Download A Hake-type theorem for integrals with respect to

A HAKE-TYPE THEOREM FOR INTEGRALS WITH
RESPECT TO ABSTRACT DERIVATION BASES IN THE
RIESZ SPACE SETTING
A. BOCCUTO,* V. A. SKVORTSOV,** F. TULONE ***
Abstract. A Kurzweil-Henstock type integral with respect to an abstract
derivation basis in a topological measure space, for Riesz space-valued functions, is studied. A Hake-type theorem is proved for this integral, by using
technical properties of Riesz spaces.
1. Introduction
The known Hake theorem in integration theory (see [27, Chapter VIII, Lemma
3.1]) states that, in contrast to the Lebesgue integral, the Perron integral on
a compact interval is equivalent to the improper Perron integral, i.e., Perron
integrability of a function f on [a, b] is equivalent to Perron integrability
on
Z
c
[a, c] with a < c < b together with the existence of the limit lim−
c→b
f . As
a
the Perron integral on the real line is known to be equivalent to the KurzweilHenstock integral (see [21, 24]), the same property is true for the last integral.
The general idea of computing the improper integral as a limit of integrals
over increasing families {Aα } of sets can be realized in the multidimensional
case and in abstract measure spaces in several ways, depending on the type of
the integral and on the family {Aα } chosen to generalize the compact intervals
of the one-dimensional case. This gives rise to various versions of the Hake
theorem. Some kinds of it for Kurzweil-Henstock type integrals in Rn and in
more general spaces were studied in [9, 17, 19, 23, 31]. A version of the Hake
theorem for Riesz space-valued functions was proved in [10]. For recent studies
about measures and integrals in the Riesz space context see also [5, 6, 11, 12].
In this paper we consider a Kurzweil-Henstock type integral with respect to an
abstract derivation basis in a topological space for Riesz space-valued functions
2010 Mathematics Subject Classification. Primary 28B15; Secondary 26A39.
Key words and phrases. Riesz space, order convergence, (D)-convergence, KurzweilHenstock integral, Hake theorem.
This work was supported by NSh-979.2012.1, RFFI-11-01-00321a, G. N. A. M. P. A. and
Universities of Palermo and Perugia.
1
2
A. BOCCUTO, V. A. SKVORTSOV AND F. TULONE
(for the related literature see for instance [1, 8, 22, 26, 35]). A similar kind of
integral was studied in [16]. We investigate what assumptions on the basis and
on the Riesz space guarantee a generalized Hake property for this integral. A
special attention is given to the kind of order convergence used in the definition
of the integral.
The main result of the paper is an extension, to the Riesz space setting, of
a Hake type theorem proved in [31] for the real-valued case. It also gives a
generalization of results of [10], where a special type of derivation basis, defined
by gauges, was considered.
2. Riesz spaces and modes of convergence
We begin with reminding some preliminary notions and results about Riesz
spaces.
A Riesz space R is said to be Dedekind complete if every nonempty subset
A ⊂ R, bounded from above, has a supremum in R. A Dedekind complete
Riesz space R is said to be super Dedekind complete if for any nonempty set
A ⊂ R, bounded from above, there exists a countable subset A∗ ⊂ A, such that
∨A = ∨A∗ .
A non-empty ordered set Λ = (Λ, ) is said to be directed if for every λ1 ,
λ2 ∈ Λ there exists λ3 ∈ Λ such that λ3 λ1 and λ3 λ2 .
A net is any function defined on a directed set Λ = (Λ, ) with values in a
Riesz space R, and we denote it by the symbol (rλ )λ∈Λ .
V An (O)-net (rλ )λ∈Λ is a monotone decreasing net of elements of R, such that
λ∈Λ rλ = 0. In particular, in the case (Λ, ) = (N, ≥), this defines the notion
of (O)-sequence.
A bounded double sequence (ai,j )i,j in R is called a regulator or a (D)sequence if (ai,j )j is an (O)-sequence for each i ∈ N.
A subset W of a Riesz space R is (P R)-bounded if there exists a strictly
increasing sequence (wn )n of positive elements of R, such that for every w ∈ W
there is n ∈ N with |w| ≤ wn (see [25, 32]).
A Riesz space R is weakly σ-distributive if
∞
^ _
(2.1)
ai,ϕ(i) = 0
ϕ∈NN i=1
for every (D)-sequence (ai,j )i,j .
We now recall some modes of convergence of nets in Riesz spaces. The most
general notion of order convergence of a net in a Riesz space R seems to be
introduced in [28] (see also [3, Definition 1.2]).
Definition 2.1. A net (xλ )λ∈Λ in R is order convergent to x ∈ R if there exist
a directed set Γ and an (O)-net (yγ )γ∈Γ such that for each γ ∈ Γ there exists
λ0 ∈ Λ with |xλ − x| ≤ yγ for every λ λ0 .
A HAKE-TYPE THEOREM
3
For a Dedekind complete Riesz space this definition is equivalent (see [3]) to
the following one:
Definition 2.2. A net (xλ )λ∈Λ in R is order convergent to x ∈ R if there exist
an (O)-net (yλ )λ∈Λ and λ0 ∈ Λ with |xλ − x| ≤ yλ for every λ λ0 .
Specifying the directed set Γ in Definition 2.1 we get more restrictive notions
which we call (O)- and (D)-convergence (see also [8, 22, 26]).
Definition 2.3. A net (xλ )λ∈Λ is (O)-convergent to x ∈ R (and we write
(O) lim xλ = x) if there exists an (O)-sequence (bn )n such that for each n ∈ N
λ∈Λ
there is λ0 ∈ Λ with |xλ − x| ≤ bn for every λ λ0 .
Definition 2.4. Let R be a weakly σ-distributive Riesz space. A net (xλ )λ∈Λ is
(D)-convergent to x ∈ R (shortly, (D) lim xλ = x) if there exists a (D)-sequence
λ∈Λ
(ai,j )i,j in R, such that for every ϕ ∈ NN there is λ0 ∈ Λ with
|xλ − x| ≤
∞
_
ai,ϕ(i)
for every λ λ0 .
i=1
It is clear that both (O)- and (D)-convergence of a net imply order convergence in the sense of Definitions 2.1 and 2.2 to the same limit (indeed, choose
as Γ the directed sets (N, ≥) for (O)-convergence and NN with the increasing
pointwise ordering for (D)-convergence). Moreover, note that order convergence
and (O)-convergence coincide if and only if the Riesz space involved is super
Dedekind complete. Indeed (see [1, p. 20]), a Dedekind complete Riesz space
is super Dedekind complete if and only if for any (O)-net (xλ )λ∈Λ there is a
strictly increasing sequence (λn )n such that (xλn )n is an (O)-sequence.
A Riesz space R is said to satisfy property (σ) if, given any sequence (un )n
in R with un ≥ 0 for all n ∈ N, there exist an element u ∈ R, u ≥ 0, and a
sequence (λn )n of positive real numbers, such that λn un ≤ u for all n ∈ N.
A Riesz space R is said to be regular if it satisfies property (σ) and for every
sequence (xn )n , (O)-convergent to x ∈ R, there exists w ∈ R such that for any
real number ε > 0 there is n0 ∈ N with
|xn − x| ≤ εw
for each n ≥ n0 .
(2.2)
Remark 1. Observe that, if (X, M, µ) is a measure space, with µ positive, σfinite and σ-additive, then the space L0 (X, M, µ) of all µ-measurable real-valued
functions, with identification up to µ-null sets, is super Dedekind complete,
weakly σ-distributive and regular (see [4, 22]).
We now turn to a relation between (O)- and (D)-convergence. The following
result is an extension of [4, Theorem 3.4] and will be useful in the sequel.
Proposition 2.1. In every Riesz space R, any (O)-convergent net is (D)convergent too.
4
A. BOCCUTO, V. A. SKVORTSOV AND F. TULONE
Proof. Let (xλ )λ∈Λ be a net in R, (O)-convergent to x ∈ R, and (bn )n be an
(O)-sequence which defines (O)-convergence. For every i, j ∈ N define ai,j := bj .
It is clear that this double sequence (ai,j )i,j is a regulator. Choose arbitrarily
ϕ ∈ NN . By hypothesis, in correspondence with n = ϕ(1) there exists λ ∈ Λ
such that, for all λ λ,
∞
_
|xλ − x| ≤ bϕ(1) = a1,ϕ(1) ≤
ai,ϕ(i) .
i=1
Thus the assertion follows.
The converse of Proposition 2.1 holds if R is a super Dedekind complete
and weakly σ-distributive Riesz space. However, as we have already mentioned
above, (D)-convergence always implies the order convergence in the sense of
Definition 2.1 and, in the case of a Dedekind complete space, also the convergence
in the sense of Definition 2.2, and here we need not the assumption of super
Dedekind completeness. So we have the following
Proposition 2.2. Let R be a Dedekind complete and weakly σ-distributive Riesz
space, (Λ, ) be a directed set, and suppose that (D) limλ∈Λ xλ = x. Then there
exist an (O)-net (yλ )λ∈Λ and λ0 ∈ Λ with |xλ − x| ≤ yλ for every λ λ0 .
We now state a technical lemma, which gives a property of regular Riesz
spaces.
(k)
Lemma 2.1. If R is a regular Riesz space and (xλ )λ∈Λ , k ∈ N, is a sequence
of nets in R (O)-convergent to x(k) , then there is w ∈ R, w ≥ 0, such that for
b = λ
bk ∈ Λ with |x(k) − x(k) | ≤ w for each
every k ∈ N there is an element λ
λ
b
λ λ.
(k)
Proof. For each k ∈ N let (bn )n be the (O)-sequences, which correspond to the
(k)
net (xλ )λ∈Λ according to Definition 2.3. As R is regular, for each k ∈ N and
(k)
for the sequences (bn )n there is a positive element wk ∈ R, such that for every
ε > 0 there exists n0 ∈ N with
|xn − x| ≤ εwk
for all n ≥ n0 .
(2.3)
Since, by hypothesis, R satisfies property (σ), there exist a sequence (ζk )k of
positive real numbers and an element w ∈ R, w ≥ 0, with ζk wk ≤ w for every
k. Choose now arbitrarily ε > 0 and k ∈ N. Using (2.3) for the sequence
(k)
bn and with ε replaced by ζk ε, we find a number n0 = n0 (ε, k) ∈ N with
(k)
bn ≤ εζk wk ≤ εw whenever n ≥ n0 . By Definition 2.3, in correspondence with
b ∈ Λ can be found, with
n0 , an element λ
(k)
b
|x − x(k) | ≤ b(k) ≤ εw for all λ λ.
λ
Taking ε = 1, we get the assertion.
n0
A HAKE-TYPE THEOREM
5
The following result (Fremlin lemma, see [18, Lemma 1C], [26, Theorem 3.2.3,
pp. 42-45]) allows us to replace a countable family of regulators with one regulator, and this technique will be useful in the proof of the main results of the
paper.
(k)
Lemma 2.2. Let R be any Dedekind complete Riesz space and (ai,j )i,j , k ∈ N,
be a sequence of regulators in R. Then for every u ∈ R, u ≥ 0, there exists a
(D)-sequence (ai,j )i,j with
q _
∞
∞
X
_
(k)
u∧
ai,ϕ(i+k)
≤
ai,ϕ(i)
k=1 i=1
i=1
for every q ∈ N and ϕ ∈ N .
N
Definition 2.5. Let R1 , R2 , R be three Dedekind complete Riesz spaces. We
say that (R1 , R2 , R) is a product triple if there exists a function · : R1 × R2 → R,
which we call product, such that for all rj , sj ∈ Rj , j = 1, 2, we have:
2.5.1): (r1 + s1 ) · r2 = r1 · r2 + s1 · r2 ,
2.5.2): r1 · (r2 + s2 ) = r1 · r2 + r1 · s2 ,
2.5.3): [r1 ≥ s1 , r2 ≥ 0] ⇒ [r1 · r2 ≥ s1 · r2 ],
2.5.4): [r1 ≥ 0, r2 ≥ s2 ] ⇒ [r1 · r2 ≥ r1 · s2 ];
2.5.5): if (aλ )λ∈Λ is any net in R2 , (O) limλ∈Λ aλ = 0 and 0 ≤ b ∈ R1 ,
then (O) lim (b · aλ ) = 0;
λ∈Λ
2.5.6): if (aλ )λ∈Λ is any net in R1 , (O) limλ∈Λ aλ = 0 and 0 ≤ b ∈ R2 ,
then (O) lim (aλ · b) = 0;
λ∈Λ
2.5.7): if α is a real number, then (αr1 ) · r2 = r1 · (αr2 ) = α(r1 · r2 ) for all
rj ∈ Rj , j = 1, 2.
3. The abstract derivation bases
We now introduce some notions involving abstract bases and some related
topics (for the case R1 = R2 = R = R see also [31]).
Let X be a Hausdorff topological space, M be the class of all Borel subsets
of X, R2 be as in Definition 2.5 and µ : M → R2 be a finitely additive positive
measure. We say that µ is an outer regular measure if for every E ∈ M there
exists a strictly positive element z ∈ R2 with the property: for any ε > 0 there
is an open set G ⊃ E with µ(G \ E) ≤ εz.
For any set I ⊂ X we use the notation int(I) and ∂I for the interior and the
boundary of I, respectively.
A derivation basis (or simply a basis) B in (X, M, µ) is a filter base on the
product space I × X, where I is a family of closed subsets of X having strictly
positive measure µ and called generalized intervals or B-intervals (see also [16,
24, 31, 33]). That is B is a nonempty collection of subsets of I × X so that
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A. BOCCUTO, V. A. SKVORTSOV AND F. TULONE
each β ∈ B is a set of pairs (I, x), where I ∈ I, x ∈ X, and B has the filter
base property: ∅ ∈
/ B and for every β1 , β2 ∈ B there exists β3 ∈ B such that
β3 ⊂ β1 ∩ β2 . So, B = (B, ⊂) is a directed set with the order given by the
”reversed” inclusion ⊂. We refer to the elements β of B as basis sets. Some
particular examples of derivation basis in different types of topological spaces
can be found in [2, 16, 24, 29, 30, 34].
In this paper we suppose that the pairs (I, x) forming each β ∈ B have the
property that x ∈ I, although it is not the case in the general theory (see for
instance [20, 24]). We assume that µ(∂I) = 0 for any B-interval I. We say that
two B-intervals I 0 , I 00 are non-overlapping if µ(I 0 ∩ I 00 ) = 0. We call B-figure a
finite union of non-overlapping B-intervals. We suppose that the intersection of
two overlapping B-intervals is a B-figure, so the intersection of two overlapping
B-figures is also a B-figure. For a set E ⊂ X and β ∈ B we put
β(E) := {(I, x) ∈ β : I ⊂ E} and β[E] := {(I, x) ∈ β : x ∈ E}.
If the basis is considered exclusively on a B-interval or a B-figure J (instead of
the whole of X), we refer to it as basis in J using the same notations B for it
and β for its elements.
From now on, we assume that the basis involved B has the following properties:
a) B ignores no point, i.e., β[{x}] 6= ∅ for any point x ∈ X and for any
β ∈ B.
b) B has a local character, that is for any family of basis sets (βτ )τ in
BS
and for any
S pairwise disjoint sets Eτ there exists β ∈ B such that
β[ τ Eτ ] ⊂ τ βτ [Eτ ].
c) B is a Vitali basis, namely for every x ∈ X and for any neighborhood
U (x) of x there exists βx ∈ B such that I ⊂ U (x) for each pair (I, x) ∈
βx .
For a fixed basis set β, a finite collection π of β, where the distinct elements
(I 0 , x0 ) and (I 00 , x00 ) in π have I 0 and I 00 non-overlapping, is called a β-partition.
Let L ⊂ X. If π ⊂ β(L), then π is
[called a β-partition in L; if π ⊂ β[L], then
π is called a β-partition on L; if
I = L, then π is called a β-partition of
(I,x)∈π
L. For a set E and a β-partition π, let π[E] := {(I, x) ∈ π : (I, x) ∈ β[E]} .
We say that a basis B has the partitioning property if the following conditions
hold:
(i) for each B-interval I0 and each finite collection I1 , . . . , In of pairwise nonoverlapping B-intervals with I1 , . . ., In ⊂ I0 there exists a finite number of Bm
[
intervals In+1 , . . . , Im such that I0 =
Ir and Ir are pairwise non-overlapping;
r=1
(ii) for each B-interval I and for any β ∈ B there exists a β-partition of I.
A HAKE-TYPE THEOREM
7
Note that condition (i) of partitioning property implies that the union of
any two B-figures is a B-figure, while condition (ii) implies the existence of a
β-partition of any B-figure. The union of all B-intervals involved in a β-partition
π is called the B-figure generated by π.
One of the simplest derivation bases is the soncalled full interval basis on
the real line R, defined by the basis sets βδ := (I, x) : I ∈ I, x ∈ I ⊂
o
(x−δ(x), x+δ(x)) , where δ is a so-called gauge, i.e., a positive function defined
on R.
The following lemma on extension of the β-partition holds.
Lemma 3.1. Let π 0 be a β-partition in a B-figure L. Then there exists a βpartition π 00 such that π = π 0 ∪ π 00 is a β-partition of L.
Proof. Let π 0 := {(Ij , xSj )}m
j=1 . By condition (i) of the partitioning property we
n
get that L = (∪m
I
)
(∪
j
r=1 Kr ), where the Kr ’s are pairwise non-overlapping
j=1
B-intervals. By virtue of the partitioning property (ii) we obtain the existence
of a β-partition πr of Kr for every index r. Thus π 00 := ∪nr=1 πr is the required
β-partition.
The β-partition π 00 of the above lemma is called a β-complementary partition
to π 0 in L. If F is the B-figure generated by the partition π 0 , then the B-figure
generated by the partition π 00 is called the B-figure complementary to F in L,
and it is denoted by the symbol Cβ (F ).
4. The HB -integral and the main results
Let (R1 , R2 , R) be a product triple. We now introduce the Kurzweil-Henstock
type integral with respect to an abstract basis in the Riesz space setting.
Definition 4.1. (see also [24]) Let B be a basis in (X, M, µ) having the partitioning property, µ : M → R2 be an outer regular measure, and L be a B-figure.
Let Π(β, L) denote, for each β ∈ B, the set of all β-partitions of L. A function f : L → R1 is said to be Kurzweil-Henstock integrable with respect to B (or
HB -integrable) on L, with HB -integral A ∈ R, if (O) lim pβ = 0, where
β∈B
o
_n X
pβ :=
f (x)µ(I) − A : π ∈ Π(β, L) , β ∈ B.
(4.1)
(I,x)∈π
Z
We denote the integral value A by (HB )
f.
L
We say that a function f : L → R1 is HB -integrable on a set E ⊂ L if the
function f · χE , where χE is
function of the set E, is HB Z the characteristic
Z
integrable on L, and we set
f · χE .
f :=
E
L
8
A. BOCCUTO, V. A. SKVORTSOV AND F. TULONE
Remark 2. Observe that in [16] the concept of HB -integral was given by requiring that
^
pβ = 0,
(4.2)
β∈B
where (pβ )β∈B is the decreasing net in (4.1), i.e., on the basis of Definition 2.2
of order convergence. However, we give our definition of HB -integral as above,
to prove our main results, because we will often deal with (O)-sequences, and
this is more natural for the techniques used in the proofs. Note that, when R
is super Dedekind complete, the definitions of HB -integral in (4.1) and (4.2)
are equivalent, since, as we have already mentioned, Definition 2.2 of order
convergence is equivalent in this case to Definition 2.3 of (O)-convergence.
The next result is a condition under which a Riesz space-valued function,
vanishing outside a µ-null set, has integral value zero. Note that, as we shall
see below, there are Riesz space-valued functions, vanishing outside a countable
subset of [0, 1], which are not Kurzweil-Henstock integrable with respect to the
basis formed by all closed subintervals of [0, 1]. The following statement is a
generalization of a result which is always true for the real case (see [21] and [31,
Proposition 1]).
Proposition 4.1. Let (R1 , R2 , R) be a product triple, assume that R satisfies
property (σ), and suppose that f : X → R1 is equal to zero µ-almost everywhere
on a B-figure L and has (P R)-bounded range. Then f is HB -integrable on L
with integral value zero.
Proof. Since the range of f is (P R)-bounded, there is a strictly increasing sequence (wn )n of positive elements of R1 , such that for each x ∈ L there is an
index n with |f (x)| ≤ wn . Let E := {x ∈ L : f (x) 6= 0} and En = {x[
∈E :
|f (x)| ≤ wn }, n ∈ N. We put Dn = En \ En−1 with E0 := ∅. Then E =
Dn .
n
In correspondence with E there is z ∈ R2 according to outer regularity of the
measure µ. By virtue of property (σ), it is possible to associate with the sequence
(wn z)n a sequence (λn )n of positive real numbers and an element 0 ≤ v ∈ R with
ελn
λn wn z ≤ v for all n ∈ N. Now, in correspondence with n ∈ N and n , there
2
ελn
exists an open set Gn ⊃ E with µ(Gn ) ≤ n z. As our basis is a Vitali basis,
2
for any x ∈ E there is a basis set βxn in L with I ⊂ Gn for each (I, x) ∈ βxn [{x}].
By the local character of the basis, we can find β in L such that
β[E] ⊂
∞ [
[
n=1 x∈Dn
βxn [{x}] ,
A HAKE-TYPE THEOREM
9
with β[L \ E] being defined arbitrarily. Take any β-partition π of L, we get:
∞ X
X
f (x) · µ(I) ≤
(I,x)∈π
≤
∞ X
X
n=1 (I,x)∈π[Dn ]
X
f (x) · µ(I) ≤
n=1 (I,x)∈π[Dn ]
∞
∞
∞
X
X
X
ε
ε wn ·µ(I) ≤
wn ·µ(Gn ) ≤
λ
w
z
≤
v = εv.
n
n
2n
2n
n=1
n=1
n=1
From this the assertion follows.
Observe that the hypothesis that R fulfils property (σ) in Proposition 4.1
in general cannot be dropped. To see this we use the full interval basis B on
X = [0, 1] mentioned above. Let R2 = R, µ be the Lebesgue measure on all
measurable subsets of [0, 1], and R1 = R = c00 be the space of all eventually
null real-valued sequences. Since RN is super Dedekind complete and every solid
subspace of a super Dedekind complete Riesz space is super Dedekind complete
too (see also [22]), then c00 is super Dedekind complete. Moreover, observe
that every c00 -valued function f has (P R)-bounded range, since c00 is (P R)bounded (see also [32]). Furthermore it is easy to check that in c00 the topology
generated by pointwise convergence is locally convex and Hausdorff. Finally,
if an increasing sequence (yn )n in c00 has supremum y ∈ c00 according to the
order convergence, then (yn )n is pointwise convergent to y. Therefore, by [36,
Corollary J], the space c00 is weakly σ-distributive.
Note that c00 does not fulfil property (σ). Indeed, for all n ∈ N, set un :=
(0, . . . , 0, 1, 0, . . .), where the value 1 is assumed at the n-th coordinate. Note
that, for any sequence (λn )n of positive real numbers, the sequence (λn un )n is
not order bounded by any element of c00 .
Example 1. Using the above sequence (un )n we consider a function f : [0, 1] →
c00 , defined by
n u , if x = 1/n, n = 1, 2, ...,
n
f (x) =
(4.3)
0,
otherwise.
This function vanishes almost everywhere (with respect to the Lebesgue measure), but is not (KH)-integrable on [0, 1] (see also [8, Example 7.33], [7, Example 2.8]). Indeed, if π is any δ-fineX
partition with an interval I tagged
X at 1/n,
then, as f is non-negative, we get
f ≥ |I|un , so the set of such
f is not
π
π
bounded in R.
We now examine some other properties of the HB -integral.
It is easy to check that the family of all HB -integrable functions on a B-figure
is a linear space, and the correspondent linearity equalities for integrals hold.
10
A. BOCCUTO, V. A. SKVORTSOV AND F. TULONE
Observe that, by means of a technique similar to that used in [15] and [16],
it is possible to check that our integral satisfies a Cauchy-type criterion.
It is not difficult to see that, if f is HB -integrable on B-figure L, then it is
HB -integrable also on any B-figure J ⊂ L. The proof is analogous to the one of
[15, Proposition 3.5] (see also [16, Proposition
3.6]).
Z
The B-figure function Φ : J →
7 (HB ) f is called the indefinite HB -integral
J
Z
of f . If we suppose that (HB )
f = 0 for each set V ⊂ L with µ(V ) = 0 (this
V
is true, for example, if R and R1 satisfy the hypotheses of Proposition 4.1), the
function Φ is additive in the following sense: Φ(A) + Φ(B) = Φ(A ∪ B) if A, B
are any two non-overlapping B-figures.
Using the mentioned above additivity of the integral and the local character
of the basis, we can give a version of the Saks-Henstock lemma for our integral,
whose proof is analogous to that of [15, Theorem 3.3] (see also [20, Theorem
3.2.1], [21, Lemma 3.8] and [24, Theorem 1.6.1] for the real-valued case).
Lemma 4.1. Under the hypotheses of Proposition 4.1, if an R1 -valued function
f is HB -integrable on a B-figure L, and Φ is its indefinite HB -integral, then
oi
h_n X
f (x)µ(I) − Φ(F ) : π ⊂ β(L) = 0,
(O) lim
β∈B
(I,x)∈π
where F is the B-figure generated by the β-partition π in L.
We now recall the following lemma.
Lemma 4.2. For any β ∈ B in a B-figure L and a closed set E ⊂ L there exists
a basis set β 0 ⊂ β in L such that I ∩ E = ∅ for each (I, x) ∈ β 0 [L \ E].
Definition 4.2. Given a B-figure L, a closed set E ⊂ L and a basis set β in
k
[
Ij is a β-halo of E if OE is generated by a
L, we say that a B-figure OE =
j=1
β[E]-partition {(Ij , xj )}kj=1 and Cβ (OE ) ∩ E = ∅, where Cβ (OE ) is the B-figure
complementary to OE in L .
The following result is a straightforward consequence of the previous lemma
and definition.
Lemma 4.3. For any β ∈ B in a B-figure L and a closed set E ⊂ L there exists
β 0 ⊂ β in L such that for any β 0 -partition π of L, the B-figure generated by the
partition π[E] is a β-halo OE .
We now turn to our main theorem, which is a Hake-type result in the context
of abstract derivation basis and extends to the Riesz space setting a result in
[31, Theorem 1], which was proved in the particular case R1 = R2 = R = R.
A HAKE-TYPE THEOREM
11
Similar versions of Hake-type theorems were proved in [9, 10, 13, 14], where the
basis defined by gauges was considered. In the formulation of the next theorem
and in its proof, intL (F ) denotes the interior of the set F with respect to the
topology in L induced by the original topology in the space X.
Theorem 4.2. Let R be a super Dedekind complete weakly σ-distributive Riesz
space satisfying property (σ), and L ⊂ X be a B-figure. Suppose that there
exist a closed set E ⊂ L and an increasing sequence of B-figures (Fk )k such
∞
[
that Fk ∩ E = ∅, k ∈ N, and
intL (Fk ) = L \ E. Assume that a function
k=1
f : L → R1 has (P R)-bounded range and is HB -integrable on the set E and on
any B-figure F ⊂ L with F ∩ E = ∅.
Furthermore, suppose that there is a positive element u ∈ R such that for
every k ∈ N there exist βbk ∈ B with
Z
X
(H1)
f (x)µ(I) −
f ≤ u
(I,x)∈πk
Sk
for every βbk -partition πk in Fk , where Sk is the B-figure Zgenerated by πk .
Then f is HB -integrable on L with integral value A +
f if and only if
E
h_nZ
(O) lim
β∈B
Cβ (OE )
oi
f − A : OE is a β-halo of E = 0.
(4.4)
Proof. We begin
with the ”only if” part. Let f be HB -integrable on L, and
Z
Z
f =A+
f . Then f is also HB -integrable on L \ E with integral value A,
L
E
and thus there exists an (O)-sequence (pn )n , with the property that for every
n ∈ N there is βn ∈ B such that, for every β-partition π of L, with β ⊂ βn , we
have
X
f (x)χL\E (x)µ(I) − A ≤ pn .
(4.5)
(I,x)∈π
Fix any β-halo OE corresponding to the above chosen β, according to Lemma 4.3.
It is a B-figure generated by π[E]. If π0 is any β 0 -partition of the complementary
B-figure Cβ (OE ) with β 0 ⊂ β, then π0 ∪ π[E] is a new β-partition of L for which
(4.5) also holds. The function f is HB -integrable on the B-figure Cβ (OE ), and
0
so there is an (O)-sequence (qm )m , such that for any m ∈ N there exists βm
∈B
0
such that for any βm -partition πm of Cβ (OE ) we get
Z
X
f (x)µ(I) −
f ≤ qm .
(4.6)
(I,x)∈πm
Cβ (OE )
12
A. BOCCUTO, V. A. SKVORTSOV AND F. TULONE
0
We can suppose that βm
⊂ β, and so πm is also a β-partition of Cβ (OE ). Then,
as above, (4.5) holds for the β-partition π = πm ∪ π[E] of L. Note that, for this
π,
X
X
f (x)χL\E (x)µ(I) =
f (x)µ(I).
(4.7)
(I,x)∈π
(I,x)∈πm
Now combining (4.5), (4.6) and (4.7) we get for any m
Z
f − A ≤ pn + qm .
(4.8)
Cβ (OE )
Since n is fixed and the left-hand side of (4.8) does not depend on m, we can
take in (4.8) the infimum with respect to m, getting
Z
f − A ≤ pn .
Cβ (OE )
Since β ⊂ βn and the β-halo for this β was chosen arbitrarily, we obtain (4.4).
So the ”only if” part is proved. Note that, in this implication, super Dedekind
completeness of R is not used.
We now turn to the ”if” part. Suppose that (4.4) holds for some A ∈ R.
Consider the increasing sequence of B-figures (Fk )k given by assumption and
put Tk = intL (Fk ) \ intL (Fk−1 ) (where F0 = ∅). The assumption of the theorem
∞
[
implies that
Tk = L \ E and Tr ∩ Ts = ∅ for every r 6= s. It is enough to
k=1
prove that f χL\E is HB -integrable with the integral value A.
Since f is HB -integrable on Fk for each k ∈ N, then, taking into account
(k)
Lemma 4.1, there is a countable family of (O)-sequences (bn )n , k ∈ N, such
(k)
that to every k, n ∈ N there corresponds a basis set βn with
Z
X
f (x)µ(I) −
f ≤ b(k)
(4.9)
n
Sk
(I,x)∈πk
(k)
for all βn -partitions πk in Fk , where Sk is the B-figure generated by πk . Thus,
if we consider the directed set (B, ⊂), the net
Z
_n X
o
f (x)µ(I) −
f : πk is a β-partition in Fk
(I,x)∈πk
β∈B
Sk
is (O)-convergent, and hence also (D)-convergent. So, for every k ∈ N there
(k)
exists a (D)-sequence (ai,j )i,j with the property that for every ϕ ∈ NN there is
∗
∗
an element βk = βk,ϕ ∈ B, such that
Z
∞
X
_
(k)
f (x)µ(I) −
f ≤
ai,ϕ(i+k)
(4.10)
(I,x)∈πk
Sk
i=1
A HAKE-TYPE THEOREM
13
for any βk∗ -partition πk in Fk .
Furthermore, by condition (H1), there exists u ∈ R, u ≥ 0, such that to every
k ∈ N there corresponds a basis set βbk with
Z
X
f ≤ u
f (x)µ(I) −
Sk
(I,x)∈πk
(4.11)
for all βbk -partitions πk in Fk . By virtue of Lemma 2.2, there is a regulator
(ai,j )i,j such that
u∧
q _
∞
X
(k)
ai,ϕ(i+k)
≤
k=1 i=1
∞
_
ai,ϕ(i)
(4.12)
i=1
for all q ∈ N and ϕ ∈ NN .
By Lemma 4.2, for each k ∈ N there is βk0 ⊂ βk∗ ∩ βbk such that I ⊂ intL (Fk )
for all (I, x) ∈ βk0 [intL (Fk )]. Now, by the local character of B, there exists βb
b k ] ⊂ β 0 [Tk ] for any k ∈ N. Now we apply Proposition 2.1 to the
in L with β[T
k
(O)-net
o
_nZ
f − A : OE is a β-halo of E
Cβ (OE )
and we obtain the existence of a regulator (ci,j )i,j with the property that, in
correspondence with the function ϕ in (4.10), there exists β 00 ∈ B such that
Z
Cβ 00 (OE )
∞
_
ci,ϕ(i)
f − A ≤
(4.13)
i=1
for every β 00 -halo OE .
e
Take now βe ⊂ βb ∩ β 00 , and pick any β-partition
π of L. From the definition
b
e
b
b
of β and since β ⊂ β, the B-figure generated by the partition π[E] is a β-halo
b for all (I, x) ∈ π[Tk ] we get I ⊂ Fk . So
of E, say OE . By construction of β,
Z
XZ
(4.10) holds with πk = π[Tk ]. Observe that S f =
f , because the
Sk
k
Sk
HB -integral is additive and the partition π is finite, and thus there are only
finitely many non-empty Sk ’s, say S1 , . . ., Sq . Moreover, we get
Z
Z
f=
S
Sk
f.
Cβe (OE )
14
A. BOCCUTO, V. A. SKVORTSOV AND F. TULONE
e
Note that, since βe ⊂ β 00 , then any β-halo
is also a β 00 -halo. Therefore, from
(4.10), (4.11), (4.12) and (4.13) we get
q
q Z
X
X
X
X
f (x)µ(I) −
f (x)χL\E (x)µ(I) − A ≤ f +
k=1 (I,x)∈π[Tk ]
(I,x)∈π
Z
+ S
Z
+
Sk
Cβe (OE )
q X
f − A ≤
k=1
Z
X
f (x)µ(I) −
Sk
k=1 (I,x)∈π[Tk ]
Sk
f +
q _
∞
∞
X
_
(k)
ai,ϕ(i+k) +
f − A ≤ u ∧
ci,ϕ(i) ≤
k=1 i=1
≤
∞
_
ai,ϕ(i) +
i=1
i=1
∞
_
ci,ϕ(i) .
i=1
From this, by Proposition 2.2, we obtain that the decreasing net
o
_n X
f (x)χL\E (x)µ(I) − A : π ∈ Π(β, L) , β ∈ B
qβ :=
(I,x)∈π
is an (O)-net and so, by super Dedekind completeness of R, it admits a sub-(O)sequence (qβn )n , implying the HB -integrabilityZ of f on L \ZE with integral value
A. Hence f is HB -integrable also on L and
f =A+
L
the proof.
f . This completes
E
Remark 3. Observe that in Lemma 2.1 we have proved that, if R is regular,
then condition (H1) is fulfilled. Thus, if in Theorem 4.2 we assume the regularity
of R, (H1) becomes superfluous.
Moreover, note that in Example 1 the condition (H1) is not fulfilled, since the
Riemann sums involved are not bounded in c00 , and the thesis of Theorem 4.2 is
not satisfied: indeed the function f in (4.3) is not Kurzweil-Henstock integrable
on [0, 1], though it is so on every subinterval [a, 1], 0 < a < 1, with integral equal
to zero.
Acknowledgement. The authors would thank the referee for his/her valuable
suggestions.
References
[1] ALIPRANTIS, Ch. D.-BURKINSHAW, O.: Locally solid Riesz spaces with applications
to Economics, Second edition, Mathematical Surveys and Monographs, 105, Am. Math.
Soc., Providence, Rhode Island, 2003.
[2] AHMED, S. I.-PFEFFER, W. F.: A Riemann integral in a locally compact Hausdorff
space, J. Austral. Math. Soc. (Series A) 41 (1) (1986), 115-137.
[3] ABRAMOVICH, Yu.-SIROTKIN, G.: On order convergence of nets, Positivity 9 (2005),
287-292.
A HAKE-TYPE THEOREM
15
[4] BOCCUTO, A.: Egorov property and weak σ-distributivity in (`)-groups, Acta Math.
(Nitra) 6 (2003), 61-66.
[5] BOCCUTO, A.-DIMITRIOU, X.-PAPANASTASSIOU, N.: Basic matrix theorems for
(I)-convergence in (`)-groups, Math. Slovaca 62 (5) (2012), 269-298.
[6] BOCCUTO, A.-DIMITRIOU, X.-PAPANASTASSIOU, N.: Schur lemma and limit theorems in lattice groups with respect to filters, Math. Slovaca 62 (6) (2012), 1145-1166.
[7] BOCCUTO, A.-MINOTTI, A. M.-SAMBUCINI, A. R.: Set-valued Kurzweil-Henstock
integral in Riesz space setting, PanAmer. Math. J. 23 (1) (2013), 57-74.
[8] BOCCUTO, A.-RIEČAN, B.-VRÁBELOVÁ, M.: Kurzweil-Henstock Integral in Riesz
Spaces, Bentham Sci. Publ., Singapore, 2009.
[9] BOCCUTO, A.-RIEČAN, B.: A note on the improper Kurzweil-Henstock integral, Acta
Univ. M. Belii Math. 9 (2001), 7-12.
[10] BOCCUTO, A.-RIEČAN, B.: A note on improper Kurzweil-Henstock integral in Riesz
spaces, Acta Math. (Nitra) 5 (2002), 15-24; Addendum to ”A note on improper KurzweilHenstock integral in Riesz spaces”, ibidem 7 (2004), 53-59.
[11] BOCCUTO, A.-RIEČAN, B.: The Concave Integral with respect to Riesz Space-valued
Capacities, Math. Slovaca 59(2009), 647-660.
[12] BOCCUTO, A.-RIEČAN, B.: On extension theorems for M -measures in (`)-groups,
Math. Slovaca 60 (2010), 65-74.
[13] BOCCUTO, A.-RIEČAN, B.-SAMBUCINI, A. R.: Convergence and Fubini Theorems
for metric semigroup-valued functions defined on unbounded rectangles, PanAmerican
Math. J. 19 (2009), 1-12.
[14] BOCCUTO, A.-SAMBUCINI, A. R.: The Henstock-Kurzweil integral for functions defined on unbounded intervals and with values in Banach spaces, Acta Math. (Nitra) 7,
(2004) 3-17.
[15] BOCCUTO, A.-SKVORTSOV, V. A.: Henstock-Kurzweil type integration of Riesz
space-valued functions and applications to Walsh series, Real Anal. Exchange 29
(2003/2004), 419-438.
[16] BOCCUTO, A.-SKVORTSOV, V. A.: On Kurzweil-Henstock type integrals with respect
to abstract derivation bases for Riesz space-valued functions, J. Appl. Funct. Anal. 1
(2006), 251-270.
[17] FAURE, C. A.-MAWHIN, J.: The Hake’s property for some integrals over multidimensional intervals, Real Anal. Exchange 20 (2) (1994/1995), 622-630.
[18] FREMLIN, D. H.: A direct proof of the Matthes-Wright integral extension theorem, J.
London Math. Soc. 11 (1975), 276-284.
[19] KURZWEIL, J.-JARNIK, J.: Differentiability and integrabilty in n dimensions with
respect to α-regular intervals, Results Math. 21 (1992), 138-151.
[20] LEE, P. Y.-VÝBORNÝ, R.: The Integral: An Easy Approach after Kurzweil and Henstock, Austral. Math. Soc. Lectures Series 14, Cambridge Univ. Press, 2000.
[21] LUKASHENKO, T. P.-SOLODOV, A.-SKVORTSOV, V. A.: Generalized integrals,
Moscow, 2010 (in Russian).
[22] LUXEMBURG, W. A. J.-ZAANEN, A. C.: Riesz Spaces I, North-Holland Publ. Co.,
Amsterdam, 1971.
[23] MULDOWNEY, P.-SKVORTSOV, V. A.: Improper Riemann Integral and Henstock
Integral in Rn , Matematicheskie Zametki 78 (2) (2005), 251-258; Translated in Mathematical Notes 78 (2) (2005), 228-233.
[24] OSTASZEWSKI, K. M.: Henstock integration in the plane, Memoirs Amer. Math. Soc.,
Providence, 63, 353, 1986.
[25] PERESSINI, A. L.: Ordered topological vector spaces, Harper & Row, New York, 1967.
16
A. BOCCUTO, V. A. SKVORTSOV AND F. TULONE
[26] RIEČAN, B.-NEUBRUNN, T.: Integral, Measure and Ordering, Kluwer, Dordrecht, and
Ister, Bratislava, 1997.
[27] SAKS, S.: Theory of the integral, Hafner Publishing Company, 1937.
[28] SCHAEFER, H. H.: Banach lattices and positive operators, Springer-Verlag, New York,
1974.
[29] SKVORTSOV, V. A.-TULONE, F.: The P-integral in the theory of series with respect
to characters of zero-dimensional groups, Vestnik Moskov. Gos. Univ. Ser. Mat. Mekh.
61 (2006), 25-29; Engl. transl. Moscow Univ. Math. Bull. 61 (2006), 27-31.
[30] SKVORTSOV, V. A.-TULONE, F.: Henstock-Kurzweil Type Integral on a Compact
Zero-Dimensional Metric Space and Representation of Quasi-Measure, Vestnik Moskov.
Gos. Univ. Ser. Mat. Mekh. Vestnik Moskov. Gos. Univ. Ser. Mat. Mekh. 67 (2012), 1117; Engl. transl. Moscow Univ. Math. Bull. 67 (2012), 55-60.
[31] SKVORTSOV, V. A.-TULONE, F.: Generalized Hake’s property for a Henstock type
integral, (2012), submitted.
[32] SWARTZ, C.: The Nikodým boundedness theorem for lattice-valued measures, Arch.
Math. 53 (1989), 390-393.
[33] THOMSON, B. S.: Derivation bases on the real line, I and II, Real Anal. Exchange 8
(1982/83), 67-207 and 278-442.
[34] TULONE, F.-ZHEREBYOV, Yu.: Absolutely continuous variational measures of
Mawhin’s type, Math. Slovaca 61 (5) (2011), 747-768.
[35] VULIKH, B. Z.: Introduction to the theory of partially ordered spaces, Wolters-Noordhoff
Sci. Publ., Groningen, 1967.
[36] WRIGHT, J. D. M.: The measure extension problem for vector lattices, Ann. Inst.
Fourier, Grenoble 21 (4) (1971), 65-85.
* Department of Mathematics and Computer Sciences
University of Perugia
via Vanvitelli, 1
I-06123 Perugia
ITALY
E-mail address: [email protected], [email protected] (corresponding author)
** Department of Mathematics
Moscow State University
RU-119992 Moscow
RUSSIA
and
Institute of Mathematics Casimirus the Great University
pl. Weyssenhoffa 11
PL-85079 Bydgoszcz
POLAND
E-mail address: [email protected]
*** Department of Mathematics and Computer Sciences
University of Palermo
via Archirafi 34
I-90123 Palermo
ITALY
E-mail address: [email protected]