The Birkhoff weak integral of functions relative to a set function in Banach spaces setting
ANCA CROITORU University ”Alexandru Ioan Cuza”
Faculty of Mathematics Bd. Carol I, 11, 700506 Ias¸i
ROMANIA croitoru@uaic.ro
ALINA GAVRILUT¸ University ”Alexandru Ioan Cuza”
Faculty of Mathematics Bd. Carol I, 11, 700506 Ias¸i
ROMANIA gavrilut@uaic.ro ALINA IOSIF
Department of Computer Science
Information Technology, Mathematics and Physics Bd. Bucures¸ti, 39, 100680 Ploies¸ti
ROMANIA
emilia.iosif@upg-ploiesti.ro
Abstract:In this paper, we define and study the Birkhoff weak integral in two cases: for vector functions relative to a non-negative set function and for real functions with respect to a vector set function. Some comparison results and classical integral properties are obtained: the linearity relative to the function and the measure, and the monotonicity with respect to the function, the measure and the set.
Key–Words:Birkhoff weak integral; integrable function; non-additive measure; vector integral; vector function.
1 Introduction
This paper deals with a subject in the field of non- additivity. In Measure Theory, the countable (or fi- nite) additivity is fundamental. However, there are many aspects of the real world where the countable (or finite) additivity does not work. For example, in problems of capacity, the efficiency of a finite set of persons working together is not the sum of the effi- ciency of every person. The capacities and the Cho- quet integrals (defined by Choquet [9] in 1953-1954), as well as fuzzy measures and non-linear fuzzy inte- grals (defined by Sugeno [44] in 1974) have important and interesting applications in potential theory, sta- tistical mechanics, economics, finance, the theory of transferable-utility cooperative games, artificial intel- ligence, data mining, decision making, computer sci- ence, subjective evaluation (e.g., [9], [22], [26], [28], [29], [33], [34], [40], [44], [45]).
By means of finite or infinite Riemann type sums, different kinds of integrals have been defined and studied for instance in [1], [2], [3], [4], [5], [6], [7], [8], [10], [11], [12], [13], [14], [16-24], [25], [27], [31], [32], [36-39], [41], [42], [43]. For example, the Birkhoff integral [2] was defined for a vector function f : T → X, relative to a complete finite measure m : A → [0, +∞), using series of type
∞
P
n=0
f (tn)m(Bn),
accordingly to a countable partition {Bn}n∈N of T and tn∈ Bn, for every n ∈ N.
In this paper, we define and study a new non- linear integral of Birkhoff type (named Birkhoff weak) for vector (real respectively) functions, with re- spect to a non-additive non-negative (vector respec- tively) set function, using finite Riemann type sums and countable partitions. Our definition can be placed between the Birkhoff integral [2] and the Gould in- tegral [25]. The paper is structured in six sections.
The first one is for Introduction. In Section 2, some preliminaries are presented. In Section 3, we define and study the Birkhoff weak integral of vector or real functions and establish some comparison results with Birkhoff inequality and Pettis integrability. Section 4 contains some integral properties for vector func- tions and Sections 5 includes some integral properties for real functions such as, the linearity relative to the function and the measure and the monotonicity with respect to the function, the measure and the set. In Section 6 we present some properties of monotonicity for the case when both the function and the set func- tion µ have real values. Finally, in Section 7, we ex- pose some conclusions.
2 Preliminaries
Let T be a nonempty set, P(T ) the family of all sub- sets of T , A a σ-algebra of subsets of T and (X, k · k) a real Banach space.
Definition 2.1 Let T be an infinite set.
(i) A countable partition ofT is a countable fam- ily of nonempty sets P = {An}n∈N ⊂ A such that Ai∩ Aj = ∅, i 6= j and S
n∈N
An= T .
(ii) IfP and P0are two countable partitions ofT , thenP0is said to be finer thanP , denoted by P ≤ P0 (or,P0 ≥ P ), if every set of P0is included in some set ofP .
(iii) The common refinement of two countable partitions P = {Ai} and P0 = {Bj} is the parti- tionP ∧ P0 = {Ai∩ Bj}. We denote by P the class of all partitions ofT and if A ∈ A is fixed, by PAwe denote the class of all partitions ofA.
Definition 2.2 [30] Let µ : A → [0, +∞) be a non- negative function, withµ(∅) = 0. µ is said to be:
(i) monotone ifµ(A) ≤ µ(B), ∀A, B ∈ A, with A ⊆ B;
(ii) finitely additive ifµ(A ∪ B) = µ(A) + µ(B) for every disjointA, B ∈ A;
(iii) a (σ-additive) measure if µ(
∞
S
n=0
An) =
∞
P
n=0
µ(An), for every sequence of pairwise disjoint sets(An)n∈N⊂ A.
Definition 2.3 [15] Let µ : A → [0, +∞) be a non- negative set function.
(i) The variation µ of µ is the set function µ : P(T ) → [0, +∞] defined by µ(E) = sup{
Pn i=1
µ(Ai)}, for every E ∈ P(T ), where the supremum is extended over all finite families of pair- wise disjoint sets {Ai}ni=1 ⊂ A, with Ai ⊆ E, for everyi ∈ {1, . . . , n}. If µ : A → X is a vector set function, then in the definition of supremum, we will considerPn
i=1kµ(Ai)k.
(ii) µ is said to be of finite variation on A if µ(T ) < ∞.
Remark 2.4 If E ∈ A, then in Definition 3, we may consider the supremum over all finite partitions {Ai}ni=1⊂ A, of E.
Definition 2.5 A property (P) about the points of T holds almost everywhere (denotedµ-ae) if there exists A ∈ P(T ) so thatµ(A) = 0 and (P) holds on T \A.e
Definition 2.6 [27] Let µ : A → [0, +∞) be a non-negative set function with µ(∅) = 0. A function f : T → X is called Riemann-Lebesgue µ-integrable (RL-µ-integrable for short) (on T ) if there exists α ∈ X having the property that for every ε > 0, there ex- ists a countable partition Pε ofT in A such that for every other countable partition P = {An}n∈N ofT inA, with P ≥ Pε and everytn ∈ An,n ∈ N, the series
∞
P
n=0
f (tn)µ(An) converges unconditionally and kP∞
n=0f (tn)µ(An) − αk < ε. In this case, we de- noteα = (RL)R
Tf dµ, which is called the Riemann- Lebesgue integral off on T relative to µ.
Remark 2.7 According to Theorem 8 of [35], if (T, A, µ) is a σ-finite measure space, then Riemann- Lebesgue integrability of f : T → X is equivalent to Birkhoff [2] integrability of f .
3 The Birkhoff weak integral of functions
In this section, we define and study the Birkhoff weak integral of real or vector functions and establish some comparison results.
In the sequel, suppose (X, k·k) is a Banach space, T is infinite and A is a σ-algebra of subsets of T . Definition 3.1 Let f : T → X and µ : A → [0, +∞) (orf : T → R and µ : A → X) with µ(∅) = 0.
I. f is called Birkhoff weak µ-integrable (on T ) (simply Bw-µ-integrable) if there exists α ∈ X so that for everyε > 0, there exist a countable partition Pε ofT in A and nε ∈ N such that for every other countable partition P = {An}n∈N of T in A, with P ≥ Pε, and everytn∈ An,n ∈ N, it holds
k
n
X
k=0
f (tk)µ(Ak) − αk < ε, ∀n ≥ nε.
In this case, α is denoted by (Bw)R
T f dµ and is called the Birkhoff weak integral of f on T relative toµ.
II. f is called Birkhoff weak µ-integrable on a set E ∈ A if the restriction f |E is Birkhoff weak µ- integrable on (E, AE, µ), where AE = {E ∩ A|A ∈ A}.
Remark 3.2 If it exists, the integral is unique.
Example. I. If f (t) = 0, for every t ∈ T , then f is Bw-µ-integrable and (Bw)R
T f dµ = 0.
II. Suppose T = {tn|n ∈ N} is countable, {tn} ∈ A and let f : T → X be such that the se- ries
∞
P
n=0
f (tn)µ({tn}) is unconditionally convergent.
Then f is Bw-µ-integrable and
(Bw) Z
T
f dµ =
∞
X
n=0
f (tn)µ({tn}).
Remark 3.3 I. If we consider the multifunction F = {f }, where f : T → X is a vector function, and µ : A → [0, +∞) with µ(∅) = 0, then by Definition 9 [10], it results that f is Bw-µ-integrable if and only if F is Bw-µ-integrable (according to [10]).
II. If we consider a real function f : T → R and the set multifunction ϕ = {µ}, where µ : A → X is a vector set function, with µ(∅) = 0, then by Definition 14-II [11], it follows that f is Bw-µ-integrable if and only if f is Bw-ϕ-integrable (according to [11]).
Theorem 3.4 Let f : T → [0, +∞) be a non- negative function and µ : A → [0, +∞) a non-negative set function with µ(∅) = 0. If f is Bw-µ-integrable, then f is RL-µ-integrable and (RL)R
T f dµ = (Bw)R
T f dµ.
Proof. Let ε > 0 be arbitrary. Since f is Bw-µ- integrable, there exist Pε⊂ A a countable partition of T and nε∈ N that satisfy the conditions of Definition 3.1. Let P = {An}n∈N ⊂ A be a countable partition of T , with P ≥ Pε and tn ∈ An, n ∈ N. Denoting sn=
n
P
k=0
f (tk)m(Ak), for every n ∈ N, according to Definition 3.1, it holds
(∗)
sn− (Bw) Z
T
f dm
< ε
2, ∀n ≥ nε. This shows that the series
∞
P
n=0
f (tn)m(An) is un- conditionally convergent. By (*) it also results that
|
∞
P
n=0
f (tn)m(An) − (Bw)R
T f dm| < ε. Conse- quently, f is RL-µ-integrable and (RL)R
Tf dm = (Bw)R
Tf dm.
Theorem 3.5 Suppose (T, A, µ) is a σ-finite measure space andµ : A → [0, +∞) is σ-additive. If f : T → X is Bw-µ-integrable, then f is Pettis integrable.
Proof. Since f is Bw-µ-integrable, for every ε >
0, there exist a countable partition Pεof T in A and nε ∈ N such that for every other countable partition
P = {An}n∈N of T in A, with P ≥ Pε, and every tn∈ An, n ∈ N, it holds
k
n
X
k=0
f (tk)µ(Ak) − (Bw) Z
T
f dµk < ε, ∀n ≥ nε.
Then for every x∗ ∈ X∗and every n ≥ nεit follows
|
n
X
k=0
(x∗◦ f )(tk)µ(Ak) − x∗((Bw) Z
T
f dµ)| < ε.
This shows that x∗ ◦ f is Bw-µ-integrable. It also results that the series P∞
n=0(x∗ ◦ f )(tn)µ(An) is convergent and |P∞
n=0(x∗ ◦ f )(tn)µ(An) − x∗((Bw)R
Tf dµ)| < ε.
As in Theorem 3.14 [7], it is demonstrated that the series P∞
n=0(x∗ ◦ f )(tn)µ(An) is uncondition- ally convergent. So, x∗ ◦ f is Birkhoff µ-integrable and therefore is in L1for every x∗ ∈ X∗. Evidently, R
T(x∗◦ f )dµ = x∗((Bw)R
T f dµ), for all x∗ ∈ X∗. Consequently, f is Pettis integrable.
4 Integral properties of vector func- tions
In this section, we present some properties of the Birkhoff weak integral for a vector function f with re- spect to a non-negative set function µ : A → [0, +∞) with µ(∅) = 0. According to Remark 3.3-I and the integral properties of the multifunction F = {f } in [10], we obtain the following properties of the inte- gral introduced in Definition 3.1.
Theorem 4.1 If f : T → X is bounded and f = 0 µ- ae, then f is Bw-µ-integrable and (Bw)R
T f dµ = 0.
Theorem 4.2 Let f : T → X be a Bw-µ-integrable function. Thenf is Bw-µ-integrable on E ∈ A if and only if f XE is Bw-µ-integrable on T , where XE is the characteristic function ofE. In this case,
(Bw) Z
E
f dµ = (Bw) Z
T
f XEdµ.
Theorem 4.3 If f, g : T → X are Bw-µ-integrable, thenf + g is Bw-µ-integrable and
(Bw)R
T(f + g)dµ = (Bw)R
T f dµ + (Bw)R
T gdµ.
Theorem 4.4 Let f, g : T → X be Bw-µ-integrable vector functions. Then the following properties hold:
(i) k(Bw)R
T f dµ − (Bw)R
T gdµk ≤
sup
t∈T
kf (t) − g(t)k · µ(T );
(ii)k(Bw)R
T f dµk ≤ supt∈Tkf (t)k · µ(T ).
Theorem 4.5 Let f : T → X be a Bw-µ-integrable function andα ∈ R. Then:
(i)αf is Bw-µ-integrable and (Bw)
Z
T
αf dµ = α(Bw) Z
T
f dµ;
(ii)f is Bw-αµ-integrable (α ∈ [0, +∞)) and (Bw)
Z
T
f d(αµ) = α(Bw) Z
T
f dµ.
Theorem 4.6 Suppose µ1, µ2 : A → [0, +∞) are non-negative set functions with µ1(∅) = µ2(∅) = 0.
Iff : T → X is both Bw-µ1-integrable andBw-µ2- integrable, thenf is Bw-(µ1+ µ2)-integrable and
(Bw) Z
T
f d(µ1+ µ2)
= (Bw) Z
T
f dµ1+ (Bw) Z
T
f dµ2.
Theorem 4.7 Suppose µ : A → [0, +∞) is finitely additive. Let f, g : T → X be vector functions, with sup
t∈T
kf (t) − g(t)k < +∞, such that f is Bw- µ-integrable and f = g µ-ae. Then g is Bw-µ- integrable and(Bw)R
Tf dµ = (Bw)R
Tgdµ.
Other properties of the integral will be presented in the sequel.
Theorem 4.8 Let f : T → X be a Bw-µ-integrable function, such that the real function kf k : T → [0, +∞) is Bw-µ-integrable. Then
k(Bw) Z
T
f dµk ≤ (Bw) Z
T
kf kdµ.
Proof. Suppose ε > 0 is arbitrary. Since f is Bw- µ-integrable, there exist Pε1 ∈ P and n1ε ∈ N so that for every P = {An}n∈N ∈ P, P ≥ Pε1, and every sn∈ An, n ∈ N, it holds
k
n
X
k=0
f (sk)µ(Ak) − (Bw) Z
T
f dµk < ε 2, for every n ∈ N, n ≥ n1ε.
Since kf k is Bw-µ-integrable, there exist Pε2 ∈ P and n2ε ∈ N so that for every P = {Bn}n∈N ∈ P, P ≥ Pε2 and every un∈ Bn, n ∈ N, it holds
|
n
X
k=0
kf (tk)kµ(Bk) − (Bw) Z
T
kf kdµ| < ε 2, for every n ∈ N, n ≥ n2ε.
Now, we consider Pε = Pε1∧ Pε2 ∈ P and nε= max{n1ε, n2ε}. Then for every P = {Cn}n∈N ∈ P, P ≥ Pε and every tn ∈ Cn, n ∈ N, we obtain the following inequalities
k
n
X
k=0
f (tk)µ(Ck) − (Bw) Z
T
f dµk < ε 2, (1)
|
n
X
k=0
kf (tk)kµ(Ck) − (Bw) Z
T
kf kdµ| < ε 2. (2) By (1) and (2) we get:
k(Bw) Z
T
f dµk ≤ k(Bw) Z
T
f dµ −
n
X
k=0
f (tk)µ(Ck)k
+ k
n
X
k=0
f (tk)µ(Ck)k
< ε 2+ |
n
X
k=0
kf (tk)kµ(Ck) − (Bw) Z
T
kf kdµ|+
+ (Bw) Z
T
kf kdµ < ε + (Bw) Z
T
kf kdµ.
Since ε > 0 is arbitrary, the conclusion follows. Definition 4.9 Let ν : A → X, ν(∅) = 0 and µ : A → [0, +∞), µ(∅) = 0. It is said that ν is absolutely continuous with respect toµ (denoted by ν µ) if for anyε > 0, there is δ > 0 such that for every E ∈ A,
µ(E) < δ ⇒ kν(E)k < ε.
Theorem 4.10 Suppose f : T → X is bounded and Bw-µ-integrable on every set E ∈ A, such that α = supt∈T kf (t)k > 0 and let ν : A → X be defined by ν(E) = (Bw)R
Ef dµ, for every E ∈ A. Then ν µ.
Proof. For every ε > 0, let δ = αε > 0 and let E ∈ A be any measurable set so that µ(E) < δ. Then, by Theorem 4.4 - (ii), we have:
kν(E)k = k(Bw) Z
E
f dµk
≤ sup
t∈E
kf (t)k · µ(E) < αδ = ε.
This shows that ν µ.
Theorem 4.11 Let f : T → X be a vector function.
(i) If f is both Bw-µ-integrable on B and Bw- µ-integrable on C, when B, C ∈ A are disjoint mea- surable sets, thenf is Bw-µ-integrable on B ∪ C and (Bw)R
B∪Cf dµ = (Bw)R
Bf dµ + (Bw)R
Cf dµ.
(ii) Iff is Bw-µ-integrable on every set E ∈ A, then the set functionν : A → X, defined by ν(E) = (Bw)R
Ef dµ, ∀E ∈ A, is finitely additive.
Proof. (i) Suppose ε > 0 is arbitrary. Since f is Bw-µ-integrable on B, there exist a partition of B, PBε = {Dn}n∈N ∈ PB and n1ε ∈ N such that for every partition of B, P = {An}n∈N ∈ PB, P ≥ PBε and tn∈ An, n ∈ N, it holds
k
n
X
k=0
f (tk)µ(Ak) − (Bw) Z
B
f dµk < ε
2, ∀n ≥ n1ε. (3) Since f is Bw-µ-integrable on C, there exist a parti- tion of C, PCε = {En}n∈N ∈ PC and n2ε ∈ N such that for every partition of C, P = {Un}n∈N ∈ PC, P ≥ PCε and sn∈ Un, n ∈ N, it holds
k
n
X
k=0
f (sk)µ(Uk) − (Bw) Z
C
f dµk < ε
2, ∀n ≥ n2ε. (4) Let PB∪Cε = {Dn, En}n∈N ∈ PB∪C and n ≥ n1ε + n2ε. Now, for every partition of B ∪ C, P = {Vn}n∈N ∈ PB∪C0 with P ≥ PB∪Cε and every un ∈ Vn, n ∈ N, we can write P = {Bn0}n∈N∪ {Cn0}n∈N, where PB0 = {Bn0}n∈N≥ PBε and PC0 = {Cn0}n∈N≥ PCε. Also, we can write Pn
k=0f (uk)µ(Vk) = Pp
i=0f (uk)µ(Bk0) + Pr
j=0f (uk)µ(Ck0), with p ≥ n1ε, r ≥ n2ε. From (3) and (4) we obtain
k
n
X
k=0
f (uk)µ(Vk) − ((Bw) Z
B
f dµ + (Bw) Z
C
f dµ)k ≤
≤ k
p
X
i=0
f (uk)µ(Bk0) − (Bw) Z
B
f dµk
+ k
r
X
j=0
f (uk)µ(Ck0) − (Bw) Z
C
f dµk < ε 2 +ε
2 = ε, which shows that f is Bw-µ-integrable on B ∪ C and
(Bw) Z
B∪C
f dµ = (Bw) Z
B
f dµ + (Bw) Z
C
f dµ.
(ii) It easily results from (i).
5 Integral properties of real func- tions
In this section, we present some properties of the Birkhoff weak integral for a real function with respect to a vector set function µ : A → X, with µ(∅) = 0.
The nonnegative set function kµk : A → [0, +∞) is defined by kµk(A) = kµ(A)k, for every A ∈ A.
Evidently, kµk(∅) = 0. According to Remark 3.3-II and the properties of the integral of f with respect to the set multifunction ϕ = {µ} in [11], the following results are obtained.
Theorem 5.1 If f : T → R is bounded and f = 0 µ- ae, then f is Bw-µ-integrable and (Bw)R
T f dµ = 0.
Theorem 5.2 f : T → R is Bw-µ-integrable on E ∈ A if and only if f XE isBw-µ-integrable on T.
Theorem 5.3 If f, g : T → R are Bw-µ-integrable andf (t)g(t) ≥ 0 for every t ∈ T , then f + g is Bw- µ-integrable and
(Bw) Z
T
(f + g)dµ = (Bw) Z
T
f dµ + (Bw) Z
T
gdµ.
Theorem 5.4 Let f, g : T → R be Bw-µ-integrable bounded functions. Then:
(i) k(Bw)R
T f dµ − (Bw)R
T gdµk ≤
supt∈T |f (t) − g(t)| · µ(T );
(ii)k(Bw)R
T f dµk ≤ supt∈T|f (t)| · µ(T ).
Theorem 5.5 Let f : T → R be a Bw-µ-integrable function andα ∈ R. Then:
(i)αf is Bw-µ-integrable and (Bw)
Z
T
αf dµ = α(Bw) Z
T
f dµ;
(ii)f is Bw-(αµ)-integrable and (Bw)
Z
T
f d(αµ) = α(Bw) Z
T
f dµ.
Theorem 5.6 Let µ1, µ2 : A → X be non-negative set functions withµ1(∅) = µ2(∅) = 0. If f : T → R is both Bw-µ1-integrable and Bw-µ2-integrable, then f is Bw-(µ1+ µ2)-integrable and (Bw)R
T f d(µ1+ µ2) = (Bw)R
T f dµ1+ (Bw)R
T f dµ2.
Theorem 5.7 Suppose µ : A → X is finitely additive and let A, B ∈ A be disjoint sets. If f : T → R is Bw-µ-integrable both on A and on B, then f is Bw-µ-integrable on A ∪ B and (Bw)R
A∪Bf dµ = (Bw)R
Af dµ + (Bw)R
Bf dµ.
Theorem 5.8 Suppose µ : A → X is finitely addi- tive. If f, g : T → R are bounded functions such thatf is Bw-µ-integrable and f = g µ-ae, then g is Bw-µ-integrable and
(Bw) Z
T
f dµ = (Bw) Z
T
gdµ.
Moreover, the following properties of the integral are obtained.
Theorem 5.9 Suppose the nonnegative function f : T → [0, +∞) is Bw-µ-integrable and Bw-kµk- integrable. Then
k(Bw) Z
T
f dµk ≤ (Bw) Z
T
f dkµk.
Proof. Let ε > 0 be arbitrary. Since f is Bw-µ- integrable, there exist Pε1 ∈ P and n1ε ∈ N such that for every P = {An}n∈N∈ P, with P ≥ Pε1, and each sn∈ An, n ∈ N, it holds
k
n
X
k=0
f (sk)µ(Ak) − (Bw) Z
T
f dµk < ε
2, ∀n ≥ n1ε. Since f is Bw-kµk-integrable, there exist Pε2 ∈ P and n2ε ∈ N such that for every P = {Bn}n∈N ∈ P, with P ≥ Pε2, and each un∈ Bn, n ∈ N, it holds
|
n
X
k=0
f (uk)kµ(Bk)k−(Bw) Z
T
f dkµk| < ε
2, ∀n ≥ n2ε. Let Pε= Pε1∧ Pε2∈ P and nε= max{n1ε, n2ε}. Then for every P = {Cn}n∈N ∈ P, P ≥ Pε, and every tn∈ Cn, n ∈ N, it results
k
n
X
k=0
f (tk)µ(Ck) − (Bw) Z
T
f dµk < ε 2, (5)
|
n
X
k=0
f (tk)kµ(Ck)k − (Bw) Z
T
f dkµk| < ε 2. (6) Now, by (5) and (6), it follows
k(Bw) Z
T
f dµk ≤ k(Bw) Z
T
f dµ −
n
X
k=0
f (tk)µ(Ck)k
+ k
n
X
k=0
f (tk)µ(Ck)k
< ε 2+ |
n
X
k=0
f (tk)kµ(Ck)k
− (Bw) Z
T
f dkµk| + (Bw) Z
T
f dkµk
< ε + (Bw) Z
T
f dkµk, ∀ε > 0.
Consequently, k(Bw)R
T f dµk ≤ (Bw)R
T f dkµk. Definition 5.10 Suppose ν : A → X and µ : A → X are vector set functions such that ν(∅) = µ(∅) = 0.
It is said thatν is absolutely continuous with respect to µ (denoted by ν µ) if for every ε > 0, there existsδ > 0 so that:
∀E ∈ A, µ(E) < δ ⇒ kν(E)k < ε.
Theorem 5.11 Let f : T → R be a bounded function such thatf is Bw-µ-integrable on every measurable setE ∈ A and α = supt∈T |f (t)| > 0. If we consider the vector set functionν : A → X, defined by ν(E) = (Bw)R
Ef dµ, for every E ∈ A, then ν µ.
Proof. Consider δ = αε > 0 for any ε > 0. Now, according to Theorem 5.4-(ii), it results:
kν(E)k = k(Bw) Z
E
f dµk ≤ sup
t∈E
|f (t)| · µ(E)
< αδ = ε,
for every measurable set E ∈ A, with µ(E) < δ. This
implies ν µ.
The following theorem is demonstrated the same as Theorem 4.11.
Theorem 5.12 Let f : T → R be a real function.
(i) If B, C ∈ A are disjoint sets and f is both Bw-µ-integrable on B and Bw-µ-integrable on C, thenf is Bw-µ-integrable on B ∪C and the following relation holds:
(Bw) Z
B∪C
f dµ = (Bw) Z
B
f dµ + (Bw) Z
C
f dµ.
(ii) Iff is Bw-µ-integrable on every set E ∈ A, then the set functionν : A → X, defined by ν(E) = (Bw)R
Ef dµ, ∀E ∈ A, is finitely additive.
6 The real case
This section contains some properties of monotonic- ity for the case when both the function f and the set function µ have real values. These properties show that the integral is monotone with respect to the func- tion, the measure and the set. In the sequel, suppose µ : A → [0, +∞) is a nonnegative set function, with µ(∅) = 0.
Theorem 6.1 Let f, g : T → R he Bw-µ-integrable functions. Then the following properties hold:
(i) Iff ≤ g, then (Bw)R
T f dµ ≤ (Bw)R
T gdµ.
(ii) Ifg ≥ 0, then (Bw)R
T gdµ ≥ 0.
Proof. (i) Consider an arbitrary ε > 0. Since f is Bw-µ-integrable, then there exist P1 ∈ P a countable partition of T and n1ε ∈ N such that for every count- able partition of T , P = {An}n∈N, P ≥ P1, and every tn∈ An, n ∈ N, it holds
|
n
X
k=0
f (tk)µ(Ak) − (Bw) Z
T
f dµ| < ε 2, (7)
∀n ≥ n1ε.
Analogously, since g is Bw-µ-integrable, there ex- ist P2 ∈ P a countable partition of T and n2ε ∈ N such that for every countable partition of T , P =
{An}n∈N, P ≥ P2, and every tn ∈ An, n ∈ N, it holds
|
n
X
k=0
g(tk)µ(Ak) − (Bw) Z
T
gdµ| < ε 2, (8)
∀n ≥ n2ε.
Now, let P0 = P1∧ P2 and n0 = max{n1ε, n2ε}, and consider P = {An}n∈N a countable partition of T , with P ≥ P0, tn ∈ An for every n ∈ N and a fixed n ≥ n0. Since f ≤ g, by (7) and (8), we get
(Bw) Z
T
f dµ − (Bw) Z
T
gdµ
≤ |
n
X
k=0
f (tk)µ(Ak) − (Bw) Z
T
f dµ|+
+
n
X
k=0
[f (tk) − g(tk)]µ(Ak)
+ |
n
X
k=0
g(tk)µ(Ak) − (Bw) Z
T
gdµ| < ε, ∀ε > 0.
Consequently, the inequality follows.
(ii) We apply (i) for f = 0.
Theorem 6.2 Let µ1, µ2 : A → [0, +∞) be non- negative set functions such that µ1(∅) = µ2(∅) = 0 and µ1(A) ≤ µ2(A), for every A ∈ A, and let f : T → [0, +∞) be a nonnegative function which is bothBw-µ1-integrable andBw-µ1-integrable. Then (Bw)R
Tf dµ1 ≤ (Bw)R
Tf dµ2.
Proof. Let ε > 0 be arbitrary. From the hypothesis, there exist P = {An}n∈Na countable partition of T and nε ∈ N such that for every tn ∈ An, n ∈ N, it holds
|
n
X
k=0
f (tk)µ1(Ak) − (Bw) Z
T
f dµ1| < ε
2, (9)
∀n ≥ nε and
|
n
X
k=0
f (tk)µ2(Ak) − (Bw) Z
T
f dµ2| < ε
2, (10)
∀n ≥ nε.
According to (9) and (10), since µ1 ≤ µ2, we obtain
(Bw) Z
T
f dµ1− (Bw) Z
T
f dµ2
≤ |
n
X
k=0
f (tk)µ1(Ak) − (Bw) Z
T
f dµ1|
+
n
X
k=0
f (tk)[µ1(Ak) − µ2(Ak)]
+ |
n
X
k=0
f (tk)µ2(Ak) − (Bw) Z
T
f dµ2| < ε.
Since ε > 0 is arbitrary, the conclusion follows.
Theorem 6.3 Suppose µ : A → [0, +∞) is mono- tone andf : T → [0, +∞) is a nonegative function.
Then the following properties hold:
(i) LetA, B ∈ A be measurable sets such that f is both Bw-µ-integrable on A and Bw-µ-integrable onB. If A ⊆ B, then (Bw)R
Af dµ ≤ (Bw)R
Bf dµ.
(ii) Suppose f is Bw-µ-integrable on every set A ∈ A and denote ν(A) = (Bw)R
Af dµ, for every A ∈ A. Then ν is monotone.
Proof. Let ε > 0 be arbitrary. Since f is Bw-µ- integrable on A, there exist Pε1 = {Cn}n∈N ⊂ A a partition of A and n1ε ∈ N such that for each partition of A, P = {En}n∈N, with P ≥ Pε1 and tn ∈ En, n ∈ N, it holds
|
n
X
k=0
f (tk)µ(Ek) − (Bw) Z
A
f dµ| < ε
2, (11)
∀n ≥ n1ε.
Now, similarly in the case of B, there exist Pε2= {Dn}n∈N ⊂ A a partition of B and n2ε ∈ N so that for all partitions of B, P = {En}n∈N, with P ≥ Pε2 and tn∈ En, n ∈ N, it holds
|
n
X
k=0
f (tk)µ(Ek) − (Bw) Z
B
f dµ| < ε
2, (12)
∀n ≥ n2ε.
Let ePε1 = {Cn, B \ A} and P = {En}n∈N a par- tition of B, with P ≥ ePε1 ∧ Pε2. It results that Pε00= {En∩ A}n∈Nis a partition of A and Pε00 ≥ Pε0. Now, consider nε = max{n1ε, n2ε} and tn ∈ En∩ A,
n ∈ N. According to (11) and (12) it follows:
(Bw) Z
A
f dµ − (Bw) Z
B
f dµ
≤ |(Bw) Z
A
f dµ −
n
X
k=0
f (tk)µ(Ek∩ A)|
+
n
X
k=0
f (tk)[µ(Ek∩ A) − µ(Ek)]
+ |
n
X
k=0
f (tk)µ(Ek) − (Bw) Z
B
f µ| < ε.
Since ε > 0 is arbitrary, the inequality results.
(ii) It immediately follows from (i).
7 Conclusion
We have defined and studied Birkhoff weak integra- bility of functions f relative to a set function µ. For a real Banach space (X, k · k), we have considered two cases:
(i) f : T → X and µ : A → [0, +∞) with µ(∅) = 0 and
(ii) f : T → R and µ : A → X with µ(∅) = 0.
Some comparison results and classical integral properties are obtained: linearity relative to the func- tion and to the measure, absolutely continuity. Also, monotonicity properties for the real case are pre- sented. As future works:
- relationships between Birkhoff weak integra- bility and other integrabilities (Gould, Dunford, Mc- Shane, Henstock-Kurzweil);
- properties of continuity, regularity and Birkhoff weak integrability on atoms.
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