Primitive of The Henstock-Bochner Integral

PRIMITIVE OF THE HENSTOCK-BOCHNER INTEGRAL
EDY NUR FALAH
SOEPARNA DARMAWIJAYA
Abstract. In this paper, we talk about the properties of primitive of Henstock integral of Banach space-valued functions and their relationship with Denjoy-Bochner integral. This paper also contains a necessary and sufficient conditions for sequence of functions which is Henstock-Bochner integrable converges to an integrable function in the same sense.

Key words and Phrases : BV, BVG*, AC, ACG*, primitive (primitive function), Henstock-Bochner integral, Denjoy-Bochner integral, convergence theorems.
1. INTRODUCTION
In this paper, we discuss about the properties of primitive of Henstock integral of Banach space-valued functions, and hereinafter referred to as Henstock-Bochner integral.
Definition 1.1. Let X be a Banach space. The function is called Henstock-Bochner integrable on and denote if there is a vector such that for every there exist a positive function δ on and for every Perron δ-fine partition on the inequality ‖∑ ‖ ‖ ∑ ‖
hold. We denote ∫ and denotes the set of all Henstock Bochner integrable functions on
Proposition 1.2. The function if and only if for every there is a positive function on and for every , 𝒬 the Perron δ-fine partition on , we have ‖ ∑ 𝒬∑ ‖
Definition 1.3. Let be a Banach space. The function is variational-integrable on denotes if there is a function such that for every there is a positive function on and an increasing function with

and if ( ), with we have ‖ ‖
And, is called -primitive of a function on .
Lemma 1.4. Let be a function. Then, the three following statements are equivalent.

(i)

(ii) There exists a function such that for any , there is a positive function on and if is a Perron δ-fine partition on , then

∑‖ ‖

(iii)

2. PRIMITIVE OF THE HENSTOCK-BOCHNER INTEGRAL
In the present paper, we will discuss the primitives of Henstock-Bochner integrable functions as well as some connections of the integrals with the concept of derivative of a Banach space-valued function.
Remark 2.1. Let then for each
Therefore, if , then there is an additive closed interval function in such a way that ∫
for each with is the collection of all closed interval on The function of is called Henstock-Bochner primitive (HB-primitive) of function on In particular, for with
∫ and ∫
Proposition 2.2. The function if and only if there is a function such that for every there is a positive function on and for every Perron δ-fine partition on the inequality ‖ ∑ ‖
hold.
Theorem 2.3. If is Henstock-Bochner integrable on with primitive then continuous, F′(x) = f(x) a.e. on and
Proof. Since with as primitive of if and only if. with as a primitive By taken { ‖ ‖ }, , we have for every and there exist such that if ( ( )) we have ‖ ‖
Hence, is continuous at all
Since with primitive , then there is an increasing function on with ( ) in such a way that ‖ ‖
or ‖ ‖ | |
a.e. on [a, b] for each .
By other word,
a.e. on [a, b].
The next proof for see [2].
Theorem 2.4. If is Henstock-Bochner integrable on with primitive , then
Proof. Since i.e., for every there is a positive function on , and if is δ-fine partition on , then we have ‖ ∑ ‖
Let for all . Then, for every , we set Eni with { [ ] ‖ ‖ }
Thus, ⋃
Now, we show for every Fix and let be any collection of non-overlapping intervals with and every [ for all . Then, we obtain
∑‖ ‖ ∑‖ ‖ ∑‖ ‖ ∑‖ ‖ ∑‖ ‖ ∑‖ ‖ ∑‖ ‖ ∑‖ ‖ ∑‖ ‖ ∑‖ ‖ ∑‖ ‖ ∑‖ ‖
Taking and ∑ we get ∑‖ ‖
for any Therefore, for every . Consequently,
Definition 2.5. Function is called Denjoy-Bochner integrable on if there exist a function such that

(i) is continuous on ,

(ii) a.e. on ,

(iii) .

The number ∫ ‖ ‖ is called Denjoy-Bochner integral (DB*-integral) of function on , and denotes .

ByTheorem 2.3., Theorem 2.4. and Definition 2.5., we obtain the following theorem.
Theorem 2.6. A function is Henstock-Bochner integrable on if and only if is Denjoy-Bochner integrable on
Proof. In view of Theorem 2.3. and Theorem 2.4., it is clear that . Let , i.e. there exist a function with a.e. on and
a.e. on ; i.e. there is a set with λ( ) = 0 and for . So, for each and there exist and if ( )
then, we obtain ‖ ‖ ‖ ‖
or ‖ ‖
Since i.e. there is a sequence of set such that ⋃ and for all i. And, , so we can assume that is closed for all i. Now, we create the sets:
Y1 = E1,
Y2 = E2 – E1,

Yn = En – E1 – E2 – … – En-1,
Then, we obtain for i ≠ j.
Yi Ei, ⋃ ⋃ and for all i.
If is the set of all and ‖ ‖ then , ⋃⋃ where .
Hence, and (for above) there is a number such that for each sequence of non-overlapping interval { } on and ∑| |
we obtain ∑‖ ‖
Since there is a sequence of opened interval { } such that ⋃ ∑| |
If then there exist for some n. In this case, we take a number δ2(x) > 0 such that ( )
Choose δ(x) = min{δ1(x), δ2(x)} for each then for each Perron δ-fine partition on by splitting addition ∑on as two subdivision ∑ and ∑ for and we obtain ‖ ∑ ‖ ‖∑ ‖ ∑‖ ‖ ∑‖ ‖ ∑ ∑ ∑
Hence,
3. CONVERGENCE THEOREMS
For each , the function is Henstock-Bochner integrable on So, is the sequence of function of Henstock-Bochner integrable on The next, a necessary and sufficient conditions for sequence of functions which is Henstock-Bochner integrable converges to an integrable function in the same sense will be presented.
Theorem 3.1. (Controlled Convergence Theorem) Let be a sequence of Henstock-Bochner integrable functions on with , , such that

(i) For each n, converges to a.e. on with as primitive of funcion ,

(ii) uniformly, i.e., there is a sequence of closed sets of such that ⋃ and uniformly for each i, and

(iii) uniformly convergence on

then is Henstock-Bochner integrable on and ∫ ∫
Proof. Since is continuous for each n and converges uniformly on then converges to a function which is continuous on By (ii) and (iii), we obtain so is uniformly for each i,
i.e. for every there exist such that for every non-overlapping interval sequence with and ∑| | we have ∑‖ ‖
for each n.
Since converges uniformly to on , then for each i. Hence, ′ exist a.e. on because ′ for almost Assume that is closed such that in which is contiguous interval sequence of on
Define a continuous function on with {
So, is linear on each interval , and we get on , uniformly and ′ ′ exist a.e. on , where {
Since for each i and converges uniformly to on then we obtain
a.e. on Moreover, since ⋃ then we obtain ′ ′
a.e. on
Hence, the theorem is proofed.
Theorem 3.2. Let is Henstock-Bochner integrable on for each and converges uniformly on to a function , then and ∫ ∫
Proof. The function converges uniformly on to a function i.e. for each there exist a natural number such that for and , we have
‖ ‖
The function for each n , i.e. for every there exist a positive function such that for both -fine partition and 𝒬 on , we have ‖ ∑ 𝒬∑ ‖
Hence,
Furthermore, since then for we obtain ‖ ∫ ∫ ‖
or by the other word, and ∫ ∫
Definition 3.3. A sequence of function is called Henstock-Bochner equiintegable (HB-equiintegrable) on if for each there exist a positive function δ on and for each δ-fine partition on we have ‖ ∑ ∫ ‖
for each .
Corollary 3.4. (Theorem on equiintegrability) Let be a sequence of equiintegrable function on and converges to a function for each then is Henstock-Bochner integrable on and ∫ ∫
Proof. Take and 𝒬 both are δ-fine partitions on where and 𝒬 {( [ ]) }. Then, we have ‖∑ ∑ ( )( ) ‖
Since converges to , i.e. for every and there exist a natural number such that for we have ‖ ‖
Take { 𝒬} such that for any two δ-fine partition and 𝒬 on we have ‖ ∑ 𝒬∑ ‖
Thus, , and for we obtain ‖ ∫ ∫ ‖
or, and ∫ ∫
4. CONCLUDING REMARKS
In general, the properties of primitive of Henstock integral for Banach space-valued functions are apply to real-valued functions. But, there are some properties of primitive of Henstock integral of real-valued function can not apply to Banach space-valued function.
Acknowledgement. The author gives thanks to Prof. Dr. Soeparna Darmawijaya as guidance in writing this paper.
REFERENCES
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[3] Lee, P, Y and Vyborny, R, Integral: An Easy Approach after Kurzweil and Henstock, Cambridge University Press, UK, 2000.
[4] Lee, P, Y, Lanzhou Lectures on Henstock Integration, World Scientific, Singapore, 1989.