# Harnark extention principle of Henstock integral for Banach-valued function 1

Document Sample

```					   Scientia Magna
Vol. 5 (2009), No. 4, 91-95

Harnark extention principle of Henstock
integral for Banach-valued function 1
Fengling Jia and Wansheng He

College of Mathematics and Statistics, Tianshui Normal University, Tianshui, Gansu, 741001,
P.R.China
E-mail: jiafengling1978@163.com

Abstract The paper mainly gives Harnark extention principle of Henstock integral on
banach-valued function, promotes Harnark extention principle of Henstock integral on Real-
valued function.
Keywords Henstock integral, Harnark extention principle.

§1. Introductions and basic deﬁnitions
Throughout this paper X denotes a real Banach Space and A partition of [a, b] is a ﬁnite
collection of interval-point pairs {I, ξ} with the intervals non-overlapping and their union [a, b],
here ξ is the associated point of I, we write D = {I, ξ}. It is said to be δ-ﬁne partition of [a, b]
if for each interval-point {I, ξ} we have ξ ∈ I ⊂ (ξ − δ(x), ξ + δ(x)).
Let f is Banach-valued function deﬁned on [a, b] and we use to (D)             f (ξ)|I| represent the
Riemann sum of f corresponding to the δ-ﬁne partition D = {I, ξ}.
Deﬁnition 1.1. The function f : [a, b] → X is Henstock integrable on [a, b] and A ∈ X
is its Henstock integral if for each ε > 0 there is gauge δ(t) on [a, b] such that for any δ-ﬁne
partition D = {I, ξ} of [a, b] we have

(D)     f (ξ)|I| − A < ε.

The function f is Henstock integrable on a set E ⊂ [a, b] if the function f · χE is Henstock
b
integrable on [a, b] and we denote (H) a f χE = (H) E f , where χE denotes the characteristic
b
function of E and we denote (H) a f χE = (H) E f .
The properties of Henstock integral of Banach-valued functions are similar to real-valued
functions, the reader is referred to [1] [2] [3] for the details.

§2. Main results
First we give Harnark extention principle of Henstock integral on Real-valued function.
1 This   work is supported by the Gansu Provincial Education Department Foundation 0708-10.
92                                               Fengling Jia and Wansheng He                                                            No. 4

Theorem 2.1.[1][2] Let f : [a, b] → R is Henstock integrable on close set E ⊂ [a, b],
∞                                                                                                 ∞
[a, b] \ E =         (ci , di ), if f (x) is Henstock integrable on each [ci , di ], and                                ω(Fi , [ci , di ]) < ∞,
i=1                                                                                                i=1
then the function f is Henstock integrable on [a, b]
b                             ∞                    di

```
DOCUMENT INFO
Shared By:
Categories:
Stats:
 views: 12 posted: 6/25/2010 language: English pages: 6
Description: Introductions and basic definitions Throughout this paper X denotes a real Banach Space and A partition of [a, b] is a finite collection of interval-point pairs {I, } with the intervals non-overlapping and their union [a, b], here is the associated point of I, we write D = {I, }. Proof. Because ... such that ...
BUY THIS DOCUMENT NOW PRICE: \$6.95 100% MONEY BACK GUARANTEED
PARTNER
ProQuest creates specialized information resources and technologies that propel successful research, discovery, and lifelong learning.