5.Nurul_Henstock-Kurzweil HENSTOCK- KURZWEIL INTEGRAL ON [a,b] Siti Nurul Afiyah Dosen STMIK STIE Asia Malang e-mail: noeroel_afy@yahoo.com ABSTRACT The theory of the Riemann integral was not fully satisfactory. Many important functions do not have a Riemann integral. So, Henstock and Kurzweil make the new theory of integral. From the background, the writer will be research about Henstock-Kurzweil integral and also theorems of Henstock- Kurzweil Integral. Henstock- Kurzweil Integral is generalized from Riemann integral. In this case the writer uses research methods literature or literature study carried out by way explore, observe, examine and identify the existing knowledge in the literature. In this thesis explain about partition which used in Henstock- Kurzweil Integral, definition and some property of Henstock- Kurzweil Integral. And some properties of Henstock- Kurzweil integral as follows: value of the Henstock- Kurzweil integral is unique, linearity of the Henstock-Kurzweil integral, Additivity of the Henstock-Kurzweil integral, Cauchy criteria, nonnegativity of Henstock-Kurzweil integral and primitive function. Keywords: Riemann Integral, � � ���� partition, Henstock-Kurzweil Integral. INTRODUCTION We have already mentioned the developments, during the 1630’s, by Fermat and Descrates leading to analytic geometry and the theory of the derivatives. However, the subject we know as calculus did not begin to take shape until the late 1660’s when Issac Newton (1642- 1727) created his theory of fluxions and invented the method of inverse tangents to find areas under curves. The reversal of the process for finding tangent lines to find areas was also discovered in the 1680’s by Leibniz (1646-1716), who was unaware of Newton unpublished work and who arrived at the discovery by a very different route. Leibniz introduced the terminology calculus differential and calculus integral, since finding tangents lines involved differences and finding areas involved summations. Thus they had discovered that integration, being a process of summation, was inverse to the operation of differentiation. During a century and a half of development and refinement of techniques, calculus consisted of these paired operations and their applications, primarily to physical problems. In the 1850s, Bernhard Riemann (1826-1866) adopted a new and different viewpoint. He separated the concept of integration from its companion, differentiation, and examined the motivating summation and limit process of finding areas by itself. He broadened the scope by considering all functions on an interval for which this process of integration could be defined: the class of integrable functions. The fundamental Theorem of calculus became a result that held only for a restricted set of integrable functions. The viewpoint of Riemann led others to invent other integration theories, the most significant being Lebesgue’s theory of integration. The theory of the Riemann integral was not fully satisfactory. Many important functions do not have a Riemann integral even after we extend the class of integrable functions slightly by allowing "improper" Riemann integrals. For example Characteristic function. In 1957, the Czech mathematician Jaroslav Kurzweil discovered a new definition of this integral elegantly similar in nature to Riemann's original definition which he named the gauge integral; the theory was developed by Ralph Henstock. Due to these two important mathematicians, it is now commonly known as Henstock-Kurzweil integral. The simplicity of Kurzweil's definition made some educators advocate that this integral should replace the Riemann integral in introductory calculus courses, but this idea has not gained traction. Concerning the background of the study, the writer formulates the statement of the problems as follows: 1. How does the concept δ-fine partition of Henstock-Kurzweil Integral? 2. How does the definition of Henstock- Kurzweil Integral? 3. How does the fundamental properties of Henstock-Kurzweil Integral? REVIEW OF THE RELATED LITERATURE 1. Supremum and Infimum We start with a straightforward definition similar to many others in this course. Read the Henstock-Kurzweil Integral on [a,b] Jurnal CAUCHY – ISSN: 2086-0382 25 definitions carefully, and note the use of ≤ and ≥ here rather than < and > . Definition 1 Let S be a subset of ℜ 1. A number ℜ∈u is said to be an upper bound of S if us ≤ for all Ss ∈ 2. A number ℜ∈w is said to be a lower bound of S if sw ≤ for all Ss ∈ Definition 2 Let S be a subset of ℜ . 1. If S is bounded above, then an upper bound u is said to be supremum (or a least upper bound) of S if no number smaller than u is an upper bound of S. 2. If S is bounded below, then a lower bound w is said to be infimum (or a greatest lower bound) of S if no number greater than w is a lower bound of S. 2. Limit of Function The essence of the concept of limit for real valued functions of a real variable is this: if L is a real number, then ( ) Lxf xx = → 0 lim means that the value f(x) can be made as close to L as we wish by taking x sufficiently close to x0. This is made precise in the following definition. Figure 1. The limit of f at x0 is L Definition 3 We say that f(x) approaches the limit L as x approaches 0x , and write ( ) Lxf xx = → 0 lim If f is defined on some deleted neighborhood of 0x , and for every 0>ε , there is a 0>δ such that ( ) ε<− Lxf If δ<−< 00 xx 3. Compact Sets Definition 4 A subset K of R is said to be compact if every open cover of K has a finite sub cover. In other words, a set K is compact if, whenever it is contained in the union of a collection G { }αG= of open sets in R , then it is contained in the union of some finite number of sets in G. Theorem 1 If K is a compact subset of R, then K is closed and bounded. Proof: We shall first show that K is bounded. For each � � , let �: ���,��. Since each � is open and since � � � ����� �, we see that the collection � �:� � � is an open cover of K. since K is compact, this collection has a finite sub cover , so there exists � � such that, � � � � � ��� � ���,�� Therefore K is bounded, since it is contained in the bounded interval ���,��. We show that K is bounded, by showing that its complement � ���� is open. To do so, let � ���� be arbitrary and for each � � , we let � ! "# � �:|# � �| % 1 �' (. Since � ) �, we have � � � � ~ �� . Since K is compact, there exists � � such that � � �� ~ �� �� Now it follows from this that � + ,� � 1 �' ,� - 1 �' . /, so that ,� � 1 �' ,� - 1 �' . � ���� . but since u was an arbitrary point in ����, we infer that ���� is open. 4. Continuity Definition 5 a) We say that f is continuous at 0x if f is defined on an open interval ( )ba, containing 0x and ( ).)(lim 0 0 xfxf xx = → b) We say that f is continuous from the left at 0x if f is defined on an open interval ( )0, xa and ( ).)( 00 xfxf =− c) We say that f is continuous from the right at 0x if f is defined on an open interval ( )bx ,0 and ( )00 )( xfxf =+ Siti Nurul Afiyah 26 Volume 2 No. 1 November 2011 Theorem 2 a) A function f is continuous at 0x if and only if f is defined on an open interval ( )ba, containing 0x and for each 0>ε , there is a 0>δ such that ε<− )()( 0xfxf , whenever δ<− 0xx b) A function f is continuous from the right at 0x if and only if f is defined on an open interval [ )bx ,0 and for each 0>ε , there is a 0>δ such that ε<− )()( 0xfxf holds whenever δ+≤≤ 00 xxx c) A function f is continuous from the left at 0x if and only if f is defined on an open interval ( ]0, xa and for each 0>ε , there is a 0>δ such that ε<− )()( 0xfxf holds whenever 00 xxx ≤≤− δ Definition 6 A function f is continuous on an open interval ( )ba, if it is continuous at every point in ( )ba, . if, in addition. ( ) ( )bfbf =− or ( ) ( )afaf =+ Then f is continuous on ( ] [ )baorba ,, , respectively, if f continuous on ( )ba, and ( ) ( )bfbf =− or ( ) ( )afaf =+ both hold, then f is continuous on [ ]ba, . Definition 7 A function f is piecewise continuous on [ ]ba, if a) ( ).0 +xf exist for all 0x in [ )ba, ; b) ( ).0 −xf exist for all 0x in ( ]ba, ; c) ( ) ( ) ( )000 xfxfxf =−=+ for all but finitely many points 0x in ( )ba, . If c) fails to hold at some 0x in ( )ba, , f has a jump discontinuity at 0x . Also, f has a jump discontinuity at a if ( ) ( )afaf ≠=+ or at b if ( ) ( )bfbf ≠− . 5. Uniform Continuity Definition 8 Let RA ⊆ , let RAf →: ,we say that f is uniformly continuous on A if for each 0>ε there is a ( ) 0>εδ such that if x, u A∈ are any number satisfying , then ( ) ( ) ε<− ufxf . Theorem 3 If f is continuous on a closed interval [ ]ba, , then f is uniformly continuous on [ ]ba, . Proof: Suppose that 0>ε . Since f is continuous on [ ]ba, , for each t in [ ]ba, there is a positive number tδ such that ( ) ( ) 2 ε <− tfxf if ttx δ2<− and [ ]bax ,∈ If ( )ttt ttI δδ +−= , , the collection [ ]{ }batIH t ,∈= Is an open covering of [ ]ba, . Since [ ]ba, is compact, the Heine-Borel theorem implies that there are finitely many points nttt ,...,, 21 an [ ]ba, such that nttt III ,...,, 21 cover [ ]ba, . Now define { } nttt δδδδ ,...,,min 21 = We will show that if δ<− 'xx and [ ]baxx ,', ∈ Then ( ) ( ) ε<− 'xfxf From the triangle inequality. ( ) ( ) ( ) ( )( ) ( ) ( )( )'' xftftfxfxfxf rr −+−=− ( ) ( )( ) ( ) ( )( )'xftftfxf rr −+−≤ Since nttt III ,...,, 21 cover [ ]ba, , x must be in one of these intervals. Suppose that rt Ix ∈ ; That is, rtr tx δ<− With rtt = , ( ) ( ) . 2 ε <− rtfxf Henstock-Kurzweil Integral on [a,b] Jurnal CAUCHY – ISSN: 2086-0382 27 Such that, ( ) ( ) .2''' rr ttrrr txxxtxxxtx δδγ ≤+<−+−≤−+−=− Therefore with rtt = and x replaced by x’ implies that ( ) ( ) . 2 ' ε <− rtfxf This imply that ( ) ( ) ε<− 'xfxf . Definition 9 (Lipschitz Functions) Let RA ⊆ , let RAf →: . If there exist a constant 0>K such that ( ) ( ) uxKufxf −≤− For all Aux ∈, , then f is said to be a Lipschitz Functions on A Theorem 4 If RAf →: is a Lipschitz Functions, then f is uniform continuous on A. Proof: If the a Lipschitz conditions satisfied with constant K, then given 0>ε , we can take K εδ =: . If Aux ∈, satisfy δ<− ux , then ( ) ( ) εε =⋅<− K Kufxf Therefore f is uniformly continuous on A. 6. Upper And Lower Integral Definition 10 If f is bounded on [ ]ba, and { }nxxxP ,...,, 10 is a partition of [ ]ba, , let ( )xfM jj xxx j ≤≤− = 1 sup And ( )xfm jj xxx j ≤≤− = 1 inf The upper sum of f over P is ( ) ( )∑ = −−= n j jjj xxMPS 1 1 And the upper integral of f over [ ]ba, , denoted by ( )dxxfb a∫ −−−− Is the infimum of all upper sums. The lower sum of f over P is ( ) ( )∑ = −−= n j jjj xxmPs 1 1 And the lower integral of f over [ ]ba, , denoted by ( )dxxfb a∫ −−− Is the supremum off all lower sums. 7. Riemann Integral Riemann integral, defined in 1854, was the first of the modern theories of integration and enjoys many of the desirable properties of an integration theory. The groundwork for the Riemann integral of a function f over the interval [ ]ba, begins with dividing the interval into smaller subintervals. With infimum and suprimum taken include all partitions P on [ ]ba, , if the upper integral and lower integral same, then f can be said integrable on [ ]ba, . And called Riemann function f on [ ]ba, and denoted by [ ]baf ,∈ This same value is called the Riemann integral function f on [ ]baf ,∈ and written ( ) ( )dxxfR b a ∫ Definition 11 Let [ ] ℜ⊂ba, . A partition of [ ]ba, is a finite set of numbers { }nxxxP ,...,, 10= such that bxax n == ,0 and ii xx <−1 for ni ,...,2,1= . For each subinterval [ ]ii xx ,1− , define its length to be [ ]( ) 11 , −− −= iiii xxxxℓ . The mesh of the partition is then the length of the largest subinterval, [ ]ii xx ,1− : ( ) { }nixxP ii ,...,2,1:max 1 =−= −µ Thus the point { }nxxx ,...,, 10 form an increasing sequence of numbers in [ ]ba, that divides the interval [ ]ba, into contiguous subintervals. Let [ ] ℜ→baf ,: , { }nxxxP ,...,, 10= be a partition of [ ]ba, , and [ ]iii xxt ,1−∈ for each i. Riemann began by considering the approximating (Riemann) sums { }( ) ( )( ),,, 1 1 1 − = = −= ∑ ii n i i n ii xxtftPfS Siti Nurul Afiyah 28 Volume 2 No. 1 November 2011 Defined with respect to the partition P and the set of sampling points { }n ii t 1= . Riemann considered the integral of f over [ ]ba, to be a “limit” of the sums { }( ),,, 1 n ii tPfS = in the following sense. Definition 12 A function [ ] ℜ→baf ,: is Riemann integrable over [ ]ba, if there is an ℜ∈A such that for all 0>ε there is a 0>δ so that if P is any partition of [ ]ba, with ( ) δµ

ε there exists [ ] +ℜ→ba,:δ such that for every δ-fine Henstock-Kurzweil partition [ ]( ){ }n iiii vuD 1 ,, == ξ of [ ]ba, , we have ( )( ) εξ <−−∑ = Auvf ii n i i 1 We denote the Henstock-KurzweilIntegral (also write as HK-integral) A by ( ) ( )dxxfHK b a∫ . Example 2 Define [ ] ℜ→1,0:f the Dirichlet’s function (= the characteristic function of the rational numbers in [ ]1,0 ), by ( )    ∉ ∈ = Qxif Qxif xf 0 1 Then ( )xf is Henstock-Kurzweil integrable on [ ]1,0 . And ( ) 0 1 0 =∫ xf To Prove this assertion, we will define an appropriate gauge εδ , First we enumerate these rational numbers as ,..., 21 rr . We define Henstock-Kurzweil Integral on [a,b] Jurnal CAUCHY – ISSN: 2086-0382 29 ( ) ,...2,12 1 == −− iforr ii εδ , and if [ ]1,0∈x is irrational we define ( ) 1=ξδ ; clearly εδ is a gauge on [ ]1,0 . If P is a δ-fine tagged partition, there can be at most two subintervals in P that have the number ir as tag, and the length of each of those subintervals is 12 −−≤ iε . Hence the contribution to ( )PfS , from subintervals with tag ir is i−≤ 2ε . Since the terms in ( )PfS , with tags at irrational points contribute 0, we readily see that ( ) εε =<≤ ∑ ∞ =1 2 ,0 i iPfS Since 0>ε is arbitrary, this shows that ( )xf is Henstock-Kurzweil integrable on [ ]1,0 . And ( ) 0 1 0 =∫ xf 3. Fundamental Properties Of Henstock- Kurzweil Integral Theorem 5 (Unique Property) if f is Henstock-Kurzweil integrable over [ ]ba, , then the value of the integral is unique. Proof: Suppose that f is Henstock-Kurzweil integrable on [ ]ba, and both real number A and B satisfy Definition 3.2.1. Fix 0>ε choose Aδ and Bδ corresponding to A and B, respectively, in the definition with 2 ' ε ε = . Let ( )BA δδδ ,min= and suppose for every δ-fine Henstock-Kurzweil partition [ ]( ){ }n iiii vuD 1 ,, == ξ of [ ]ba, , Then ( )( ) ( )( ) ( )( ) ( )( ) εεε ξξ ξξ =+< −−+−−≤       −−−      −−=− ∑∑ ∑∑ == == '' 11 11 BuvfuvfA AuvfBuvfBA n i iii n i iii n i iii n i iii Since ε was arbitrary, it follows that BA = . Thus, the value of the integral is unique. Theorem 6 (Linearity of the Henstock- Kurzweil integral) If f and g are Henstock-Kurzweil integrable on [ ]ba, , then so are gf + and fα where α is real. Furthermore, ( ) ∫ ∫∫ +=+ b a b a b a gfgf and ( ) ∫∫ = b a b a ff αα Proof: Let A and B denote respectively the integrals of f and g on [ ]ba, . given 0>ε , there is a ( ) 01 >ξδ such that for any δ1-fine division [ ]( )ξ;, vuD = we have ( )( ) 2 ε ξ <−−∑ Auvf Similarly, there is a ( ) 02 >ξδ such that for any δ2- fine division [ ]( )ξ;, vuD = we have ( )( ) 2 ε ξ <−−∑ Buvg Now put ( ) ( ) ( )( )ξδξδξδ 21 ,min= . Note that any δ-fine division is also δ1-fine and δ2-fine. Therefore for any δ-fine division [ ]( )ξ;, vuD = we have ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) f g v u A B f v u A g v u B ξ ξ ξ ξ ε + − − + ≤ − − + − − < ∑ ∑ ∑ The proof is complete. Example 3 valuate ∫ + 1 0 gf where ( ) 2xxf = and ( ) xxg = . Solution: By theorem 3.3.3, dxxdxxdxxxgf ∫∫∫∫ +=+=+ 1 0 1 0 2 1 0 2 1 0 Now, we evaluate value dxx∫ 1 0 2 Consider a division 1...0 10 =<<<= nxxx and { }nξξξ ,...,, 21 . And this time choose the intermediate points ( ) 2 1 2 11 2 3 1       ++= −− iiiii xxxxξ , then ( ) ( ) ( ) iiiiiiii xxxxxxxx =<      ++<=≤ −−−− 2 1 2 2 1 2 11 22 1 2 11 3 1 0 For ni ,,2,1 ⋅⋅⋅= ; that is ( )iii xx ,1−∈ξ for each i . So 3 1 1 0 2 =∫ dxx And then , we evaluate value dxx∫ 1 0 Consider a division 1...0 10 =<<<= nxxx and { }nξξξ ,...,, 21 . And this time choose the points Siti Nurul Afiyah 30 Volume 2 No. 1 November 2011 ( )1 2 1 −+= iii xxξ , Clearly [ ]iii xx ,1−∈ξ For ni ,,2,1 ⋅⋅⋅= Now ( ) ( )( ) ( )( ) ( ) ( ) 2 1 1 2 1 1 2 1 2 1 2 1 2 1 , 2 0 2 1 2 1 2 1 1 11 1 =⋅= ⋅= −= −= −+=−= ∑ ∑∑ = − − = −− = n n n i ii ii n i iiii n i i xx xx xxxxxxfDfS ξ So, 2 1 1 0 =∫ dxx that, 6 5 2 1 3 1 1 0 1 0 2 1 0 2 1 0 =+=+=+=+ ∫∫∫∫ dxxdxxdxxxgf Theorem 7 (Additivity of the henstock- Kurzweil Integral) Let bca << . If f is Henstock-Kurzweil integrable on [ ]ca, and on [ ]bc, and ∫∫∫ += b c c a b a fff Proof: Let A denote the integral of f on [ ]ca, and B that of f on [ ]bc, . Given 0>ε , there is a ( ) 01 >ξδ , defined on [ ]ca, , such that for any δ1-fine division [ ]( )ξ;, vuD = of [ ]ca, we have ( )( ) 2 ε ξ <−−∑ Auvf Similarly, there is a ( ) 02 >ξδ defined on [ ]bc, such that for any δ2-fine division 0 �1�,23;5� of [ ]bc, we have ( )( ) 2 ε ξ <−−∑ Buvf Define ( ) ( )( )ξξδξδ −= c,min 1 when [ )ca,∈ξ , ( )( )c−ξξδ ,min 2 when ( ]bc,∈ξ , and ( ) ( )( )cc 21 ,min δδ when c−ξ . Note for any δ-fine division D of 16,73, c is always a division point of D. therefore for any δ-fine division 0 �1�,23;5� of 16,73 with Σ over D, writing 21 Σ+Σ=Σ where 1 Σ is the partial sum over [ ]ca, and 2 Σ over [ ]bc, we have ( ) ( ) ( ) ( ) ( ) ( ) ( )1 2 f v u A B f v u A f v u B ξ ξ ξ ε − − + ≤ − − + − − < ∑ ∑ ∑ Hence f is Henstock-Kurzweil integrable to A+B on [ ]ba, . Alternatively, Let [ ]ca,χ denote the characteristic function of [ ]ca, and [ ]caff ,1 χ= . Similarly, let [ ]bcff ,2 χ= . Then it follows from Theorem 3.3.1 that ( ) ∫ ∫∫∫ +=+= c a b c b a b a fffff 21 Example 7 Let ( ) xxf = and Let 1 2 1 0 << .and, If f is Henstock-Kurzweil integrable on [ ]      = 2 1 ,0, ca and on [ ]      = 1, 2 1 , bc and ∫∫∫ += b c c a b a fff Solution: Consider a division 1...0 10 =<<<= nxxx and { }nξξξ ,...,, 21 . And this time choose the points ( )1 2 1 −+= iii xxξ , Clearly [ ]iii xx ,1−∈ξ For ni ,,2,1 ⋅⋅⋅= Now ( ) ( )( ) ( )( ) ( ) ( ) ( )22 2 0 2 1 2 1 2 1 1 11 1 2 1 2 1 2 1 2 1 , ab xx xx xxxxxxfDfS n n i ii ii n i iiii n i i −= −= −= −+=−= ∑ ∑∑ = − − = −− = ξ With same procedure we get; on [ ]      = 2 1 ,0, ca Henstock-Kurzweil Integral on [a,b] Jurnal CAUCHY – ISSN: 2086-0382 31 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 2 2 1 1 2 2 0 2 2 1 , 2 1 2 1 2 1 2 n n i i i i i i i i i n i i i n S f D f x x x x x x x x x x c a ξ − − − = = − = = − = + − = − = − = − ∑ ∑ ∑ So 0>ε , there is a ( ) 01 >ξδ , defined on 10, � : 3, such that for any δ1-fine division 0 �1�,23;5� of 10, �: 3 we have ( ) 28 1 2 1 22 ε<−− ac And on 1;,73 1�: ,13, we get ( ) ( )( ) ( )( ) ( ) ( ) ( )22 2 0 2 1 2 1 2 1 1 11 1 2 1 2 1 2 1 2 1 , cb xx xx xxxxxxfDfS n n i ii ii n i iiii n i i −= −= −= −+=−= ∑ ∑∑ = − − = −− = ξ 0>ε , there is a ( ) 02 >ξδ , defined on 1�: ,13, such that for any δ1-fine division 0 �1�,23;5� of 1�: ,13 we have ( ) 28 3 2 1 22 ε<−− cb therefore for any δ-fine division 0 �1�,23;5� of 10,13 with Σ over D, writing 21 Σ+Σ=Σ where 1Σ is the partial sum over [ ]ca, and 2 Σ over 1�: ,13 we have ( ) ( ) ( ) ( ) ( ) 2 2 1 2 1 1 3 2 8 8 1 3 8 8 b a f v u f v uξ ξ ε  − − + ≤    − − + − − <∑ ∑ Hence f is Henstock-Kurzweil integrable to � : on 10,13 Lemma 8 (Cauchy Criteria) A function is Henstock-kurzweil integrable on [ ]ba, if and only if for every 0>ε , there is a ( ) 0>ξδ such that for any δ-fine division [ ]( )ξ;, vuD = and [ ]( )';','' ξvuD = we have ( )( ) ( )( ) εξξ <−−− ∑∑ ''' uvfuvf Where the first sum is over D and the second over D’. Proof ( )⇒ we will prove that if A function is Henstock- kurzweil integrable on [ ]ba, Then for every 0>ε , there is a ( ) 0>ξδ such that for any δ- fine division [ ]( )ξ;, vuD = and [ ]( )';','' ξvuD = we have ( )( ) ( )( ) εξξ <−−− ∑∑ ''' uvfuvf A function is Henstock-kurzweil integrable on [ ]ba, Then for every 0>ε , there is a ( ) 0>ξδ such that for any δ-fine division [ ]( )ξ;, vuD = and [ ]( )';','' ξvuD = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ' ' ' ' ' 'f v u f v u f v u A A f v u f v u A A f v u ξ ξ ξ ξ ξ ξ ε − − − = − − + − − ≤ − − + − − < ∑ ∑ ∑ ∑ ∑ ∑ Analogous to the situation for real-valued sequences, the condition that ( )( ) ( )( ) εξξ <−−− ∑∑ ''' uvfuvf ( )⇐ We have already proved that the integrability of f implies the Cauchy criterion. So, assume the Cauchy criterion holds. We will prove that f is Henstock-kurzweil integrable . if for every 0>ε , there is a ( ) 0>ξδ such that for any δ-fine division [ ]( )ξ;, vuD = and [ ]( )';','' ξvuD = we have ( )( ) ( )( ) εξξ <−−− ∑∑ ''' uvfuvf Then A function is Henstock-Kurzweil integrable on [a,b]. For each Ν∈k , choose a 0>kδ so that for any two division [ ]( )ξ;, vuD = and [ ]( )';','' ξvuD = , and corresponding sampling points, we have ( )( ) ( )( ) k uvfuvf 1 ''' <−−− ∑∑ ξξ Replacing k δ by { }kδδδ ,...,,min 21 , we may assume that 1+ ≥ kk δδ . Next for each k, fix a partition [ ]( )kkkk vuD ξ;,= and set of sampling point { }n ii 1=ξ . Note for kj > Thus ( )( ) ( )( ) { }kj uvfuvf jjjkkk ,min 1 <−−− ∑∑ ξξ , Siti Nurul Afiyah 32 Volume 2 No. 1 November 2011 Which implies that sequence ( )( )∑ ∞ = − 1k kkk uvf ξ is a Cauchy sequence in R, and hence converges. Let A be a limit of this sequence. it follows from the previous inequality that ( )( ) k Auvf 1 <−−∑ ξ It remains to show that A satisfies Definition 3.2.1 Fix 0>ε and let division [ ]( )ξ;, vuD = . Then ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ε ξξξ ξξξ ξ =+< −−+−−−≤ −−+−−−= −− ∑∑∑ ∑∑∑ ∑ === === kk Auvfuvfuvf Auvfuvfuvf Auvf n k kkk n k kkk n i iii n k kkk n k kkk n i iii 11 111 111 It now follows that f is Henstock-Kurzweil integrable on [ ]ba, Theorem 9 If f is Henstock-Kurzweil integrable on [ ]ba, , then so it is on a subinterval [ ]dc, of [ ]ba, . Proof : Since f is Henstock-Kurzweil integrable on [ ]ba, , the Cauchy condition holds. Take any two δ-fine divisions of [ ]dc, . say D1 and D2, and denote by s1 and s2 respectively the Riemann sums of f over D1 and D2. similarly, take another δ-fine division D3 of [ ] [ ]bdca ,, ∪ and denote by s3 the corresponding Riemann sums. Then the union 31 DD ∪ forms a δ-fine division of [ ]ba, . Here the division points and associated points of 31 DD ∪ are the union of those from D1 and D3. The Riemann sum of f over 31 DD ∪ is 31 ss + . And similarly that over 32 DD ∪ is 32 ss + .Therefore by the Cauchy condition we have ( ) ( ) ε<+−+≤− 323121 ssssss Hence the result follows from lemma 3.3.7 with [ ]ba, replaced by [ ]dc, Theorem 10 (Nonnegativity of The Henstock- Kurzweil integral) If f and g are Henstock-Kurzweil integrable on [ ]ba, and if ( ) ( )xgxf ≤ for almost all in x in [ ]ba, , then ∫∫ ≤ b a b a gf Proof: In view of theorem 3.3.9 we may assume that ( ) ( )xgxf ≤ for all x. Given 0>ε , as in the proof of theorem 3.2.1, there is a ( ) 0>ξδ such that for any δ-fine division [ ]( )ξ;, vuD = we have ( )( ) εξ <−− ∫∑ b a fuvf , ( )( ) εξ <−− ∫∑ b a guvg It follows that ( )( ) ( )( ) εξξε +<−<−<− ∫∑∑∫ b a b a guvguvff Since ε is arbitrary, we have the required inequality. Theorem 11 If f is Henstock-Kurzweil integrable on [ ]ba, with the primitive F, then for every 0>ε , there is a ( ) 0>ξδ such that for any δ-fine division [ ]( )ξ;, vuD = we have ( ) ( ) ( )( ) εξ <−−−∑ uvfuFvF We shall make a few remarks. Before proof, from the computational point of view, we may regard ( )( )uvf −ξ as an approximation of ( ) ( )uFvF − . Then the difference ( ) ( ) ( )( )uvfuFvF −−− ξ is an error. The definition of the Henstock-Kurzweil integral says that the absolute error is also small, whereas Henstock’Lemma. In fact, the two are equivalent by theorem 3.3.8. Another way of putting it is that taking any partial sum 1Σ of Σ we still have ( ) ( ) ( ) ( )1F v F u f v uξ εΣ − − − < That is to say, the selected error is again small, and indeed it is equivalent to the above two. Proof: Given 0>ε , there is a ( ) 0>ξδ such that for any δ-fine division [ ]( )ξ;, vuD = we have ( ) ( ) ( )( ) 4 εξ <−−−Σ uvfuFvF Let 1Σ be a partial sum of Σ and E1 the union of [ ]vu, from 1Σ . Suppose E2. thus we can choose a δ-fine division [ ]( )ξ;,2 vuD = of E2 such that Henstock-Kurzweil Integral on [a,b] Jurnal CAUCHY – ISSN: 2086-0382 33 ( ) ( ) ( )( ) 42 εξ <−−−Σ uvfuFvF Where 2Σ is over D2. Now writing 213 Σ+Σ=Σ we have ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) 1 3 2 2 F v F u f v u F v F u f v u F v F u f v u ξ εξ ξ Σ − − − ≤ Σ − − − +Σ − − − < Consequently the result follows. Example 8 Let ( ) x xf 1= for 10 ≤< x . Given 0>ε , we shall construct ( )ξδ so that f is Henstock- Kurzweil integrable on [ ]1,0 . Consider a division 1...0 10 =<<<= nxxx and { }nξξξ ,...,, 21 With 01 =ξ and iii xx ≤≤− ξ1 for ni ,...,2= . Note that the primitive of x 1 is x2 . Then we can write ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 2 1 1 1 2 2 2 2 2 1 1 . n i i i i n i i i x i n i i i i i f x x x dx x x x x x x x x ξ ξ − = − = − − = − − ≤ − − + − − ≤ + − − ∑ ∑∫ ∑ We shall prove that above is less than ε for suitable fine−δ divisions. Suppose ( ) ξξδ c− for 10 ≤< ξ and 210 << c so that 01 =ξ always. If the above division is fine−δ and [ ]vu, is a typical interval [ ]ii xx ,1− in the division with 0≠u and vu ≤≤ ξ , then ( ) cvuv 220 ≤<−< ξδ , re-arranging we get ( )cuv 211 −≤ , and finally ( ) ( ).2122 ccucvuvuv −≤<− Now choose c so that 210 << c and ( ) 2212 ε≤− cc . In addition, put ( ) 160 2εδ ≤ . Then for the given fine−δ division the above inequality is less than ( ) ( ) εδ <− − + ∑ = − n i ii xx c c 2 1 21 2 02 For example, when 10 ≤< ε we may choose 6 ε =c . Hence the function is Henstock-Kurzweil integrable on [ ]1,0 . CONCLUSION From the discussion we get conclusion that: 1. δ-fine Henstock-Kurzweil partition [ ]( ){ }n iiii vuD 1 ,, == ξ we write ( ) ( )( )ii n i i uvfDfS −= ∑ =1 , ξ where D be a finite collection of interval-point pairs [ ]( ){ }n iiii vu 1 ,, =ξ , where [ ]( ){ } n iii vu 1 , = are non-overlapping subintervals of [ ]ba, . Let ( )ξδ be a positive function on [ ]ba, , ( ) [ ] +ℜ→baei ,:.. ξδ . And if [ ] ( )( ) ( ) ( )( )iiiiiiiii Bvu ξδξξδξξδξξ +−=⊂∈ ,,, for all ni ,...,3,2,1= . 2. A function [ ] ℜ→baf ,: is said to be Henstock-Kurzweil integrable on [ ]ba, if there exists a real number fS such that for every 0>ε there exists [ ] +ℜ→ba,:δ such that for every δ-fine Henstock-Kurzweil partition [ ]( ){ }n iiii vuD 1 ,, == ξ of [ ]ba, , we have ( ) ., ε<− fSDfS 3. And the fundamental properties of Henstock- Kurzweil integral as follows: value of the Henstock- Kurzweil integral is unique, linearity of the Henstock-Kurzweil integral, Additivity of the Henstock-Kurzweil integral, Cauchy criteria, nonnegativity of Henstock- Kurzweil integral, and primitive function. BIBLIOGRAPHY [1] Ding and G.J. Ye., (2009). Generalized Gronwall-Bellman Inequalities Using the Henstock-Kurzweil Integral, Southeast Asian Bulletin of Mathematic(2009)33: 103-713. [2] Douglas S Kurtz, Charles W Swartz. (2004). Theories Of Integration; The integrals of Riemann, Lebesgue, Henstock-Kurzweil and Mc. Shane. London: World Scientific Publishing Co.Pc.Ltd. [3] G. Bartle, Robert. And Donald R. Sherbert. (1994). Introduction To Real Analysis, 2nd Edition. New York: Wiley. [4] Gordon, Russel A. (1955). The integrals of Lebesgue, Denjoy, Perron, and Henstock, Lyrary of Congress Cataloging in Publication Data. American Mathematical Society. Siti Nurul Afiyah 34 Volume 2 No. 1 November 2011 [5] Hutahean, effendi. (1989). Fungsi Riil. ITB: Bandung [6] Lee, P.-Y. (1989). Lanzhou Lectures on Henstock . Singapore: World Scientific. [7] Mark Bridger, Real Analysis, A Constructive Approach, Northeastern University. Department of Mathematics Boston, M.A, A John Willey and Sons, Inc., Publication [8] Przynski, William and Philip. (1987). Introduction to Mathematical Analysis. mcGraww Hill International. [9] Slavik, Antonin and Stefan Schwabik, Praha. (2006). Henstock-Kurzweil and McShane product integration; Descriptive definitions. October 23, 2006 [10] Rahman, Hairur. (2008). Pengantar Analisis Real. Malang: UIN Malang.