Kuelbs-Steadman spaces for Banach space-valued measures

We introduce Kuelbs-Steadman-type spaces for real-valued functions, with respect to countably additive measures, taking values in Banach spaces. We investigate their main properties and embeddings in $L^p$-type spaces, considering both the norm associated to norm convergence of the involved integrals and that related to weak convergence of the integrals.


Introduction
Kuelbs-Steadman spaces have been the subject of many recent studies (see e.g. [22,23,26] and the references therein). The investigation of such spaces arises from the idea to consider the L 1 spaces as embedded in a larger Hilbert space with smaller norm, and containing in a certain sense the Henstock-Kurzweil integrable functions. This allows to give several applications to Functional Analysis and other branches of Mathematics, for instance Gaussian measures (see also [29]), convolution operators, Fourier transforms, Feynman integral, quantum mechanics, differential equations and Markov chains (see also [22,23,26]). This approach allows also to develop a theory of Functional Analysis which includes Sobolev-type spaces, in connection with Kuelbs-Steadman spaces rather than with classical L p spaces.
In this paper we extend the theory of Kuelbs-Steadman spaces to measures µ defined on a σ-algebra and with values in a Banach space X. We consider an integral for real-valued functions f with respect to X-valued countably additive measures. In this setting, a fundamental role is played by the separability of µ. This condition is satisfied, for instance, when T is a metrizable separable space, not necessarily with a Schauder basis (such spaces exist, see for instance [22]), and µ is a Radon measure. In the literature, some deeply investigated particular cases are when X = R n and µ is the Lebesgue measure, and when X is a Banach space with a Schauder basis (see also [22,23,26]). Since the integral of f with respect to µ is an element of X, in general it is not natural to define an inner product, when it is dealt with norm convergence of the involved integrals. Moreover, when µ is a vector measure, the spaces L p [µ] do not satisfy all classical properties as the spaces L p with respect to a scalar measure (see also [20,35,40]). However, it is always possible to define Kuelbs-Steadman spaces as Banach spaces, which are completions of suitable L p spaces. We introduce them and prove that they are normed spaces, and that the embeddings of KS p [µ] into L q [µ] are continuous and dense. Moreover, we show that the norm of KS p spaces is smaller than that related to the space of all Henstock-Kurzweil integrable functions (the Alexiewicz norm). Furthermore, we prove that KS p spaces are Köthe function spaces and Banach lattices, extending to the setting of KS p [µ]-spaces some results proved in [40] for spaces of type L p [µ]. Furthermore, when X ′ is separable, it is possible to consider a topology associated to weak convergence of integrals and to define a corresponding norm and an inner product. We introduce the Kuelbs-Steadman spaces related to this norm, and prove the analogous properties investigated for KS p spaces related to norm convergence of the integrals. In this case, since we deal with a separable Hilbert space, it is possible to consider operators like convolution and Fourier transform, and to extend the theory developed in [22,23,26] to the context of Banach space-valued measures.

Vector measures, (HKL)and (KL)-integrals
Let T = ∅ be an abstract set, P(T ) be the class of all subsets of T , Σ ⊂ P(T ) be a σ-algebra, X be a Banach space and X ′ be its topological dual. For each A ∈ Σ, let us denote by χ A the characteristic function of A, defined by A vector measure is a σ-additive set function µ : Σ → X. By the Orlicz-Pettis theorem (see also [14,Corollary 1.4]), the σ-additivity of µ is equivalent to the σ-additivity of the scalar-valued set function x ′ µ : A → x ′ (µ(A)) on Σ for every x ′ ∈ X ′ . For a literature on vector measures, see also [12,14,20,28,31,35,36] and the references therein.
The completion of Σ with respect to µ is defined by Observe that from (1) and (2) it follows that every µ-measurable real-valued function is also x ′ µmeasurable for every x ′ ∈ X ′ . Moreover, it is readily seen that every Σ-measurable real-valued function is also µ-measurable.
We say that µ is Σ-separable (or separable) if there is a countable family B = (B k ) k in Σ such that, for each A ∈ Σ and ε > 0, there is k 0 ∈ N such that (see also [38]). Such a family B is said to be µ-dense.
Observe that µ is Σ-separable if and only if Σ is µ-essentially countably generated, namely there is a countably generated σ-algebra Σ 0 ⊂ Σ such that for each A ∈ Σ there is B ∈ Σ 0 with µ(A∆B) = 0. The separability of µ is satisfied, for instance, when T is a separable metrizable space, Σ is the Borel σ-algebra of the Borel subsets of T , and µ is a Radon measure (see also [5,Theorem 4.13], [ From now on, we assume that µ is separable, and B = (B k ) k is a µ-dense family in Σ with Now we recall the Henstock-Kurzweil (in short, (HK))-integral for real-valued functions, defined on abstract sets, with respect to (possibly infinite) non-negative measures. For a related literature, see also [2,4,6,7,8,9,10,15,16,18,21,24,30,37,39] and the references therein. When we deal with the (HK)integral, we assume that T is a compact topological space and Σ is the σ-algebra of all Borel subsets of T . We will not do these assumptions to prove the results which do not involve the (HK)-integral.
Let ν : Σ → R ∪ {+∞} be a σ-additive non-negative measure. A decomposition of a set A ∈ Σ is a finite collection {(A 1 , ξ 1 ), (A 2 , ξ 2 ), . . . , (A N , ξ N )} such that A j ∈ Σ and ξ j ∈ A j for every j ∈ {1, 2, . . . , N }, and ν(A i ∩ A j ) = 0 whenever i = j. A decomposition of subsets of A ∈ Σ is called a partition An example is when T 0 is a locally compact and Hausdorff topological space, and T = T 0 ∪ {x 0 } is the one-point compactification of T 0 . In this case, we will suppose that all involved functions f vanish on x 0 . For instance this is the case, when T 0 = R n is endowed with the usual topology and x 0 is a point "at the infinity", or when T is the unbounded interval if the sum exists in R, with the convention 0 · (+∞) = 0. Note that for any gauge δ there exists at least one δ-fine partition D such that S(f, D) is well-defined.
A function f : T → R is said to be Henstock-Kurzweil integrable (briefly, (HK)-integrable) on a set A ∈ Σ if there is an element I A ∈ R such that for every ε > 0 there is a gauge δ on A with |S(f, D)−I A | ≤ ε whenever D is a δ-fine partition of A such that S(f, D) exists in R, and we write Observe that, if A, B ∈ Σ, B ⊂ A and f : T → R is (HK)-integrable on A, then f is also (HK)-integrable on B and on A \ B, and (see also [4, Propositions 5.14 and 5.15], [39, Lemma 1.10 and Proposition 1.11]). From (5) used with We say that a Σ-measurable function f : and for every A ∈ Σ there is x where the symbols (L) and (HK) in (8) denote the usual Lebesgue (resp. Henstock-Kurzweil) integral of a real-valued function with respect to an (extended) real-valued measure. A Σ-measurable function f : T → R is said to be weakly (KL) (resp. weakly (HKL)) µ-integrable if it satisfies only condition (7) (see also [11,12,36]). We recall the following facts about the (KL)-integral.

Construction of the Kuelbs-Steadman spaces and main properties
We begin with giving the following technical results, which will be useful later.
Proposition 3.1. Let (a k ) k and (η k ) k be two sequences of non-negative real numbers, such that a = sup k a k < +∞, and and p > 0 be a fixed real number. Then, Proof. We have η k a p k ≤ a p η k for all k ∈ N, and hence getting (12).
be two sequences of real numbers, (η k ) k be a sequence of positive real numbers, satisfying (11), and p ≥ 1 be a fixed real number. Then, Proof. It is a consequence of Minkowski's inequality (see also [25,Theorem 2.11.24]).
For 1 ≤ p ≤ ∞, let us define a norm on L 1 [µ] by setting The following inequality holds.
Proof. By (12) used with Taking the supremum in (17) getting the assertion. (14) is a norm.

Now we prove that
Proof. Observe that, by definition, We prove that f = 0 µ-almost everywhere. It is enough to take 1 ≤ p < ∞, since the case p = ∞ will follow from (15). For k ∈ N, let a k be as in (16). As the η k 's are strictly positive, from it follows that a k = 0 for every k ∈ N. Hence, Proceeding by contradiction, suppose that f = 0 µ-almost everywhere. Since By the separability of µ, in correspondence with ε and B there is B k0 ∈ B satisfying (3), that is From (20) and (21) we deduce which contradicts (18). Therefore, µ(E + ) = 0. Now, suppose that µ(E − ) = 0. By proceeding analogously as in (22), replacing f with −f and f * with the function f * defined by f * (t) = min{−f (t), 1}, t ∈ T , we find an x ′ 1 ∈ X ′ with x ′ 1 ≤ 1, an n ∈ N, a B ∈ Σ, an ε > 0 and a B k1 ∈ B with µ (B k1 ) < M , and getting again a contradiction with (18). Thus, µ(E − ) = 0, and f = 0 almost everywhere. The triangular property of the norm can be deduced from Proposition 3.2 for 1 ≤ p < ∞ and is not difficult to see for p = ∞, and the other properties are easy to check.  (14) (see also [3,19,23,26,29,42]). Observe that, to avoid ambiguity, we take the completion of L 1 [µ] rather than that of L p [µ], but since the embeddings in Theorem 3.5 are continuous and dense, the two methods are substantially equivalent.
By proceeding similarly as in [ Proof. We first consider the case 1 ≤ p < ∞. Let f ∈ L q [µ], with 1 ≤ q < ∞, and M be as in (4). Note that M q−1 q ≤ M , since M ≥ 1. As |E k (t)| = E k (t) ≤ 1 and |E k (t)| q ≤ E k (t) for any k ∈ N and t ∈ T , taking into account (12) and Jensen's inequality (see also [5,Exercise 4.9]), we deduce where M is as in (4). Now, let 1 ≤ p < ∞ and q = ∞. We have The proof of the case p = ∞ is analogous to that of the case 1 ≤ p < ∞. Therefore, f ∈ KS p [µ], and the embeddings in (23) and (24) [27, §4.4]).

Choose arbitrarily ε > 0 and
in correspondence with ε and g we find a Σ-simple function s, with s − g L 1 [µ] ≤ ε M + 1 . By (23) and , and hence we obtain getting the last part of the assertion. Thus, the embeddings in (23) and (24)  , respectively. (b) If f is (HKL)-integrable, then for each x ′ ∈ X ′ and k ∈ N, E k f is both Henstock-Kurzweil and Lebesgue integrable with respect to |x ′ µ|, since f is Σ-measurable, and the two integrals coincide, thanks to the (HK)-integrability of the characteristic function χ E for each E ∈ Σ and the monotone convergence theorem (see also [4,39]). Thus, taking into account (17), for every p with 1 ≤ p < ∞ we have The next result deals with the separability of Kuelbs-Steadman spaces, which holds even for p = ∞, differently from L p spaces.
Proof. Observe that, by our assumptions, µ is separable, and this is equivalent to the separability of the spaces L p [µ] with 1 ≤ p < ∞ (see also [19,Proposition 2.3], [38,Propositions 1A and 3]). Now, let H = {h n : n ∈ N} be a countable subset of L 1 , dense in L 1 [µ] with respect to the norm . In correspondence with ε and g there getting the claim. Now we prove that KS p [µ] spaces are Banach lattices and Köthe function spaces. First, we recall some properties of such spaces (see also [32,34]).
A partially ordered Banach space X which is also a vector lattice is a Banach lattice if x ≤ y for every x, y ∈ X such that |x| ≤ |y|.
A weak order unit of X is a positive element e ∈ X such that, if x ∈ X and x ∧ e = 0, then x = 0. Let X be a Banach lattice and ∅ = A ⊂ B ⊂ X. We say that A is solid in B if for each x, y with x ∈ B, y ∈ A and |x| ≤ |y|, it is x ∈ A.
Let λ be an extended real-valued measure on Σ. A Banach space X consisting of (classes of equivalence of) λ-measurable functions is called a Köthe function space with respect to λ if, for every g ∈ X and for each measurable function f with |f | ≤ |g| λ-almost everywhere, it is f ∈ X and f ≤ g , and χ A ∈ X for every A ∈ Σ with λ(A) < +∞. Theorem 3.9. If p ≥ 1, then KS p [µ] is a Banach lattice with a weak order unit and a Köthe function space with respect to a control measure λ of µ.
Finally, we prove that χ T is a weak order unit of KS p [µ]. First, note that χ T ∈ L p [µ], and hence and hence f = 0 µ-almost everywhere. This ends the proof.
Note that, by the definition of the (KL)-integral, the norm defined in (14) corresponds, in a certain sense, to the topology associated with norm convergence of the integrals (µ-topology, see also [20,Theorem 2.2.2]). However, with this norm, it is not natural to define an inner product in the space KS 2 , since m is vector-valued.
On the other hand, when X ′ is separable and {x ′ h : h ∈ N} is a countable dense subset of X ′ , with x ′ h ≤ 1 for every h, it is possible to deal with the topology related to weak convergence of integrals (weak µ-topology, see also [20, Proposition 2.1.1]), whose corresponding norm is given by where E k , k ∈ N, is as in (14), and (η k ) k , (ω h ) h are two fixed sequences of strictly positive real numbers, ω h = 1. Note that, in general, weak µ-topology does not coincide with µ-topology, but there are some cases in which they are equal (see also [40,Theorem 14]). Analogously in Proposition 3.3, it is possible to prove the following Now we give the next fundamental result.
We prove that f = 0 µ-almost everywhere. It will be enough to prove the assertion for 1 ≤ p < ∞, since the case p = ∞ follows from (28). Arguing analogously as in (18), we get By contradiction, suppose that f = 0 µ-almost everywhere.
then E + , E − ∈ Σ, since f is Σ-measurable, and we have µ(E + ) = 0 or µ(E − ) = 0. Suppose that µ(E + ) = 0. By the Hahn-Banach theorem, there is Without loss of generality, we can assume x ′ h0 ≤ 1. Now, the proof continues analogously as that of Theorem 3.4, by replacing the linear continuous functional x ′ 0 in (22) with x ′ h0 found in (30), by finding another element x ′ h1 ∈ X ′ with |x ′ h1 µ(E − )| > 0, and by arguing again as in (22). The triangular property of the norm is straightforward for p = ∞, and for 1 ≤ p < ∞ is a consequence of the inequality which holds whenever (b k,h ) k,h , (c k,h ) k,h are two double sequences of real numbers, and (η k ) k , (ω h ) h are two sequences of positive real numbers, such that ∞ h=1 ω h = ∞ k=1 η k = 1. The inequality in (31), as that in (13), follows from Minkowski's inequality. The other properties are easy to check. Now, in correspondence with the norm defined in (27), we define the following bilinear functional ·, · : Arguing similarly as in Theorem 3.11, it is possible to see that the functional ·, · KS 2 [wτ µ] in (32) is an inner product, and · KS 2 [wτ µ] = ( ·, · KS 2 [wτ µ] ) 1/2 .
As in Theorem 3.5, it is possible to prove the following Theorem 3.12. For each p, q with 1 ≤ p ≤ ∞ and 1 ≤ q ≤ ∞, it is L q [µ] ⊂ KS p [wτ µ] continuously and densely, and the space of all Σ-simple functions is dense in KS p [wτ µ]. Moreover, KS p [wτ µ] is a separable Banach lattice with a weak order unit and a Köthe function space with respect to a control measure λ of µ.

Conclusions
We have introduced Kuelbs-Steadman spaces related to integration for scalar-valued functions with respect to a σ-additive measure µ, taking values in a Banach space X. We have endowed them with the structure of Banach space, both in connection with norm convergence of integrals and in connection with weak convergence of integrals (KS p [µ] and KS p [wτ µ], respectively). A fundamental role is played by the separability of µ. We have proved that these spaces are separable Banach lattices and Köthe function spaces, and can be embedded continuously and densely in the spaces L q [µ]. When X ′ is separable, we have endowed KS 2 [wτ µ] with an inner product. In this case, KS 2 [wτ µ] is a separable Hilbert space, and hence it is possible to deal with operators like convolution and Fourier transform, and to extend to Banach space-valued measures the theory investigated in [22,23,26].