, then ρ (r ) does have. If ρ is p-convex (resp. concave), then ρ (r ) is pr-convex (resp. concave). We set L (r ) ρ = L ρ (r) . A question implicit in Sect. 3.2 is whether any p-convex function quasi-norm with the weak Fatou property is equivalent to a function p-norm with the Fatou property. For function norms the answer to this question is positive, and its proof relies on using the associated gauge ρ given by In fact, we have the following analogue of [6,Theorem 2.2 of Chapter 1].

Lemma 11 Let ρ be a function quasi-norm fulfiling (F.7). Then ρ is a function norm with the Fatou property.
Proof It is a routine checking. In the non-locally convex setting, it is hopeless to try to obtain full information for ρ from the associated function norm ρ . Nonetheless, the following is a partial positive answer to the aforementoned question.

Proposition 10
Let 0 < p < ∞ and let ρ be a function quasi-norm over a σ -finite measure space (Ω, Σ, μ). Suppose that ρ is p-convex, has the weak Fatou property, and that for every . Then ρ is equivalent to a function p-norm with the Fatou property. In fact, there is G ⊂ L + 0 (μ) such that ρ is equivalent to the function quasi-norm λ given by The function quasi-norm ρ (1/ p) is 1-convex and, then, equivalent to a function norm σ . The properties of ρ yields that σ satisfies (F.7) and has the weak Fatou property. By Theorem 1, σ is equivalent to the function norm σ . Consequently, ρ is equivalent to the function quasi-norm λ = σ ( p) . Hence, the result holds with G = {g ∈ L + 0 (μ) : σ (g) ≤ 1}.
Proof It suffices to prove the result for f . The Fatou property yields that if the result holds for a non-decreasing sequence ( f n ) ∞ n=1 , then it also holds for lim n f n . Consequently, we can suppose that μ(Ω) < ∞ and that f is a measurable simple function. Given a measurable simple positive function f we denote by M f the set consisting of all E in the product σalgebra Σ ⊗ T such that the result holds for f + tχ E for every t ≥ 0. The absolute continuity and the Fatou property yields that M f is a monotone class for any measurable simple function f . Therefore, if R denotes the algebra consisting of all finite disjoint unions of measurable rectangles, the monotone class theorem yields that R ⊆ M f implies Σ ⊗ T ⊆ M f . Let C r denote the cone consisting of all positive functions measurable with respect to R. Given n ∈ N, let C[n] be the cone consisting of all measurable non-negative functions which take at most n − 1 different positive values. It is straightforward to check that the result holds for all functions in C r (which is clearly equal to C r + C [1]). Suppose that the result holds for all In other words, the result holds for all functions in C r + C[n + 1]. By induction, the result holds for every f ∈ C := ∪ ∞ n=1 C r + C[n]. Since C is the cone consisting of all measurable simple non-negative functions, we are done.
Proposition 11 allows us to iteratively apply function quasi-noms to measurable functions defined on product spaces. A Minkowski-type inequality is an inequality that compares the gauges that appear when iterating in different ways.

Definition 13
Let ρ and λ be locally absolutely continuous function quasi-norms with the Fatou property over σ -finite measure spaces (Ω, Σ, μ) and (Θ, T , ν) respectively. Given We say that the pair (ρ, λ) has the Minkowski's integral inequality (MII for short) property if there is a constant C such that The following result is obtained from the corresponding one for function norms [32]. We do not know whether a direct proof which circumvent using lattice convexity is possible.

Theorem 2 Let ρ and λ be locally absolutely continuous L-convex function quasi-norms with the Fatou property. Then (ρ, λ) has the MII property if and only if there is 0 < p ≤ ∞ such that λ is lattice p-convex and ρ is lattice p-concave.
Proof Pick 0 < s < ∞ such that ρ (s) and λ (s) are 1-convex. Since

λ) has the MII property if and only if (ρ (s) , λ (s) ) does have. It turn, by [32,Theorems 2.3 and 2.5], (ρ (s) , λ (s) ) has the MII property if and only if there is q ∈ (0, ∞] such that λ (s) is lattice q-convex and ρ (s) is lattice q-concave.
This latter condition is equivalent to the existence of p ∈ (0, ∞] (related with q by q = sp) as desired.
Given 0 < p < ∞ and a σ -finite measure space (Ω, Σ, μ), the Lebesgue space L p (μ) is absolutely continuous and lattice p-convex. Moreover, if μ is infinite-dimensional, then L p (μ) is not lattice q-concave for any q < p. Consequently, we have the following.

Proposition 12
Let 0 < p < ∞ and ρ be a locally absolutely continuous L-convex function quasi-norm over an infinite-dimensional σ -finite measure space. Given another σ -finite measure space (Ω, Σ, μ) such that L 0 (μ) is infinite-dimensional, the pair (ρ, L p (μ)) has the MII property if and only if ρ is lattice p-concave.

Theorem 3 Let ρ be a locally absolutely continuous L-convex function quasi-norm, and let
(Ω, Σ, μ) an infinite-dimensional σ -finite measure space. Then the pair (ρ, L 1,∞ (μ)) has the MII property if and only if ρ is p-concave for some p < 1.

Conditional expectation in quasi-Banach function spaces
Given a sub-σ -algebra Σ 0 ⊆ Σ, we denote by L + 0 (μ, Σ 0 ) the set consisting of all nonnegative Σ 0 -measurable functions. Given f ∈ L + 0 (μ) there is a unique g ∈ L + 0 (μ, Σ 0 ) such that A f dμ = A g dμ for all A ∈ Σ 0 . We say that g is the conditional expectation of f with respect to Σ 0 , and we denote E( f , Σ 0 ) := g.
This terminology follows that used in [12]. We remark that Ellis and Halperin imposed leveling function norms to satisfy the above definition with C = 1. Not imposing conditional expectations to be contractive turns the notion stable under equivalence.

Lemma 13 Leveling function quasi-norms satisfy (F.7).
Proof Suppose that ρ is a leveling function quasi-norm over a σ -finite measure space (Ω, Σ, μ). Given A ∈ Σ(μ) with μ(A) > 0, let Σ 0 be the smallest σ -algebra contain- It is known that, if q ≥ 1, L q (μ) has the conditional expectation property. Locally convex Lorentz and Orlicz spaces do have. More generally, we have the following. Recall that a measure space is said to be resonant if either is non-atomic or it consists of equi-measurable atoms.

Theorem 4 Let ρ be a rearrangement invariant function norm over a resonant measure space. If ρ satisfies (F.7), then it is leveling.
Proof By Calderón-Mitjagin Theorem (see [8,29], and also [6,Theorem 2.2]), L ρ is an interpolation space between L 1 and L ∞ . Since both L 1 and L ∞ are leveling, the result follows by interpolation.

Function quasi-norms over N
Suppose that ρ is a function quasi-norm over N endowed with the counting measure. In this particular case, ρ is locally dominating, and the space of integrable simple functions is the space c 00 consisting of all eventually null sequences. Concerning the density of c 00 in L ρ we have the following.

Proposition 13
Let ρ be a function quasi-norm over N. Then ρ is not minimal if and only if ∞ is a subspace of L ρ , in which case L ρ has block basic sequence equivalent to the unit vector system of ∞ .
Before tackling the proof of Proposition 13 we give an auxiliary lemma that will be used a couple of times.

Lemma 14
Let ρ be a function quasi-norm over N and let (a n ) ∞ n=1 be a sequence in L ρ . Then (a n ) ∞ n=1 does not belong to L b ρ if and only there is an increasing sequence (m k ) ∞ k=1 of non-negative integers such that Proof Use that (a n ) ∞ n=1 ∈ L ρ \ L b ρ if and only if the series ∞ n=1 a n e n does not converge.

Proof of Proposition 13 Assume that
n=1+m 2k−1 a n e n , k ∈ N, then inf k x k ρ > 0 and sup m m k=1 x k ρ < ∞. So, (x k ) ∞ k=1 is a block basic sequence as desired.

Corollary 4
Let ρ be a function quasi-norm over N. If ρ satisfies a lower p-estimate for some p < ∞, then ρ is minimal and L-convex.
Proof Our assumptions yields that ∞ is not finitely represented in L ρ by means of block basic sequences. Then, result follows from Proposition 13 and [21,Theorem 4.1].
Notice that function quasi-norms over N are closely related to unconditional bases. In fact, if ρ is a function quasi-norm over N, then the unit vector system (e n ) ∞ n=1 is an unconditional is an unconditional basis of a quasi-Banach space X, then the mapping defines a function quasi-norm over N, and the linear map given by x n → e n extends to an isomorphism from X onto L b ρ .

Definition 16 A function quasi-norm over N is said to be
is a rearrangement of f = (a n ) ∞ n=1 , i.e., there is a permutation π of N such that b n = a π(n) for all n ∈ N.
The symmetry of ρ allows us to safely define ρ( f ) for any countable family of non-negative scalars f = (a j ) j∈J . In the language of bases, if ρ is a symmetric function-quasi-norm, then the unit vector system is a 1-symmetric basis of L b ρ .

Definition 17
Given a quasi-Banach space X and a sequence f = (a n ) ∞ n=1 in [0, ∞] N we define a n x n : N ∈ N, x n ≤ 1 if a n < ∞ for all n ∈ N, and λ X ( f ) = ∞ otherwise.

Proposition 14
Let X be a quasi-Banach space. Then λ X is a symmetric function quasi-norm with modulus of concavity at most that of X. Moreover, (i) λ X is locally absolutely continuous.
(ii) λ X has the Fatou property.
(iv) If X and Y are isomorphic, then λ X and ρ Y are equivalent.
Proof We will prove (vii), and we will leave the other assertions, which are reformulations of results from [38], as an exercise for the reader. Let X be a p-convex quasi-Banach lattice. Recall that the lattice structure of the space allows to define the absolute value |x| of any vector x ∈ X (cf. [24,Chapter 1]). Notice that 1 is a p-convex lattice, that is, we have Hence, the lattice defined by the quasi-norm

Definition 18
Let X be a quasi-Banach space. We denote G (X) = L λ X , and we say that G (X) is the galb of X. The positive cone of G (X) will be denoted by G + (X), and G b (X) stands for the closure of c 00 in G (X).
Roughly speaking, it could be said that the galb of a space is a measure of its convexity. The notion of galb was introduced and developed by Turpin, within the more general setting of "espaces vectoriels à convergence", in a series of papers [36,37] and a monograph [38]. In this section, we restrict ourselves to galbs of locally bounded spaces and touch only a few aspects of the theory and summarize without proofs the properties that are more relevant to our work.
Proposition 15 (see [38]) Let X be a quasi-Banach space. Then G (X) ⊆ 1 , and G (X) = 1 if and only if X is locally convex.
Proposition 17 (see [38]) Let X be a quasi-Banach space and 0 < p ≤ 1. Then X is p-convex if and only if p ⊆ G (X).
Proposition 18 (see [36]) Let X be a quasi-Banach space. Then the mapping a n x n is well-defined, and defines a bounded bilinear map.
It is natural to wonder whether the map B defined as in Proposition 18 can be extended to a continuous bilinear map defined on G (X) × ∞ (X). In fact, the authors of [23], perhaps taking for granted that the answer to this question is positive, defined a sequence (a n ) ∞ n=1 to be in the galb of X if ∞ n=1 a n x n converges for every bounded sequence (x n ) ∞ n=1 . If we come to think of it, we obtain the following.

Lemma 15
Let X be a quasi-Banach space and let f = (a n ) ∞ n=1 ∈ F N . Then, f ∈ G b (X) if and only if ∞ n=1 a n x n converges for every bounded sequence (x n ) ∞ n=1 in X.
Proof Let G denote the set consisting of all sequences f = (a n ) ∞ n=1 ∈ F N such that ∞ n=1 a n x n converges for every bounded sequence (x n ) ∞ n=1 in X. It is routine to check that G is a closed subspace of G (X) which contains c 00 . Consequently, Then, by Lemma 14, there are δ > 0 and an increasing sequence (m k ) ∞ k=1 of non-negative integers such that ρ((|a n |) m 2k n=1+m 2k−1 ) > δ for all k ∈ N. Consequently, there is (x n ) ∞ n=1 in the unit ball of ∞ (X) such that m 2k n=1+m 2k−1 a n x n ≥ δ, k ∈ N.
We infer that ∞ n=1 a n x n does not converge. Corollary 5 Let X be a quasi-Banach space. Then the mapping a n x n is well-defined, and defines a continuous bilinear map.

then B can not be extended to a continuous bilinear map defined on G × ∞ (X).
Proof It follows from Lemma 15 and, alike the proof of Proposition 18, the Open Mapping Theorem.
In light of Corollary 5, the following question arise.

Question 1 Is G (X) minimal for any quasi-Banach space X?
Corollary 4 alerts us of the connection between Question 1 and the existence of lower estimates for λ X . Lattice concavity also plays a key role when studying galbs of vector-valued spaces.

Definition 19
We say that a symmetric function quasi-norm λ over N galbs a quasi-Banach space X if λ dominates λ X , i.e., L λ ⊆ G (X). We say that λ galbs a function quasi-norm ρ if it galbs L ρ . If λ galbs itself, we say that λ is self-galbed. Remark 1 Given 0 < p ≤ 1, the function quasi-norm defining p is self-galbed. More generally, λ X is self-galbed for any quasi-Banach space X (see Proposition 16).

Proposition 19
Let ρ and λ be locally absolutely continuous L-convex function quasi-norms with the Fatou property. Suppose that λ galbs a quasi-Banach space X. If there is 0 < p < ∞ such that λ is p-concave and ρ is p-convex, then λ galbs L ρ (X).
Proof By Theorem 2, the pair (λ, ρ) has the MII property for some constant C. Since λ galbs X, there is a constant K > 0 such that λ K -dominates λ X . Therefore, if (a n ) ∞ n=1 is a sequence in L λ , and f 1 , . . . , f N belong the unit ball of L ρ (X), we have ρ N n=1 a n f n ≤ ρ λ X (a n f n ) N n=1 ≤ K ρ λ (a n f n ) N n=1 ≤ C K λ ρ (a n f n ) N n=1 ≤ C K λ (a n ) N n=1 ≤ C K λ((a n ) ∞ n=1 ).
Proposition 19 gives, in particular, that if λ is a 1-concave function quasi-norm which galbs X, then it galbs L 1 (μ, X). As we plan to develop an integral for functions belonging to a suitable subspace of L 1 (μ, X), the following question arises. Question 2 Is G (X) 1-concave for any quasi-Banach space X?
Note that a positive answer to Question 2 would yield a positive answer to Question 1. To properly understand Question 2, we must go over the state-of-the-art of the theory galbs.
Proof By a stantard convexification technique we can suppose that r = 1. Let ( f j ) J j=1 be a finite family consisting of non-negative sequences. We will prove that To that end, it suffices to prove that if G < ∞ and 0 < t < H , then, t < G. Assume without loss of generality that λ ϕ ( f j ) > 0 for all j. Then, pick (t j ) J j=1 such that J j=1 t j = t and 0 < t j < ρ( f j ). Then, if f j = (a j,n ) ∞ n=1 , a j,n < ∞ for all n ∈ N, and ∞ n=1 ϕ a j,n t j > 1, j = 1, . . . , J . Consequently, Therefore, t < G.
The lattice convexity of spaces of galbs is also quite unknown. It is known that if the gauge λ ϕ associated with an Orlicz function ϕ is function quasi-norm, so that ϕ is a quasi-Banach lattice, then there is p > 0 such that sup 0<u,t≤1 (see [19,Proposition 4.2]). Moreover, if (4) holds for a given p, then ϕ is a p-convex lattice. Therefore, ϕ is L-convex. The behavior of general spaces of galbs is unknown.

Question 3
Is λ X an L-convex function quasi-norm for any quasi-Banach space X?
Note that Proposition 14 (vii) partially solves in the positive Question 3.

Definition 20
Let X and Y be quasi-Banach spaces and λ be a symmetric minimal function quasi-norm over N with the Fatou property. We define It is clear that · X⊗ λ Y is a semi-quasi-norm whose modulus of concavity is at most that of λ, and that x ⊗ y X⊗ λ Y ≤ C x y for all x ∈ X and y ∈ Y, where C = λ(e 1 ).

Definition 21
Let X and Y be quasi-Banach spaces and λ be a symmetric minimal function quasi-norm with the Fatou property. The quasi-Banach space built from · X⊗ λ Y will be called the topological tensor product of X and Y by λ, and will be denoted by X ⊗ λ Y. The canonical norm-one bilinear map from X × Y to X ⊗ λ Y given by (x, y) → x ⊗ y will be denoted by T λ [X, Y].

Proposition 21
Let X, Y, U and V be quasi-Banach spaces, and let λ be a symmetric minimal function quasi-norm with the Fatou property.
(vi) Let ρ be a symmetric minimal function quasi-norm with the Fatou property. If ρ dominates λ, then there is a bounded linear map I : And, if X 0 and Y 0 are dense subspaces of X and Y respectively, we can pick x j ∈ X 0 and y j ∈ Y 0 for all j ∈ N.
. To be precise, if ( y j ) n j=1 is a basis of Y, the map R : X n → X ⊗ λ Y given by (x j ) n j=1 → n j=1 x j ⊗ y j is an isomorphism. (ix) If λ galbs X and Y has the point separation property, then · X⊗ λ Y is a quasi-norm on X ⊗ Y.
Let us prove (iii). Let C be such that  B(x, y). Given τ = n j=1 x k ⊗y k ∈ X ⊗ Y we have We infer that B 0 'extends' to an operator as desired.
Now we prove (iv). Let τ ∈ X⊗Y. The mere definitions of the semi-quasi-norms involved give For statement (v), it suffices to consider the case when V = Y and S v = Id Y . Let I : U → X and P : X → U be such that P Let us prove (vii). Assume without lost of generality that λ is function p-norm for some 0 < p ≤ 1. If (5) holds, then ∞ j=1 x j ⊗ y j is a Cauchy series. Therefore, it converges to τ ∈ X ⊗ λ Y. The continuity of the quasi-norm · X⊗ λ Y yields Conversely, let τ ∈ X ⊗ λ Y and ε > 0. Assume that X 0 and Y 0 are dense subspaces of X and Y respectively. Pick (τ n ) ∞ n=1 in X 0 ⊗ Y 0 such that lim n τ − τ n X⊗ λ Y = 0, and pick a sequence (ε n ) ∞ n=1 of positive numbers with Passing to a subsequence we can suppose that τ n − τ n−1 X⊗ λ Y < ε n for all n ∈ N, with the convention τ 0 = τ . Therefore, for all n ∈ N, we can write x j,n ⊗ y j,n , R n := λ x j,n y j,n j n j=1 < ε n .
Hence, we can safely define τ = ( j,n)∈N x j,n ⊗ y j,n , and we have Now we prove (viii). The mapping R is linear and bounded, and R(X n ) spans X ⊗ λ Y. Since λ galbs X, there is a bounded linear map S : X ⊗ λ Y → X n such that S(x ⊗ y j ) = x e j for all x ∈ X and j = 1, . . . , n. Taking into account that S • R = Id X n , we are done.
Finally, let V be finite-dimensional subspace of Y. Since V is complemented in Y, X ⊗ λ V is complemented in X ⊗ λ Y via the canonical map. Hence, it suffices to consider the case when Y is finite dimensional. In this particular case, statement (ix) follows from (viii).

Topological tensor products as spaces of functions and integrals for spaces of vector-valued functions
Let us give another approach to the proof of Proposition 21 (ix). Given quasi-Banach spaces X and Y, let B : X×Y → ∞ (Ball(Y * ), X), where Ball(Y * ) denotes the unit ball of the space Y * , be defined by B(x, y)(y * ) = y * (y)x. Since B is linear and bounded, if λ galbs X, there is a bounded linear map If Y has the point separation property, then B λ is one-to-one on X ⊗ Y. Consequently, no vector in X ⊗ Y is norm-zero. Note the injectivity of B λ on X ⊗ Y does not implies the injectivity of B λ on its closure X ⊗ λ Y. That is, we can not, a priori, identify vectors in X ⊗ λ Y with functions defined over Ball(Y * ). More generally, if Y embeds in F Ω for some set Ω, then X ⊗ Y embeds into X Ω , and it is natural to wonder if the character of the members of X ⊗ Y is preserved when taking the completions, that is, if we can regard the vectors in X ⊗ λ Y as X-valued functions defined on Ω. In this section, we address this question in the case when Y is a Köthe space. Given a quasi-Banach space X and a σ -finite measure space (Ω, Σ, μ) we have a canonical linear map It is routine to check that J [X, μ] is one-to-one. Suppose that λ is a symmetric function quasi-norm and ρ is a function quasi-norm over (Ω, Σ, μ) such that λ is p-concave and ρ is p-convex for some 0 < p < ∞. Then λ is minimal (see Corollary 4). So, we can safely define X ⊗ λ L ρ . If, moreover, λ galbs X, then λ also galbs L ρ (X) (see Proposition 19). Hence, if ρ has the weak Fatou property, there is a bounded linear canonical map

Consider the range
of this operator endowed with the quotient topology. If J [ρ, X, λ] is one-to-one, then L λ ρ (X) is a space isometric to X⊗ λ L ρ which embeds continuously into L ρ (X). This is our motivation to studying the injectivity of J [ρ, X, λ]. Vogt [40] gave a positive answer to this question in the case when λ is the function quasi-norm associated with p for some 0 < p ≤ 1 and ρ is the function quasi-norm associated with L q (μ) for some p ≤ q ≤ ∞. A detailed analysis of the proof of [40,Satz 4] reveals that it depends heavily on the fact that λ is both p-convex and p-concave and ρ is both q-convex and q-concave. So, it is hopeless to try to extend this result using analogous ideas. In this paper, we use an approach based on conditional expectations.
Before going on, let us mention that if λ is the 1 -norm restricted to nonnegative sequences (and ρ and X are 1-convex), then a routine computation yields that J [ρ, X, λ] is an isometric embedding when restricted to X ⊗ S(μ). We infer that J [ρ, X, λ] is an isometric embedding and that L λ ρ (X) consists of all strongly measurable functions in L ρ (X).

Lemma 16
Let λ be a minimal symmetric function quasi-norm. For i = 1, 2, let ρ i be a function quasi-norm with the weak Fatou property over a σ -finite measure space (Ω i , Σ i , μ i ), and let X i be a quasi-Banach space galbed by λ. Suppose that the bounded linear operators S : X 1 → X 2 , T : L ρ 1 → L ρ 2 and R : Then, R restricts to a bounded linear map from L λ ρ 1 (X 1 ) → L λ ρ 2 (X 2 ).
Proof Our assumptions yield a commutative diagram We infer that R maps the range of the map J [ρ 1 , commutes. Since both L λ ρ 1 (X 1 ) and L λ ρ 2 (X 2 ) are endowed with the quotient topology and S ⊗ λ T is continuous, so is R[λ].
In other words, (λ, X) is amenable if and only if for every σ -finite measure μ there is an operator So, we must regard it as 'integral' for functions in L λ 1 (μ, X). Loosely speaking, that (λ, X) is amenable means that there is an integral for functions in L λ 1 (μ, X). Definition 23 Let X be a quasi-Banach space. We say that a net (T i ) i∈I in L(X) is a bounded approximation of the identity if sup i T i < ∞ and lim i T i (x) = x for all x ∈ X. We say that X has the BAP if it has a bounded approximation of the identity consisting of finite-rank operators.
Note that if a net (T i ) i∈I in L(X) is uniformly bounded then the set {x ∈ X : lim i T i (x) = x} is closed. This yields the following elementary result.

Lemma 17
Let X be a quasi-Banach space. Let (P i ) i∈I be a net consisting of uniformly bounded projections with P j • P i = P i if i ≤ j and ∪ i∈I P i (X) is dense in X. Then (P i ) i∈I is a bounded approximation of the identity.
If ρ satisfies (F.7), then for every A ∈ Σ(μ) we have a bounded linear map Theorem 5 Let λ be a 1-concave symmetric function quasi-norm, let ρ be a leveling function quasi-norm with the weak Fatou property over a σ -finite measure space (Ω, Σ, μ), and let X be a quasi-Banach space. Suppose that (λ, X) is amenable. Then J [ρ, X, λ] is one-to-one.
Proof Let A ∈ Σ(μ). By Lemma 13, ρ satisfies (F.7). Therefore, for each quasi-Banach space Y there is a bounded linear operator commutes. Using that (λ, X) is amenable we obtain the commutative diagram Suppose that μ(Ω) < ∞. Let Σ 0 be a finite sub-σ -algebra. If Σ 0 is generated by the partition (A j ) n j=1 of Ω consisting of nonzero measure sets, then By Proposition 21 (viii), there is an isomorphism S : Therefore, . Combining this identity with the commutative diagrams (6) associated with each set A j yields a bounded linear map R : commutes. The operators Id X ⊗ λ E(ρ, Σ 0 ) are uniformly bounded projections. Let (Σ i ) i∈I a non-decreasing net of finite σ -algebras whose union generates Σ. By Lemma 17, (Id X ⊗ λ E(ρ, Σ i )) i∈I is a bounded approximation of the identity. We infer that J [ρ, X, λ] is one-toone, as wanted, in the particular case that μ(Ω) < ∞.
In general, let R[A, X]: L ρ (X) → L ρ (A, X) be the canonical projection on a set A ∈ Σ(μ).
n=1 is a bounded approximation of the identity. Since J [ρ| A n , X, λ] is one-to-one (by the previous particular case), it follows that J [ρ, X, λ] is one-to-one.
We emphasize that the applicability of Theorem 5 depends on the existence of amenable pairs. In the optimal situation, we would be able to choose λ to be the smallest symmetric function quasi-norm which galbs the quasi-Banach space X. Thus, the following question arises.

Question 4
Let X be a quasi-Banach space. Is (λ X , X) amenable?
As long as there is no general answer to Question 4, we will focus on the spaces of galbs that have appeared in the literature. We next prove that for all of them Question 4 has a positive answer.
Proof Assume that ϕ(1) = 1. Assume by contradiction that there is a σ -finite measure space (Ω, Σ, μ), a positive sequence α = (a j ) ∞ j=1 in ϕ , a sequence ( f j ) ∞ j=1 in the unit ball of L 1 (μ), and a sequence (x j ) ∞ j=1 in the unit ball of X such that ∞ j=1 a j x j f j = 0 in L ϕ (X) and x := ∞ j=1 a j x j Ω f j dμ = 0.
The following claim will be used a couple of times. Claim. If (Ω k ) ∞ k=1 is a non-decreasing sequence in Σ(μ) such that Ω \ ∪ ∞ k=1 Ω k is a null set, then ∞ j=1 a j x j Ω k f j dμ = 0 for some k ∈ N. Proof of the claim. Since lim k Ω k f j dμ = Ω f j dμ for all j ∈ N and λ ϕ is dominating, we have This limit readily gives our claim.
The claim allow us assume that μ(Ω) < ∞. By Proposition 21 (vii), we can assume that f j ∈ S(μ) for all j ∈ N. Also, we can assume without lost of generality that λ ϕ (α) < 1, so that ∞ j=1 ϕ(a j ) < 1. Set We have Ω F 0 dμ < ∞. Therefore, F 0 < ∞ a.e. By Severini-Egorov theorem, lim m F m = 0 quasi-uniformly. By Proposition 2, there is an increasing sequence (J n ) ∞ n=1 such that, if a j x j f j , n ∈ N, then lim n G n = 0 a.e. Taking into account the claim, we can assume without lost of generality that lim m F m = 0 uniformly and that lim n G n = 0 pointwise. Pick 0 < ε < 1. There is m 0 ∈ N such that λ ϕ ((a j ) ∞ m 0 +1 ) < ε, i.e., Let m ≥ m 0 be such that , ω ∈ Ω.

The fundamental theorem of calculus
Let X be a quasi-Banach space and let λ be a symmetric function quasi-norm such that (λ, X) is amenable. If d ∈ N, A ⊆ R d is measurable, and μ is the Lebesgue measure on A, we set L λ 1 (A, X) = L λ 1 (μ, X) and, for f ∈ L λ 1 (A, X), Given d ∈ N, we denote by Q the set consisting of all d-dimensional open cubes. If y ∈ R d , the set Q[y] consisting of all Q ∈ Q such that y ∈ Q is a directed set when ordered by inverse inclusion. We denote by "Q ∈ Q → y" the convergence with respect to that directed set.
The following nonlocally convex version of the Lebesgue differentiation theorem for vector-valued integrals (see, e.g., [7,Proposition 5.3] for the classical locally convex version) improves [1,Theorem 5.2]. Theorem 7 Let X be a quasi-Banach space and λ be a symmetric function quasi-norm. Suppose that λ is p-concave for some 0 < p < 1 and that (λ, X) is amenable. Then, for any locally λ-integrable function f : R d → X, 1 |Q| If κ is the modulus of concavity of X, the quasi-triangle inequality Hence, the set of functions f that satisfy (7) is a vector space. Since this set contains By Theorem 3, the pair (λ, L 1,∞ (R d )) has the MII property. Hence, where the constant C 1 does not depend on f . In turn, since M is bounded from where the constant C 2 does not depend on f either. Consequently, if We close with an application to the theory of Lipschitz functions. Derivatives and integrals are relevant tools within the study of Lipschitz maps whose target space is a Banach space (see, e.g., [3,Sect. 14]). So, the lack of an integration theory as powerful as Bochner-Lebesgue integral is a drawback to extend to quasi-Banach spaces results achieved in the locally convex setting. The theory of integration presented here allows us to obtain a partial result for functions taking values in an Orlicz space.

Conflict of interest
The authors declare that they have no conflict of interest.
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