INTERNATIONAL
PUBLICATIONS USA
PanAmerican Mathematical Journal
Volume 23(2013), Number 1, 57–74
Set-valued Kurzweil-Henstock integral in Riesz space setting1
A. Boccuto
University of Perugia
Department of Mathematics and Computer Science
via Vanvitelli, 1 I-06123 Perugia (Italy)
[email protected]
A. M. Minotti
University of Perugia
Department of Mathematics and Computer Science
via Vanvitelli, 1 I-06123 Perugia (Italy)
[email protected]
A. R. Sambucini2
University of Perugia
Department of Mathematics and Computer Science
via Vanvitelli, 1 I-06123 Perugia (Italy)
[email protected]
Communicated by Carlo Bardaro
(Received April 23, 2012; Revised Version Accepted September 21, 2012)
Abstract
A multivalued integral in Riesz spaces is given using the Kurzweil-Henstock approach.
Some of its properties and a comparison with the Aumann integral are also investigated. As a particular case we examine also the setting of Banach lattices.
2010 AMS Subject Classification: 28B20, 46G10.
Key words: Riesz space, Banach lattice, Kurzweil-Henstock integral, Aumann integral, multivalued integral.
1
Introduction
Multivalued analysis is a very powerful tool in the study of several problems in many areas of
Mathematics. For example, we recall here subdifferentials of convex functionals, Calculus of
1 Supported
by GNAMPA of CNR and University of Perugia
author: Anna Rita Sambucini
2 corresponding
57
58
A. Boccuto, A. M. Minotti and A. R. Sambucini
Variations, degree theory, fixed points, set-valued random processes, optimal control theory,
game theory, Pareto optimization, economic analysis (competitive equilibrium, coalition
economics, etc.) and so on.
The classical theory of set-valued integration comes from Aumann using Bochner selections, and several investigations and extensions of this approach were given. For example,
there are many recent studies on multivalued integrals of the type of Debreu, Sugeno, Choquet, Pettis, Birkhoff, Kurzweil-Henstock and on pseudo-integrals. Among the authors, we
quote Balder and Sambucini ([4, 5, 6]), Hess, Castaing and his school ([20, 3, 54, 25, 34]),
Cascales, Kadets and Rodrı́guez ([16, 18, 17, 19]), Labuschagne ([37]), Martellotti and Sambucini ([41, 42]), Boccuto and Sambucini ([13, 14]), Grbić, Štajner-Papuga and Štrboja ([32])
and Bongiorno, Di Piazza and Musial ([22, 23, 15]). The last quoted results have applications in differential or integral problems taking into account integrals of highly oscillating
functions in multi-valued setting (see for example [21, 49, 50, 24, 52]).
However most of these results are given for multifunctions with values in suitable subsets
of (separable) Banach spaces while in the setting of Riesz spaces-valued multifunctions it
seems that less results are known. This motivated our approach since these spaces have
become an important tool in many branches of applied mathematics, in particular in Economic Theory (see [2, 1, 32, 36]), where commodities, prices, utility functions and equilibria
in an exchange economy are studied in the context of Riesz spaces. Moreover, also in this
setting, the Kurzweil-Henstock integral is a powerful tool since it integrates all derivatives
as, for example, highly oscillating functions, which the Lebesgue integral fails to integrate
and its multivalued version does not need a priori the notion of measurability, so this fact
looks interesting for many economic problems that can be reduced to differential inclusions.
Consequently, the Kurzweil-Henstock integral allows to solve different types of differential
equations and inclusions, with several applications for example to integral inclusions and
Komlós-type theorems by means of a more natural approach and weaker hypotheses than
classical integrals for single functions or multifunctions, and considers situations, in which
it would not be advisable to deal with integrable selections, or their existence is not known
(see also [48, 49, 51]).
Other possible applications are stochastic processes since they can be viewed as functions with values in the Riesz space L0 (T, Σ, µ) of all measurable functions with identification up to µ-null sets, where one of the convergences involved is the almost everywhere convergence, which is not generated by any topology. For a literature, see for example [10, 11, 38, 46, 47, 53]. In particular in [9] there are some investigations on Itô and
Stratonovich-type integrals.
In this paper we continue the study of the Kurzweil-Henstock integral for set valued
functions started in [13, 14] and we consider the more general setting of Riesz spaces in
view of possible applications to economic analysis. In general, in Riesz spaces we cannot
follow the ideas found in [22, 23], that are given for the Henstock-Pettis integral of multifunctions taking values in suitable subsets of separable Banach spaces with respect to the
Lebesgue measure, since there exist Riesz spaces without non-trivial linear functionals, continuous with respect to the order (for example Lp , 0 ≤ p < 1). For this motivation we have
introduced a new multivalued integral following the ideas in [33, 13, 10].
We deal with multifunctions with values in Riesz spaces with respect to positive, additive
and regular measures. The set-valued integral Φ(F, E) for Riesz space-valued multifunctions
59
Set-valued Kurzweil-Henstock integral
F that we introduce is in some sense the set of limits with respect to (o)-convergence of
Riemann sums obtained by γ-fine partitions. Properties like convexity, closedness, boundedness are investigated. We prove that in the case of simple multifunctions this integral agrees
with the closure of the usual integral obtained with Kurzweil-Henstock-type selections in
the Aumann approach.
In suitable Banach lattices endowed with a continuous norm, we extend the comparison
with the Aumann integral also to general bounded multivalued mappings. All the results
given in the paper are true for any product triple (R1, R2, R) though they are formulated
in the particular case of an algebra (R, R, R) only for the sake of simplicity.
In the particular case when the triples involved are (R, R, R), we obtain some equivalence
results between our approach and the classical one, based on the norm topology. Examples
of Banach lattices of this type are the spaces of all bounded functions in Lp (µ), 1 ≤ p < +∞
with µ positive, finite and regular.
2
Preliminaries
Let R be a Riesz space. A decreasing sequence (bn )n in R, such that ∧nbn = 0, is called an
(o)-sequence. A bounded double sequence (ai,j )i,j in R is a (D)-sequence or a regulator if
(ai,j )j is an (o)-sequence for all i ∈ N.
A Riesz space R is said to be Dedekind complete if every W
nonempty subset R1 of R, bounded
from above, has a lattice supremum in R denoted by R1 . A Dedekind complete Riesz
space R is said to be super Dedekind complete if every non-empty subset R1 ⊂ R, R1 6= ∅,
bounded from above, contains a countable subset having the same supremum as R1. An
element e ∈ R+ is an order unit if for every r ∈ R there exists n ∈ N such that |r| ≤ ne.
A Dedekind complete Riesz space R is said to !
be weakly σ-distributive if for every (D)∞
_
^
sequence (ai,j )i,j in R one has:
ai,ϕ(i) = 0.
N
i=1
ϕ∈N
Let (an)n be a sequence in R. We refer to [40, 46] for the definitions of (D)-limit and
(o)-limit.
If R is weakly σ-distributive, then the two notions of convergence coincide for sequences and
the limit is unique (see also [11, Proposition 2.22]).
We now recall the well-known Fremlin Lemma (see also [31]):
Lemma 2.1. Let (aki,j )i,j , k ∈ N, be any countable family of (D)-sequences. Then for each
fixed element u ∈ R, u ≥ 0, there exists a (D)-sequence (ai,j )i,j with
!!
s
∞
∞
X
_
_
k
u∧
≤
ai,ϕ(i+k)
ai,ϕ(i) for all ϕ ∈ NN and s ∈ N.
k=1
i=1
i=1
A norm k · k : R → R satisfying
(a) |y| ≤ |x| in R implies that kyk ≤ kxk;
(b) k |x| k = kxk for all x ∈ R;
60
A. Boccuto, A. M. Minotti and A. R. Sambucini
is called a lattice norm and (R, k · k) is called a normed Riesz space. A normed Riesz space
which is complete with respect to the norm is called a Banach lattice. The norm k · k
on a Banach lattice R is order continuous if inf{kxk : x ∈ A} = 0 for every downwards
directed set A ⊂ R with infimum zero. Note that every Dedekind complete normed Riesz
space with an order continuous norm is super Dedekind complete (see [29, 354Y (c)]). If
R is a Banach lattice with an order unit e which is an interior point of R+ , then by [43,
Corollary 1.2.24], the originary norm k · k is equivalent to the e-norm, defined by setting
k · ke := inf{α ∈ R+ : | · | ≤ αe} (see also [43]).
We say that (R1, R2, R) is a product triple if there exists a map · : R1 × R2 → R, which is
compatible with the operations of sum, scalar product, order, suprema and infima (see also
[46, Assumption 5.2.1]).
A Dedekind complete Riesz space R is called an algebra if (R, R, R) is a product triple.
All the results given in the paper are formulated in an algebra (R, R, R) for the sake of
simplicity, but they are true for any product triple (R1, R2, R), with obvious modifications.
Given a compact Hausdorff topological space T and its Borel σ-algebra Σ, we say that
a positive finitely additive measure µ : Σ → R is regular, if for every E ∈ Σ there exists a
N
(D)-sequence (aE
i,j )i,j , such that for all ϕ ∈ N there are a compact set K and an open set
U with K ⊂ E ⊂ U and
µ(U \ K) ≤
∞
_
aE
i,ϕ(i) .
(1)
i=1
We now prove σ-additivity of R-valued regular measures.
Theorem 2.2. Assume that R is a Dedekind complete and weakly σ-distributive Riesz
space, and let µ : Σ → R be a regular measure defined on the Borel σ-algebra of a compact
Hausdorff space T . Then µ is σ-additive.
Proof. Since µ is positive, all we have to prove is that, for every disjoint sequence (Ak )k in
∞
∞
[
X
Σ, µ
Ak ≤
µ(Ak ). To this aim, let us denote by A the union of all Ak ’s, and observe
k=1
k=1
that for each k ∈ N there is a regulator (aki,j )i,j such that, for every ϕ ∈ NN , it is possible to
∞
_
find a compact set Ck and an open set Uk with Ck ⊂ Ak ⊂ Uk and µ(Uk \ Ck ) ≤
aki,ϕ(i+k).
i=1
Also, setting u = 2µ(T ), by Lemma 2.1 it is possible to find a regulator (bi,j )i,j such that
!!
N
∞
∞
X
_
_
k
u∧
≤
ai,ϕ(i+k)
bi,ϕ(i) for all N ∈ N and ϕ ∈ NN .
(2)
k=1
i=1
i=1
We now claim that, for every compact set C ⊂ A,
µ(C) ≤
∞
X
µ(Ak ).
(3)
k=1
Choose arbitrarily ϕ ∈ NN . In correspondence with ϕ, there exist open sets Uk , k ∈ N, with
∞
_
Ak ⊂ Uk and µ(Uk ) ≤ µ(Ak ) +
aki,ϕ(i+k) for all k. Since C is compact, and the union of
i=1
61
Set-valued Kurzweil-Henstock integral
all Uk ’s contains C, there is N ∈ N such that C ⊂
N
[
Uk . Observe that
k=1
µ(C)
≤
N
X
µ(Uk ) ≤
k=1
|µ(C) −
N
X
k=1
µ(Ak )| =
N
X
µ(C) − µ
N
X
∞
_
k=1
i=1
µ(Ak ) +
k=1
!
aki,ϕ(i+k) , and
!
Ak ≤ 2µ(T ) = u.
N
[
k=1
From the above inequalities and (2), we get
µ(C) −
N
X
∞
_
µ(Ak ) ≤
k=1
bi,ϕ(i).
(4)
i=1
Thus (3) follows from (4) and weak σ-distributivity of R, by arbitrariness of ϕ ∈ NN .
Now, thanks to regularity of µ, in correspondence with A a regulator (ci,j )i,j can be
found, such that for every ϕ ∈ NN there is a compact set C ⊂ A with
µ(A) ≤ µ(C) +
∞
_
ci,ϕ(i).
(5)
i=1
From (3) and (5) we obtain µ(A) ≤
of R, we finally get the assertion.
P∞
k=1 µ(Ak )+
W∞
i=1 ci,ϕ(i) .
Again by weak σ-distributivity
We now turn to the concept of Kurzweil-Henstock integrability in the Riesz space setting.
From now on we shall assume that
(H0) R is super Dedekind complete and weakly σ-distributive;
(H1) (T, d) is a compact metric space.
Definition 2.3. A gage is any map γ : T → R+ .
Definition 2.4. A decomposition Π of T is a finite family Π = {(Ei , ti) : i = 1, . . ., k} of
pairs such that ti ∈ Ei , Ei ∈ Σ and µ(Ei ∩ Ej ) = 0 for i 6= j. The points ti , i = 1, . . ., k,
Sk
are called tags. If moreover i=1 Ei = T , Π is called a partition. Given a gage γ, we say
that Π is γ-fine (Π ≺ γ) if d(w, ti) < γ(ti ) for every w ∈ Ei and i = 1, . . . , k .
Remark 2.5. Observe that, under the above hypotheses, by [44, Proposition 1.7], for every
E ∈ Σ and γ : T → R+ there exists a γ|E -fine partition of E.
Definition 2.6. [10, Definition 3.1] A function f : T → R is (KH)-integrable (or, in short,
integrable) if there exist I ∈ R and an (o)-sequence (bn )n such that for all n ∈ N there is a
gage γ : T → R+ with
X
f − I ≤ bn
Π
62
A. Boccuto, A. M. Minotti and A. R. Sambucini
P
Pq
for every γ-fine partition of T , Π = {(Ei, ti), i = 1, . . . , q}. Here Π f := i=1 f(ti ) µ(Ei )
is the Riemann sum of f Ron Π. When this is the case, the element I is determined uniquely,
and will be denoted by T f dµ. In an analogous way we can define the (KH)-integral on
any set E ∈ Σ.
Observe that, thanks to super Dedekind completeness of R and [7, Theorem 3.4], our
definition of (KH)-integrability, formulated in terms of (o)-sequences, is equivalent to the
corresponding one given in [10] and [46] in terms of regulators. Moreover, by [44, Lemma
1.10] and [7, Theorem 3.4] again, we get that integrability is inherited by Σ-measurable
subsets, and that the integral is additive on disjoint sets and is a linear positive functional
(see also [10, Propositions 3.2 and 3.5]).
The following concept of Kurzweil-Henstock integrability was presented in [26] in the Banach
space context (see also [12]).
Definition 2.7. A function f : T → R is (KH)-norm integrable (or, in short, norm
integrable) if there exists I ∈ R such that, for every ε > 0 there is a gage γ : T → R+ such
that for every γ-fine partition of T , Π = {(Ei, ti), i = 1, . . ., q}, we have:
X
f − I ≤ ε.
Π
In Riesz spaces in general it is not easy to compare Bochner and Kurzweil-Henstock
integrability: we only know that simple functions are integrable in both senses and their
integral is the obvious one ([10, Theorem 3.7]), but it is possible to construct a Bochner
integrable function which is not (KH)-integrable, as the following example shows.
Example 2.8. Let R2 = R and R = R1 = c00 be the space of eventually null real-valued
sequences. For n ∈ N, let un := (0, . . . , 0, 1, 0, . . .), where the value 1 is assumed at the n-th
coordinate. The function f : [0, 1] → R, defined by
un if x = 1/n
f(x) =
(6)
0
otherwise
vanishes almost everywhere (with respect to the Lebesgue measure), so its Bochner integral
is null, but we claim that f is not (KH)-integrable on [0, 1].
Indeed, fix arbitrarily δ : [0, 1] → R+ and n ∈ N, n ≥ 2. For every i = 1, . . . , n − 1,
1
let ξi =
and choose an interval ]yi , xi[ such that ξi ∈]yi, xi[, xi − yi < δ(ξi ),
n+1−i
[yi , xi] ∩ [yj , xj ] = ∅ for all i 6= j, 0 < y1 and xn−1 < 1. We have:
0 < y1 < x1 < y2 < x2 < . . . < yn−1 < xn−1 < 1.
Let x0 = 0, yn = 1, and let us divide each of the intervals [xi−1, yi], i = 1, . . ., n, into tagged
subintervals, in such a way to obtain, together with the elements ([yi , xi], ξi), i = 1, . . . , n−1,
a δ-fine partition: this is possible, by virtue of the Cousin Lemma ([39, Theorem 2.3.1]).
Since f = 0 on each of the intervals [xi−1, yi ], i = 1, . . . , n, we have:
p
X
j=1
(tj − tj−1) f(ηj ) =
n−1
X
i=1
(xi − yi ) f(ξi ).
63
Set-valued Kurzweil-Henstock integral
(n)
Let λi
= xn+1−i − yn+1−i, i = 2, . . . , n: then we get
p
X
(tj − tj−1) f(ηj ) =
j=1
n−1
X
(n)
λn+1−i
f(ξi ) =
i=1
n−1
X
(n)
λn+1−i un+1−i ≥ λ(n)
n un .
i=1
P
(n)
Since λn is strictly positive for every n, the sequence
Πn f n is unbounded in R. If f
was (KH)-integrable on [0, 1], then there would exist a gage δ0 : [0, 1] → R+ such that
(
)
_ X
f : Π is a δ0 −fine partition of [0, 1] ∈ R :
Π
this is a contradiction. Hence, f is not (KH)-integrable on [0, 1] with respect to (o)sequences.
Remark 2.9. Observe that c00 is super Dedekind complete, because it is a solid subspace
of the super Dedekind complete Riesz space RN (see [40]).
Moreover, note that c00 is weakly σ-distributive. Indeed, any regulator (ai,j )i,j is
bounded by an element of c00, hence all coordinates of the vectors ai,j are null, except
a finite set independent of i and j: thus weak σ-distributivity follows from the same property of the finite-dimensional Euclidean spaces.
However, if we consider our function f with values in (c0 , k · k∞ ), then, since it is Bochner
integrable, it is also (KH)-norm integrable on [0, 1]. In fact, by [28, Theorem 1K], f is Mc
Shane integrable and all Mc Shane integrable functions are (KH)-norm integrable by [26,
Theorem 8]. So f is also (KH)-norm integrable in c00 with respect to k · k∞ , since c00 is
dense in c0 .
3
Multivalued Kurzweil-Henstock integral
We now introduce some definitions and properties.
Definition 3.1. Let C ⊂ R be any non-empty subset, and define
U(C, r) := {z ∈ R : there exists x ∈ C with |x − z| ≤ r},
we shall call it an r-neighborhood of C. The set C ⊂ R is said to be closed if C = cl(C),
where
[ \
cl(C) :=
U(C, bn)
(7)
(bn )n n
is the closure of C, and the union is meant with respect to the totality of (o)-sequences
(bn )n in R.
Observe that every order interval [a, b] in any Dedekind complete Riesz space R is obviously closed. Let cfb(R) be the family of all non-empty, convex, bounded and closed subsets
64
A. Boccuto, A. M. Minotti and A. R. Sambucini
of R. Analogously as in [20] we define ⊕Pon cfb(R) by A ⊕ B := cl(A + B). Moreover, if
n
Ai ∈ cfb(R), i = 1, . . ., n, we denote by i=1 Ai the set
n
X
Ai := cl(A1 + · · · + An ).
(8)
i=1
However, these definitions make sense if we prove closedness of the closure: to this aim, our
technique requires to replace a countable family of (o)-sequences with a single one. This is
possible in super Dedekind complete spaces, thanks to the following result, which is parallel
to Lemma 2.1.
(n)
Lemma 3.2. ([8, Lemma 2.8]) If (σp )p , n ∈ N, is an order equibounded countable family
of (o)-sequences in R, then there is an (o)-sequence (βr )r with the property that for every
n, r ∈ N there exists p = p(n, r), with
σp(n) ≤ βr .
(9)
Proposition 3.3. If A is a bounded set, then
cl(cl(A)) = cl(A).
Proof. First of all we observe that, if A is bounded, then cl (A) is bounded too. In fact, if
A ⊂ [−u, u] for some positive element u, then cl (A) ⊂ cl ([−u, u]) = [−u, u]. We now prove
that cl( cl (A)) ⊂ cl (A), since the converse inclusion is obvious.
Let z ∈ cl( cl (A)). There exist an (o)-sequence (bn)n and a sequence (αn)n in cl (A) such
that |z − αn| ≤ bn for all n.
(n)
Since αn ∈ cl (A), for every n ∈ N there is an (o)-sequence (σp )p with the property that
(n)
to every p ∈ N there corresponds an,p ∈ A such that |an,p − αn| ≤ σp . Without loss of
(n)
generality, we can suppose that σp ≤ 2 u for each n, p ∈ N. Thus, by virtue of Lemma
(n)
3.2, in correspondence with the (o)-sequences (σp )p , n ∈ N, there is an (o)-sequence (βr )r ,
(n)
with the property that for every n, r ∈ N there exists p = p(n, r), with σp ≤ βr . So, taking
r = n, we get
|z − an,p | ≤ |z − αn| + |αn − an,p| ≤ bn + σp(n) ≤ bn + βn .
So we have found an (o)-sequence (dn := bn + βn )n such that for every n ∈ N there is an
element an,p(n) ∈ A with |z − an,p(n)| ≤ dn, which means that z ∈ cl (A).
So, if A, B ∈ cfb(R), then A ⊕ B ∈ cfb(R). Moreover the following result holds.
Proposition 3.4. If R is a weakly σ-distributive Banach lattice with an order continuous
norm, then the norm closure and the closure coincide.
Proof. Since R is super Dedekind complete, by [29, 367X (u), p. 372] and [29, 367B (f), p.
361] a non-empty set C ⊂ R is closed for the norm topology if and only if x ∈ C whenever
there is an order bounded sequence (xn )n in C with x = lim inf xn = lim sup xn . Thus the
assertion follows.
65
Set-valued Kurzweil-Henstock integral
Definition 3.5. A multifunction F is said to be bounded, if there exists a positive element
L ∈ R with F (t) ⊂ [−L, L] for all t ∈ T .
We now define a multivalued integral in the Riesz space setting.
Definition 3.6. Let F : T → 2R be a multifunction with non-empty values, and E ∈ Σ.
We call (∗)-integral of F on E the set
Φ(F, E) =
{ z ∈ R : there exists an (o)-sequence (bn )n : for all n ∈ N
there is a gage γ : T → R+ such that
for every γ-fine partition Pγ := {(Ei, ti ) : i = 1, . . . , k}
of E there exists c ∈
k
X
F (ti ) µ(Ei) with |z − c| ≤ bn }.
i=1
We have the following results.
Proposition 3.7. The set Φ(F, E) can be described as follows:
Φ(F, E) =
[ \[
(bn )n n
\
γ {(Ei ,ti )}∈Πγ
U
k
X
!
F (ti )µ(Ei), bn .
(10)
i=1
If u is a positive upper bound for |F |, then u µ(T ) is an upper bound for Φ(F, E) for every
E ∈ Σ. If F is constant in E, say F ≡ C, then Φ(F, E) is the closure of C µ(E). The set
Φ(F, E) is convex provided that it is non-empty and the multifunction F is convex-valued.
Proof.
The first formula is an easy consequence of definitions of (∗)-integral and of
bn -neighborhood. For the second part, thanks to boundedness we have:
Φ(F, E) ⊂
[ \[
(bn )n n
⊂
[ \
(bn )n
\
U ([−uµ(T ), uµ(T )], bn) ⊂
γ {(Ei ,ti )}∈Πγ
[−uµ(T ) − bn, uµ(T ) + bn] = [−uµ(T ), uµ(T )].
n
If F is constant, F ≡ C on E, all Riemann sums of f coincide with Cµ(E), hence the first
formula proved here shows the assertion. The proof of convexity follows usual lines.
It is easy to check that, if F is single-valued (namely F (t) is a singleton) and integrable,
then the (∗)-integral is the set whose unique element coincides with the Kurzweil-Henstock
integral given in Definition 2.6. However note that, in the case of Example 2.8, we get
Φ({f}, E) = ∅, since the Riemann sums are not eventually bounded in R.
Finally we prove closedness of Φ(F, E).
Proposition 3.8. If F is bounded, then the set Φ(F, E) is closed.
66
Proof.
A. Boccuto, A. M. Minotti and A. R. Sambucini
All we have to prove is that
[ \ U Φ(F, E), bn ⊂ Φ(F, E).
(bn )n n
Take arbitrarily z ∈
S
(bn )n
T
. There is a sequence (zn )n in Φ(F, E),
U
Φ(F,
E),
b
n
n
(o)-convergent to z, and hence order bounded. For each n ∈ N there is an (o)-sequence
(n)
(σp )p with the property that to every p ∈ N there corresponds a gage γn,p such that
Pk
for any γn,p -fine partition {(Ei , ti) : i = 1, . . ., k} there is wn,p ∈ i=1 F (ti) µ(Ei ) with
(n)
|zn −wn,p | ≤ σp . Since F is bounded, the double sequence (zn −wn,p)n,p is order bounded.
(n)
So, without loss of generality, we can suppose that there is u ∈ R, u ≥ 0, with σp ≤ u
for each n, p ∈ N. Thus, by virtue of Lemma 3.2, in correspondence with the (o)-sequences
(n)
(σp )p , n ∈ N, there is an (o)-sequence (βr )r , in accordance with that Lemma: so with
every n, r ∈ N we can associate p ∈ N satisfying (9). Let (bn)n be an (o)-sequence with
|zn − z| ≤ bn for all n ∈ N. Thus, in correspondence with n and p, taking r = n, there is a
gage γn,p with the property that for all γn,p -fine partitions {(Ei, ti) : i = 1, . . ., k} there is
Pk
wn,p ∈ i=1 F (ti) µ(Ei ) with
|z − wn,p | ≤ |z − zn | + |zn − wn,p | ≤ bn + σp(n) ≤ bn + βn .
This implies that z ∈ Φ(F, E), and hence the set Φ(F, E) is closed.
The next goal, which we will obtain in the next section, is to prove integrability and
to evaluate the integrals of simple measurable multifunctions. We begin with the following
technical lemma.
Lemma 3.9. Let E ∈ Σ be fixed. Then there exists an (o)-sequence (aE
n )n such that for
every n ∈ N there is a gage γnE with
µ(DE ∆E) ≤ aE
n
for every γnE -fine partition Π = {(Di , ui), i = 1, ..., q} of T , where DE :=
(11)
[
Di .
ui ∈E
Proof.
Since µ is regular, in correspondence with E there is a (D)-sequence (aE
i,j )i,j
satisfying the condition of regularity. Moreover, thanks to super Dedekind completeness
and weak σ-distributivity of R, there exists a sequence (ϕn )n in NN such that
aE
n :=
∞
_
aE
i,ϕn (i)
i=1
is an (o)-sequence (see also [7, Theorem 3.4]).
Fix arbitrarily n ∈ N, and let Kn , Un be a compact and an open set respectively, such
that µ(Un \ Kn) ≤ aE
n . Arguing analogously as in [46, Proposition 5.2.11], we get the
existence of a gage γnE : T → R+ with

for all t ∈ Kn ,
 Un
B(t, γnE (t)) := {w ∈ T : d(t, w) < γnE (t)} ⊂
Un \ Kn for all t ∈ Un \ Kn ,

T \ Kn
for all t ∈ T \ Un .
67
Set-valued Kurzweil-Henstock integral
So, if Π = {(Di , ui), i = 1, . . ., q} is any γnE -fine partition of T , then DE ∆E ⊂ Un \ Kn and
hence we get
µ(E∆DE ) ≤ µ(Un \ Kn ) ≤ aE
n.
We now state the following
Proposition 3.10. Let F be a bounded multifunction. Then, for every E ∈ Σ we have:
Φ(F, E) = Φ(F 1E , T ).
Proof.
If z ∈ Φ(F 1E , T ), then there are an (o)-sequence (bn)n and a sequence of gages
(γn )n, such that
P for every n ∈ N and every γn -fine partition Π = {(Ti , ti) : i = 1, . . ., q}
there is cn ∈ Π F 1E with |z − cn| ≤ bn .
We now prove that z ∈ Φ(F, E). Indeed, fixed n, any γn -fine partition Π0 = {(Di , ti) :
i = 1, . . . , s} of E can be extended (thanks to the Cousin Lemma)
P to a γn -fine partition
Π =P
{(Ti , ti) : i = 1, . . . , q} of T , and clearly P
the Riemann sum i F (ti )µ(Di ) coincides
with i F (ti )1E (ti )µ(Ti ). Hence there is c0n ∈ i F (ti )µ(Di ) satisfying |z − c0n | ≤ bn. This
means that z ∈ Φ(F, E). By arbitrariness of z, the inclusion Φ(F 1E , T ) ⊂ Φ(F, E) is proved.
We now prove the converse inclusion. By Lemma 3.9, since µ is regular, in correspondence
E
+
with E, there are an (o)-sequence (aE
n )n and a sequence of gages (γn : T → R )n such
E
E
that µ(E∆(∪{Di : ui ∈ E})) ≤ an for every n ∈ N and every γn -fine partition of T ,
Π = {(Di , ui), i = 1, . . . , q}.
Let z ∈ Φ(F, E). By definition, there exist an (o)-sequence (bn)n and a sequence of gages
(γn : P
T → R+ )n such that for every n ∈ N and every γn -fine partition P of E there exists
cn ∈ P F with |z − cn | ≤ bn.
We now claim that z ∈ Φ(F 1E , T ). To this aim, let L be an upper bound for F , and for
∗
every n ∈ N set a∗n := 4 L · aE
n + bn : we prove that the (o)-sequence (an )n satisfies the claim.
For each n, let γnE and γn be as above and set γn∗ := γnE ∧ γn . Let Π = {(Di , ui), i =
e = ∪i{Di : ui ∈ E}.
1, . . . , r} be a γn∗ -fine partition of T , and D
∗
e (see Remark 2.5) and ΠE := {(Di ∩ E, ui) :
Let Π0 be a γn -fine partition of E \ D
∗
(Di , ui) ∈ Π, ui ∈ E} ∪ Π0 . By
P construction ΠE is a γn -fine partition of E and then also
γn -fine. So, there exists cn ∈ ΠE F with |z − cn| ≤ bn . Observe that
X
Π
F 1E
X
=
r
X
F 1E (ui)µ(Di ) =
i=1
F
=
ΠE
X
X
F (ui )µ(Di );
ui ∈E
F (ui )µ(Di ∩ E) ⊕
X
ui ∈E
F.
Π0
Then we have the following inclusions:
X
X
F ⊂ U(
F (ui )µ(Di ∩ E), L · aE
n );
X
ΠE
ui ∈E
F (ui)µ(Di ) =
ui ∈E
X
F (ui)[µ(Di ∩ E) + µ(Di \ E)] =
ui ∈E
=
X
ui ∈E
F (ui)µ(Di ∩ E) ⊕
X
ui ∈E
F (ui)µ(Di \ E);
68
A. Boccuto, A. M. Minotti and A. R. Sambucini
X
F (ui)µ(Di ∩ E)
ui ∈E
X
⊂
U(
X
U(
X
F (ui)µ(Di ), L · aE
n ) = U(
ui ∈E
F
⊂
ΠE
X
F 1E , L · aE
n );
Π
F 1E , 4 L · aE
n ).
Π
Then we obtain
z
∈ U(
X
F, bn) ⊂ U(
ΠE
X
F 1E , a∗n).
Π
and this proves that z ∈ Φ(F 1E , T ).
3.1
Comparison with the Aumann integral
We recall that, in the context of Banach spaces, the Aumann integral is defined via Bochner
integrable selections. In our setting this is not possible, as we showed in Example 2.8.
So we introduce the Aumann integral via Kurzweil-Henstock integrable selections and
compare it with the previous integral for multifunctions given in Definition 3.6.
For a multifunction F : T → 2R \ {∅} let
SF1 = {f : f(t) ∈ F (t) µ − a.e. and f is (KH)-integrable }
be the set of all (KH)-integrable selections of F in the sense of Definition 2.6.
Definition 3.11. If SF1 is non-empty, then for every E ∈ Σ we define the Aumann integral (shortly (A)-integral ) of F as
(A)
Z
F dµ =
Z
E
fdµ, f ∈
SF1
.
E
As in the single-valued case, we obtain the following result.
Pn
Theorem 3.12. If F = k=1 Ck 1Ek , where Ck is non-empty and bounded, Ek ∈ Σ for all
k = 1, . . ., n, and the Ek ’s are pairwise disjoint, then for every A ∈ Σ we get
Φ(
n
X
k=1
Ck 1Ek , A) =
n
X
Φ(Ck 1Ek , A) =
k=1
n
X
Z
Ck µ(Ek ∩ A) = cl (A)
F dµ .
k=1
A
Proof. First of all observe that in this case all the sets involved are non-empty, since the
Ck ’s are non-empty. The equality
n
X
k=1
Φ(Ck 1Ek , A) =
n
X
k=1
Ck µ(Ek ∩ A)
(12)
69
Set-valued Kurzweil-Henstock integral
follows immediately from Proposition
3.10.
Pn
Moreover observe that, if F = k=1 Ck 1Ek , then for every gage γ and for all γ-fine partitions
Π = {(Br , tr ) : r = 1, . . . w} of A we have:
X
F
=
Pn
k=1 Ck µ(A
∩ Ek ) ⊂ (A)
Φ(
F (tr )µ(Br ) =
r=1
Π
Since
w
X
n
X
R
A
F |Ek .
(13)
k=1 Π
F dµ ⊂ cl (A)
Ck 1Ek , A) ⊂
k=1
n X
X
n
X
R
A
F dµ , we have only to prove that
Φ(Ck 1Ek , A)
(14)
k=1
R
Pn
and finally that (A) A F dµ ⊂ Φ(F, A). In order to deduce (14), let z ∈ Φ( k=1 Ck 1Ek , A):
then there is an (o)-sequence (bp )p with the property that for any p ∈ N there is a gage
γ
for all γp -fine partitions Π∗ = {(Dm , ξm ) : m = 1, . . . , q} of A there is yp ∈
p such
P
Pthat
n
k=1 Ck 1Ek ) such that |yp − z| ≤ bp , namely z = yp + αp with |αp | ≤ bp .
Π∗ (
0
We consider only the γ-fine partitions Π∗ = {(Dm
, tm ) : m = 1, . . . , s} of A such that for
∗
every k = 1, . . ., n the family Π |Ek is also a partition of Ek . By (13) and (12), and using
commutativity of our addition of closed sets, we have
!
!
s
n
n
s
X
X
X
X
0
yp ∈
Ck 1Ek (tm )µ(Dm ) =
Ck
1Ek (tm )µ(Dm ) =
=
m=1
n
X
k=1
Ck µ(Ek ∩ A) =
k=1
So, z − αp ∈
n
X
k=1
n
X
m=1
Φ(Ck 1Ek , A).
k=1
Φ(Ck 1Ek , A). Thanks to arbitrariness of p ∈ N we deduce
k=1
z
∈
n
n
\ X
[ \ X
U
Φ(Ck 1Ek , A), bp ⊂
U
Φ(Ck 1Ek , A), bp =
p
=
n
X
k=1
(bp )p
p
k=1
Φ(Ck 1Ek , A).
k=1
R
R
R
Finally, let us prove that (A) A F dµ ⊂ Φ(F, A). If z ∈ (A) A F dµ, then z = A g dµ
with g ∈ SF1 . By Definition 2.6, there exists an (o)-sequence (bn )n such that for every n ∈ N
there is a gage γ such that
X
g − z ≤ bn
(15)
Π
P
P
whenever Π is a γ-fine partition. Since Π g ∈ Π F , then we get that (bn )n is a ”good”
(o)-sequence to prove that z ∈ Φ(F, A). Thus the assertion follows.
70
A. Boccuto, A. M. Minotti and A. R. Sambucini
Note that, if F =
n
X
Ck 1Ek , where Ck ∈ cfb(R) for each k = 1, . . . , n, then Φ(F, E) ∈
k=1
cfb(R).
Observe that the conditions known in the literature, under which SF1 is non empty, are
not useful in our approach, since all results like Kuratowski and Ryll-Nardzewski selection
theorems, or the Castaing representation (namely F = cl({fn}), where (fn )n is a suitable
sequence of measurable selections of F ([20, Chapter 3]), are given in separable Banach
spaces. So it is still an open problem to compare the (*)- and Aumann integrals for not necessarily simple multivalued functions. However, we give here some partial answers. As we
observed above, all the given results are still valid in an abstract product triple (R1 , R2, R).
So, from now on we consider a product triple of the type (R, R, R), where we assume
that
(H) R is a weakly σ-distributive Banach lattice with an order continuous norm, and has an
order unit e in the interior of R+ .
An example of such a triple is obtained by choosing R as the subset of all bounded
functions in Lp (X, B, µ), 1 ≤ p < +∞, endowed with the usual order, where X is a compact
metric space, B is the Borel σ-algebra and µ is a positive, finite and regular measure. In
fact R is super Dedekind complete, because it is solid in the super Dedekind complete space
L0 (X, B, µ) (see also [40]) and is a weakly σ-distributive Banach lattice (see also [43]).
We consider here the multivalued integral given by
Definition 3.13. Let F : T → 2R \{∅} be a multifunction, and E ∈ Σ. We call (k·k)-integral
of F on E the set
Φk·k (F, E) = {z ∈ R : for every ε ∈ R+ there is a gage γ : T → R+ with
c∈
inf
P
Πγ
F
kz − ck ≤ ε for every γ-fine partition P := {(Ei, ti ) : i = 1, . . ., k}
of E.}
The following is a comparison with other (KH) integrals:
Proposition 3.14. For every E ∈ Σ we have that
(3.14.1) Φ(F, E) = Φk·k (F, E) = Φk·ke (F, E), for every bounded multifunction F : T → cfb(R);
(3.14.2) f : T → R is (KH)-integrable iff f is (KH)-norm integrable.
Proof. Since every (o)-sequence is norm converging to 0, then we get Φ(F, E) ⊂ Φk·k (F, E).
By [43, Corollary 1.2.14] the two norms k · k and k · ke are equivalent (∼), so Φk·k (F, E) =
Φk·ke (F, E). Therefore we have only to prove that
Φk·ke (F, E) ⊂ Φ(F, E).
Let z ∈ Φk·ke (F, E). So, in correspondence with n ∈ N there exists a gage γn such that for
P
e
every γn -fine partition Π of E there exists cn ∈ Π F with |z − cn| ≤ . Now, in order to
n
Set-valued Kurzweil-Henstock integral
71
see that z ∈ Φ(F, E) it is sufficient to observe that (e/n)n is an (o)-sequence.
For the second part, observe that f is (KH)-integrable if and only if Φ({f}) 6= ∅. So, the
assertion follows from order continuity of the norm and k · k ∼ k · ke .
1
We denote by SF,k·k
the set of all (KH)-norm integrable selections of F . Thanks to
1
(3.14.2), it is clear that the Aumann integral of F is non-empty iff SF,k·k
is.
Remark 3.15. In the above considered setting, when R is even reflexive and separable, if
F : T → cfb(R) admits a Castaing representation and is integrably bounded (namely F is
bounded by an element of L1 ), then SF1 6= ∅. In fact by [42, Proposition 3.1], F admits
1
Pettis integrable selections (that is SF,P
e 6= ∅). In separable Banach spaces, Pettis and
Mc Shane integrability are equivalent [28, Corollary 4C], and Mc Shane integrability is in
general stronger than Kurzweil-Henstock norm integrability.
In non separable Banach spaces, results of existence of selections are given in [18]. We recall
that the space (cwk(X), h) of all weakly compact, convex subsets of a Banach space X can be
studied using the Rådstrőm embedding (see [45]) and, in particular, the Hőrmander embedding j : cwk(X) → l∞ (BX ∗ ) given by j(A) := δ ∗ (·, A) = supx∈A x∗ (x), where l∞ (BX ∗ ) is the
Banach space of all bounded real valued functions defined on BX ∗ , endowed with the supremum norm k · k∞ (see [20]). By [18, Theorem 2.5, Proposition 4.4], if j ◦ F : T → l∞ (BX ∗ )
1
is Pettis integrable (in the sense of [18]), then SF,P
e 6= ∅ and so, using [28, Corollary 4D]
we get that Pettis integrability implies Mc Shane integrability and non-emptiness of SF1 .
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