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@upgploiesti.ro
Abstract:In this paper, we define and study the Birkhoff weak integral in two cases: for vector functions relative to a nonnegative 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; nonadditive 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 19531954), as well as fuzzy measures and nonlinear 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 transferableutility 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], [1624], [25], [27], [31], [32], [3639], [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 (t_{n})m(B_{n}),
accordingly to a countable partition {Bn}_{n∈N} of T and t_{n}∈ B_{n}, 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 nonadditive nonnegative (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 σalgebra 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∩ A_{j} = ∅, i 6= j and S
n∈N
An= T .
(ii) IfP and P^{0}are two countable partitions ofT , thenP^{0}is said to be finer thanP , denoted by P ≤ P^{0} (or,P^{0} ≥ P ), if every set of P^{0}is included in some set ofP .
(iii) The common refinement of two countable partitions P = {Ai} and P^{0} = {Bj} is the parti tionP ∧ P^{0} = {A_{i}∩ B_{j}}. We denote by P the class of all partitions ofT and if A ∈ A is fixed, by P_{A}we 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 (σadditive) 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(A_{i})}, for every E ∈ P(T ), where the supremum is extended over all finite families of pair wise disjoint sets {A_{i}}^{n}_{i=1} ⊂ A, with A_{i} ⊆ 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(A_{i})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 {A_{i}}^{n}_{i=1}⊂ A, of E.
Definition 2.5 A property (P) about the points of T holds almost everywhere (denotedµae) 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 nonnegative set function with µ(∅) = 0. A function f : T → X is called RiemannLebesgue µintegrable (RLµintegrable 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 = {A_{n}}_{n∈N} ofT inA, with P ≥ P_{ε} and everyt_{n} ∈ A_{n},n ∈ N, the series
∞
P
n=0
f (t_{n})µ(A_{n}) 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 σfinite 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 σalgebra 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 µintegrable (on T ) (simply Bwµintegrable) if there exists α ∈ X so that for everyε > 0, there exist a countable partition P_{ε} ofT in A and n_{ε} ∈ N such that for every other countable partition P = {A_{n}}_{n∈N} of T in A, with P ≥ Pε, and everytn∈ A_{n},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 µintegrable on a set E ∈ A if the restriction f _{E} is Birkhoff weak µ integrable on (E, AE, µ), where AE = {E ∩ AA ∈ A}.
Remark 3.2 If it exists, the integral is unique.
Example. I. If f (t) = 0, for every t ∈ T , then f is Bwµintegrable and (Bw)R
T f dµ = 0.
II. Suppose T = {tnn ∈ N} is countable, {t_{n}} ∈ 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 Bwµintegrable and
(Bw) Z
T
f dµ =
∞
X
n=0
f (t_{n})µ({t_{n}}).
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 Bwµintegrable if and only if F is Bwµintegrable (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 14II [11], it follows that f is Bwµintegrable if and only if f is Bwϕintegrable (according to [11]).
Theorem 3.4 Let f : T → [0, +∞) be a non negative function and µ : A → [0, +∞) a nonnegative set function with µ(∅) = 0. If f is Bwµintegrable, then f is RLµintegrable and (RL)R
T f dµ = (Bw)R
T f dµ.
Proof. Let ε > 0 be arbitrary. Since f is Bwµ integrable, there exist Pε⊂ A a countable partition of T and nε∈ N that satisfy the conditions of Definition 3.1. Let P = {A_{n}}_{n∈N} ⊂ A be a countable partition of T , with P ≥ Pε and tn ∈ A_{n}, n ∈ N. Denoting s_{n}=
n
P
k=0
f (t_{k})m(A_{k}), for every n ∈ N, according to Definition 3.1, it holds
(∗)
s_{n}− (Bw) Z
T
f dm
< ε
2, ∀n ≥ n_{ε}. This shows that the series
∞
P
n=0
f (t_{n})m(A_{n}) 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 RLµintegrable and (RL)R
Tf dm = (Bw)R
Tf dm.
Theorem 3.5 Suppose (T, A, µ) is a σfinite measure space andµ : A → [0, +∞) is σadditive. If f : T → X is Bwµintegrable, then f is Pettis integrable.
Proof. Since f is Bwµintegrable, for every ε >
0, there exist a countable partition P_{ε}of T in A and nε ∈ N such that for every other countable partition
P = {A_{n}}_{n∈N} of T in A, with P ≥ Pε, and every t_{n}∈ A_{n}, 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 )(t_{k})µ(Ak) − x^{∗}((Bw) Z
T
f dµ) < ε.
This shows that x^{∗} ◦ f is Bwµintegrable. It also results that the series P∞
n=0(x^{∗} ◦ f )(t_{n})µ(An) is convergent and P∞
n=0(x^{∗} ◦ f )(t_{n})µ(A_{n}) − x^{∗}((Bw)R
Tf dµ) < ε.
As in Theorem 3.14 [7], it is demonstrated that the series P∞
n=0(x^{∗} ◦ f )(t_{n})µ(A_{n}) is uncondition ally convergent. So, x^{∗} ◦ f is Birkhoff µintegrable and therefore is in L^{1}for 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 nonnegative set function µ : A → [0, +∞) with µ(∅) = 0. According to Remark 3.3I 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 Bwµintegrable and (Bw)R
T f dµ = 0.
Theorem 4.2 Let f : T → X be a Bwµintegrable function. Thenf is Bwµintegrable on E ∈ A if and only if f XE is Bwµintegrable on T , where XE is the characteristic function ofE. In this case,
(Bw) Z
E
f dµ = (Bw) Z
T
f X_{E}dµ.
Theorem 4.3 If f, g : T → X are Bwµintegrable, thenf + g is Bwµintegrable 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 Bwµintegrable 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 ≤ sup_{t∈T}kf (t)k · µ(T ).
Theorem 4.5 Let f : T → X be a Bwµintegrable function andα ∈ R. Then:
(i)αf is Bwµintegrable and (Bw)
Z
T
αf dµ = α(Bw) Z
T
f dµ;
(ii)f is Bwαµintegrable (α ∈ [0, +∞)) and (Bw)
Z
T
f d(αµ) = α(Bw) Z
T
f dµ.
Theorem 4.6 Suppose µ_{1}, µ_{2} : A → [0, +∞) are nonnegative set functions with µ1(∅) = µ2(∅) = 0.
Iff : T → X is both Bwµ1integrable andBwµ2 integrable, thenf is Bw(µ_{1}+ µ_{2})integrable and
(Bw) Z
T
f d(µ_{1}+ µ_{2})
= (Bw) Z
T
f dµ_{1}+ (Bw) Z
T
f dµ_{2}.
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 µintegrable and f = g µae. Then g is Bwµ 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 Bwµintegrable function, such that the real function kf k : T → [0, +∞) is Bwµintegrable. Then
k(Bw) Z
T
f dµk ≤ (Bw) Z
T
kf kdµ.
Proof. Suppose ε > 0 is arbitrary. Since f is Bw µintegrable, there exist P_{ε}^{1} ∈ P and n^{1}_{ε} ∈ N so that for every P = {An}_{n∈N} ∈ P, P ≥ P_{ε}^{1}, and every s_{n}∈ A_{n}, n ∈ N, it holds
k
n
X
k=0
f (s_{k})µ(A_{k}) − (Bw) Z
T
f dµk < ε 2, for every n ∈ N, n ≥ n^{1}ε.
Since kf k is Bwµintegrable, there exist P_{ε}^{2} ∈ P and n^{2}_{ε} ∈ N so that for every P = {Bn}_{n∈N} ∈ P, P ≥ P_{ε}^{2} and every un∈ B_{n}, n ∈ N, it holds

n
X
k=0
kf (t_{k})kµ(B_{k}) − (Bw) Z
T
kf kdµ < ε 2, for every n ∈ N, n ≥ n^{2}ε.
Now, we consider Pε = P_{ε}^{1}∧ P_{ε}^{2} ∈ P and n_{ε}= max{n^{1}_{ε}, n^{2}_{ε}}. Then for every P = {C_{n}}_{n∈N} ∈ P, P ≥ Pε and every tn ∈ C_{n}, 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 (t_{k})kµ(C_{k}) − (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 (t_{k})kµ(C_{k}) − (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 Bwµintegrable on every set E ∈ A, such that α = sup_{t∈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 = _{α}^{ε} > 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 Bwµintegrable on B and Bw µintegrable on C, when B, C ∈ A are disjoint mea surable sets, thenf is Bwµintegrable on B ∪ C and (Bw)R
B∪Cf dµ = (Bw)R
Bf dµ + (Bw)R
Cf dµ.
(ii) Iff is Bwµintegrable 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 Bwµintegrable on B, there exist a partition of B, P_{B}^{ε} = {Dn}_{n∈N} ∈ P_{B} and n^{1}_{ε} ∈ N such that for every partition of B, P = {An}_{n∈N} ∈ P_{B}, P ≥ P_{B}^{ε} and t_{n}∈ A_{n}, n ∈ N, it holds
k
n
X
k=0
f (t_{k})µ(A_{k}) − (Bw) Z
B
f dµk < ε
2, ∀n ≥ n^{1}_{ε}. (3) Since f is Bwµintegrable on C, there exist a parti tion of C, P_{C}^{ε} = {En}_{n∈N} ∈ P_{C} and n^{2}_{ε} ∈ N such that for every partition of C, P = {U_{n}}_{n∈N} ∈ P_{C}, P ≥ P_{C}^{ε} and sn∈ U_{n}, n ∈ N, it holds
k
n
X
k=0
f (s_{k})µ(U_{k}) − (Bw) Z
C
f dµk < ε
2, ∀n ≥ n^{2}_{ε}. (4) Let P_{B∪C}^{ε} = {D_{n}, E_{n}}_{n∈N} ∈ P_{B∪C} and n ≥ n^{1}_{ε} + n^{2}_{ε}. Now, for every partition of B ∪ C, P = {V_{n}}_{n∈N} ∈ P_{B∪C}^{0} with P ≥ P_{B∪C}^{ε} and every un ∈ V_{n}, n ∈ N, we can write P = {Bn^{0}}_{n∈N}∪ {C_{n}^{0}}_{n∈N}, where P_{B}^{0} = {B_{n}^{0}}_{n∈N}≥ P_{B}^{ε} and P_{C}^{0} = {C_{n}^{0}}_{n∈N}≥ P_{C}^{ε}. Also, we can write Pn
k=0f (u_{k})µ(V_{k}) = Pp
i=0f (u_{k})µ(B_{k}^{0}) + Pr
j=0f (u_{k})µ(C_{k}^{0}), with p ≥ n^{1}_{ε}, r ≥ n^{2}_{ε}. 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(B_{k}^{0}) − (Bw) Z
B
f dµk
+ k
r
X
j=0
f (u_{k})µ(C_{k}^{0}) − (Bw) Z
C
f dµk < ε 2 +ε
2 = ε, which shows that f is Bwµintegrable 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 3.3II 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 Bwµintegrable and (Bw)R
T f dµ = 0.
Theorem 5.2 f : T → R is Bwµintegrable on E ∈ A if and only if f X_{E} isBwµintegrable on T.
Theorem 5.3 If f, g : T → R are Bwµintegrable andf (t)g(t) ≥ 0 for every t ∈ T , then f + g is Bw µintegrable 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 Bwµintegrable bounded functions. Then:
(i) k(Bw)R
T f dµ − (Bw)R
T gdµk ≤
sup_{t∈T} f (t) − g(t) · µ(T );
(ii)k(Bw)R
T f dµk ≤ sup_{t∈T}f (t) · µ(T ).
Theorem 5.5 Let f : T → R be a Bwµintegrable function andα ∈ R. Then:
(i)αf is Bwµintegrable 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, µ_{2} : A → X be nonnegative set functions withµ_{1}(∅) = µ_{2}(∅) = 0. If f : T → R is both Bwµ1integrable and Bwµ2integrable, 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 Bwµintegrable both on A and on B, then f is Bwµintegrable 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 Bwµintegrable and f = g µae, then g is Bwµintegrable 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 Bwµintegrable and Bwkµk integrable. Then
k(Bw) Z
T
f dµk ≤ (Bw) Z
T
f dkµk.
Proof. Let ε > 0 be arbitrary. Since f is Bwµ integrable, there exist P_{ε}^{1} ∈ P and n^{1}_{ε} ∈ N such that for every P = {An}_{n∈N}∈ P, with P ≥ P_{ε}^{1}, and each s_{n}∈ A_{n}, n ∈ N, it holds
k
n
X
k=0
f (sk)µ(Ak) − (Bw) Z
T
f dµk < ε
2, ∀n ≥ n^{1}_{ε}. Since f is Bwkµkintegrable, there exist P_{ε}^{2} ∈ P and n^{2}_{ε} ∈ N such that for every P = {Bn}_{n∈N} ∈ P, with P ≥ P_{ε}^{2}, and each u_{n}∈ B_{n}, n ∈ N, it holds

n
X
k=0
f (u_{k})kµ(B_{k})k−(Bw) Z
T
f dkµk < ε
2, ∀n ≥ n^{2}_{ε}. Let Pε= P_{ε}^{1}∧ P_{ε}^{2}∈ P and nε= max{n^{1}_{ε}, n^{2}_{ε}}. Then for every P = {Cn}_{n∈N} ∈ P, P ≥ Pε, and every t_{n}∈ C_{n}, n ∈ N, it results
k
n
X
k=0
f (t_{k})µ(C_{k}) − (Bw) Z
T
f dµk < ε 2, (5)

n
X
k=0
f (t_{k})kµ(C_{k})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 (t_{k})kµ(C_{k})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 Bwµintegrable on every measurable setE ∈ A and α = sup_{t∈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 = _{α}^{ε} > 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 Bwµintegrable on B and Bwµintegrable on C, thenf is Bwµintegrable 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 Bwµintegrable 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 Bwµintegrable 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 Bwµintegrable, then there exist P1 ∈ P a countable partition of T and n^{1}_{ε} ∈ N such that for every count able partition of T , P = {An}_{n∈N}, P ≥ P1, and every tn∈ A_{n}, n ∈ N, it holds

n
X
k=0
f (t_{k})µ(A_{k}) − (Bw) Z
T
f dµ < ε 2, (7)
∀n ≥ n^{1}_{ε}.
Analogously, since g is Bwµintegrable, there ex ist P2 ∈ P a countable partition of T and n^{2}_{ε} ∈ N such that for every countable partition of T , P =
{A_{n}}_{n∈N}, P ≥ P2, and every tn ∈ A_{n}, n ∈ N, it holds

n
X
k=0
g(tk)µ(Ak) − (Bw) Z
T
gdµ < ε 2, (8)
∀n ≥ n^{2}_{ε}.
Now, let P0 = P_{1}∧ P_{2} and n0 = max{n^{1}_{ε}, n^{2}_{ε}}, and consider P = {A_{n}}_{n∈N} a countable partition of T , with P ≥ P0, tn ∈ A_{n} for every n ∈ N and a fixed n ≥ n_{0}. Since f ≤ g, by (7) and (8), we get
(Bw) Z
T
f dµ − (Bw) Z
T
gdµ
≤ 
n
X
k=0
f (t_{k})µ(A_{k}) − (Bw) Z
T
f dµ+
+
n
X
k=0
[f (t_{k}) − g(t_{k})]µ(A_{k})
+ 
n
X
k=0
g(t_{k})µ(A_{k}) − (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 µ_{1}(∅) = µ_{2}(∅) = 0 and µ_{1}(A) ≤ µ_{2}(A), for every A ∈ A, and let f : T → [0, +∞) be a nonnegative function which is bothBwµ_{1}integrable andBwµ_{1}integrable. Then (Bw)R
Tf dµ_{1} ≤ (Bw)R
Tf dµ_{2}.
Proof. Let ε > 0 be arbitrary. From the hypothesis, there exist P = {An}_{n∈N}a countable partition of T and n_{ε} ∈ N such that for every tn ∈ A_{n}, n ∈ N, it holds

n
X
k=0
f (t_{k})µ_{1}(A_{k}) − (Bw) Z
T
f dµ_{1} < ε
2, (9)
∀n ≥ n_{ε} and

n
X
k=0
f (t_{k})µ2(A_{k}) − (Bw) Z
T
f dµ2 < ε
2, (10)
∀n ≥ n_{ε}.
According to (9) and (10), since µ1 ≤ µ_{2}, we obtain
(Bw) Z
T
f dµ_{1}− (Bw) Z
T
f dµ_{2}
≤ 
n
X
k=0
f (t_{k})µ_{1}(A_{k}) − (Bw) Z
T
f dµ_{1}
+
n
X
k=0
f (t_{k})[µ_{1}(A_{k}) − µ_{2}(A_{k})]
+ 
n
X
k=0
f (t_{k})µ_{2}(A_{k}) − (Bw) Z
T
f dµ_{2} < 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 Bwµintegrable on A and Bwµintegrable onB. If A ⊆ B, then (Bw)R
Af dµ ≤ (Bw)R
Bf dµ.
(ii) Suppose f is Bwµintegrable 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 Bwµ integrable on A, there exist P_{ε}^{1} = {C_{n}}_{n∈N} ⊂ A a partition of A and n^{1}_{ε} ∈ N such that for each partition of A, P = {En}_{n∈N}, with P ≥ P_{ε}^{1} and tn ∈ E_{n}, n ∈ N, it holds

n
X
k=0
f (t_{k})µ(E_{k}) − (Bw) Z
A
f dµ < ε
2, (11)
∀n ≥ n^{1}_{ε}.
Now, similarly in the case of B, there exist P_{ε}^{2}= {D_{n}}_{n∈N} ⊂ A a partition of B and n^{2}_{ε} ∈ N so that for all partitions of B, P = {E_{n}}_{n∈N}, with P ≥ P_{ε}^{2} and tn∈ E_{n}, n ∈ N, it holds

n
X
k=0
f (tk)µ(Ek) − (Bw) Z
B
f dµ < ε
2, (12)
∀n ≥ n^{2}_{ε}.
Let eP_{ε}^{1} = {C_{n}, B \ A} and P = {E_{n}}_{n∈N} a par tition of B, with P ≥ eP_{ε}^{1} ∧ P_{ε}^{2}. It results that P_{ε}^{00}= {E_{n}∩ A}_{n∈N}is a partition of A and P_{ε}^{00} ≥ P_{ε}^{0}. Now, consider nε = max{n^{1}_{ε}, n^{2}_{ε}} and tn ∈ E_{n}∩ 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 (t_{k})µ(E_{k}∩ A)
+
n
X
k=0
f (t_{k})[µ(E_{k}∩ A) − µ(E_{k})]
+ 
n
X
k=0
f (t_{k})µ(E_{k}) − (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, HenstockKurzweil);
 properties of continuity, regularity and Birkhoff weak integrability on atoms.
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