The one-dimensional Henstock-Kurzweil integral

Chpter 1 The one-dimensionl Henstock-Kurzweil integrl 1.1 Introduction nd Cousin s Lemm The purpose o this monogrph is to study multiple Henstock-Kurzweil integrls. In the present chpter, we shll irst present nd prove certin results or the one-dimensionl Henstock-Kurzweil integrl. Unless mentioned otherwise, the ollowing conventions nd nottions will be used throughout this monogrph. R, R +, nd N denote the rel line, the positive rel line, nd the set o positive integers respectively. An intervl in R is set o the orm [α,β], where < α < β <, nd [,b] denotes ixed intervl in R. Deinition 1.1.1. (i) Two intervls [u,v], [s,t] in R re sid to be non-overlpping i (u,v) (s,t) =. (ii) I {[u 1,v 1 ],...,[u p,v p ]} is inite collection o pirwise non-overlpping subintervls o [,b] such tht [,b] = p [u k,v k ], we sy tht {[u 1,v 1 ],...,[u p,v p ]} is division o [,b]. (iii) A point-intervl pir (t,[u,v]) consists o point t R nd n intervl [u,v] in R. Here t is known s the tg o [u,v]. (iv) A Perron prtition o [,b] is inite collection {(t 1,[u 1,v 1 ]),...,(t p,[u p,v p ])} o point-intervl pirs, where {[u 1,v 1 ],...,[u p,v p ]} is division o [,b], nd t k [u k,v k ] or k = 1,...,p. (v) A unction δ : [,b] R + is known s guge on [,b]. (vi) Let δ be guge on [,b]. A Perron prtition {(t 1,[u 1,v 1 ]),...,(t p,[u p,v p ])} o [,b] is sid to be δ-ine i [u k,v k ] (t k δ(t k ),t k +δ(t k )) or k = 1,...,p. 1

2 HENSTOCK-KURZWEIL INTEGRATION ON EUCLIDEAN SPACES The ollowing exmple shows tht tgs ply n importnt role in our study. Exmple 1.1.2. We deine guge δ on [0,1] by setting { t i 0 < t 1, δ(t) = 1 2 i t = 0. Then the ollowing sttements re true. (i) {[0, 1 3 ],[1 3, 1 2 ],[1 2,1]} is division o [0,1]. (ii) {(0,[0, 1 3 ]),(1 2,[1 3, 1 2 ]),(1,[1 2,1])} is δ-ine Perron prtition o [0,1]. (iii) {(0,[0, 1 3 ]),(1 2,[1 3, 1 2 ]),(1 2,[1 2,1])} is not δ-ine Perronprtition o[0,1]. The ollowing nturl question rises rom Exmple 1.1.2. Question 1.1.3. I δ is n rbitrry guge on [,b], is it possible to ind δ-ine Perron prtition o [, b]? In order to proceed urther, we need the ollowing result. Lemm 1.1.4. Let δ be guge on [,b] nd let < c < b. I P 1 nd P 2 re δ-ine Perron prtitions o [,c] nd [c,b] respectively, then P 1 P 2 is δ-ine Perron prtition o [,b]. Proo. Exercise. The ollowing theorem gives n irmtive nswer to Question 1.1.3. Theorem 1.1.5 (Cousin s Lemm). I δ is guge on [,b], then there exists δ-ine Perron prtition o [,b]. Proo. Proceeding towrds contrdiction, suppose tht [, b] does not hve δ-ine Perron prtition. We divide [,b] into [, +b 2 ] nd [+b 2,b] so tht [,b] is the union o two non-overlpping intervls in R. In view o Lemm 1.1.4, we cn choose n intervl [ 1,b 1 ] rom the set {[, +b 2 ],[+b 2,b]} so tht [ 1,b 1 ] does not hve δ-ine Perron prtition. Using induction, we construct intervls [ 1,b 1 ],[ 2,b 2 ],... in R so tht the ollowing properties re stisied or every n N: (i) [ n,b n ] [ n+1,b n+1 ]; (ii) no δ-ine Perron prtition o [ n,b n ] exists; (iii) b n n = b 2 n.

The one-dimensionl Henstock-Kurzweil integrl 3 Since properties (i) nd (iii) hold or every n N, it ollows rom the Nested Intervl Theorem [6, Theorem 2.5.3] tht [ k,b k ] = {t 0 } or some t 0 R. On the other hnd, since [ k,b k ] = {t 0 } nd δ(t 0 ) > 0, it ollows rom property (iii) tht there exists N N such tht {(t 0,[ N,b N ])} is δ-ine Perron prtition o [ N,b N ], contrdiction to (ii). This contrdiction completes the proo. Let C[, b] denote the spce o rel-vlued continuous unctions on [, b]. A simple ppliction o Theorem 1.1.5 gives the ollowing clssicl result. Theorem 1.1.6. I C[,b], then is uniormly continuous on [,b]. Proo. Let ε > 0 be given. Using the continuity o on [,b], or ech x 0 [,b] there exists δ 0 (x 0 ) > 0 such tht (x) (x 0 ) < ε 2 whenever x (x 0 δ 0 (x 0 ),x 0 +δ 0 (x 0 )) [,b]. We wnt to prove tht there exists η > 0 with the ollowing property: s,t [,b] with s t < η = (s) (t) < ε. Deine guge δ on [,b] by setting δ = 1 2 δ 0. In view o Cousin s Lemm, we my select nd ix δ-ine Perron prtition {(t 1,[u 1,v 1 ]),...,(t p,[u p,v p ])} o [,b]. I s,t [,b] with t s < η := min{δ(t i ) : i = 1,...,p}, then there exists j {1,...,p} such tht t t j < δ(t j ) nd so s t j s t + t t j < 2δ(t j ) = δ 0 (t j ). Thus (t) (s) (t) (t j ) + (t j ) (s) < ε. Thereore, is uniormly continuous on [, b]. Following the proo o Theorem 1.1.6, we obtin the ollowing corollry. Corollry 1.1.7. I C[,b], then is bounded on [,b].

4 HENSTOCK-KURZWEIL INTEGRATION ON EUCLIDEAN SPACES 1.2 Deinition o the Henstock-Kurzweil integrl Let P = {(t 1,[u 1,v 1 ]),...,(t p,[u p,v p ])} be Perron prtition o [,b]. I is rel-vlued unction deined on {t 1,...,t p }, we write p S(,P) = (t i )(v i u i ). i=1 We irst deine the Riemnn integrl. Deinition 1.2.1. A unction : [,b] R is sid to be Riemnn integrble on [,b] i there exists A 0 R with the ollowing property: given ε > 0 there exists constnt guge δ on [,b] such tht or ech δ-ine Perron prtition P o [,b]. S(,P) A 0 < ε (1.2.1) The collection o ll unctions tht re Riemnn integrble on [, b] will be denoted by R[,b]. Once we omit the word constnt rom Deinition 1.2.1, we obtin the ollowing modiiction o the Riemnn integrl. Deinition 1.2.2. A unction : [,b] R is sid to be Henstock- Kurzweil integrble on [,b] i there exists A R with the ollowing property: given ε > 0 there exists guge δ on [,b] such tht or ech δ-ine Perron prtition P o [,b]. S(,P) A < ε (1.2.2) The collection o ll unctions tht re Henstock-Kurzweil integrble on [,b] will be denoted by HK[,b]. It is esy to deduce rom Deinitions 1.2.1 nd 1.2.2 tht i R[,b], then HK[,b]. In this cse, Cousin s Lemm shows tht there is unique number stisying Deinitions 1.2.1 nd 1.2.2. Theorem 1.2.3. There is t most one number A stisying Deinition 1.2.2. Proo. Suppose tht A 1 nd A 2 stisy Deinition 1.2.2. We clim tht A 1 = A 2. Let ε > 0 be given. Since A 1 stisies Deinition 1.2.2, there exists guge δ 1 on [,b] such tht S(,P 1 ) A 1 < ε 2

The one-dimensionl Henstock-Kurzweil integrl 5 or ech δ 1 -ine Perron prtition P 1 o [,b]. Similrly, there exists guge δ 2 on [,b] such tht S(,P 2 ) A 2 < ε 2 or ech δ 2 -ine Perron prtition P 2 o [,b]. Deine guge δ on [,b] by setting δ(x) = min{δ 1 (x),δ 2 (x)}. (1.2.3) According to Cousin s Lemm (Theorem 1.1.5), we my ix δ-ine Perron prtition P o [, b]. Since (1.2.3) implies tht the δ-ine Perron prtition P is both δ 1 -ine nd δ 2 -ine, it ollows rom the tringle inequlity tht A 1 A 2 S(,P) A 1 + S(,P) A 2 < ε. Since ε > 0 is rbitrry, we conclude tht A 1 = A 2. Theorem 1.2.3 tells us tht i HK[,b], then there is unique number A stisying Deinition 1.2.2. In this cse the number A, denoted by, (x) dx or (t) dt, is known s the Henstock- Kurzweil integrl o over [,b]. It is cler tht i R[,b], then HK[,b] nd the number stisies Deinition 1.2.1. In this cse the unique number, denoted by, (x) dx or (t) dt, is known s the Riemnn integrl o over [,b]. The ollowing exmple shows tht the inclusion R[,b] HK[,b] is proper. Exmple 1.2.4. Let Q be the set o ll rtionl numbers, nd deine the unction : [0,1] R by setting Then HK[0,1]\R[0,1]. 1 i x [0,1] Q, (x) = 0 otherwise. Proo. Let (r n ) n=1 be n enumertion o [0,1] Q nd let ε > 0. We deine guge δ on [0,1] by setting ε 2 i x = r n+1 n or some n N, δ(x) = 1 i x [0,1]\Q.

6 HENSTOCK-KURZWEIL INTEGRATION ON EUCLIDEAN SPACES I P is δ-ine Perron prtition o [0,1], then S(,P) 0 = (t)(v u)+ t Q [0,1] = (t)(v u) < t Q [0,1] = ε. ε 2 k t [0,1]\Q (t)(v u) Since ε > 0 is rbitrry, we conclude tht HK[0,1]nd 1 0 = 0. It remins to prove tht R[0, 1]. Proceeding towrds contrdiction, suppose tht R[0,1]. Since R[0,1] HK[0,1] nd 1 0 = 0, we hve 1 0 = 0. Hence or ε = 1 there exists constnt guge δ 1 on [0,1] such tht S(,P 1 ) < 1 or ech δ 1 -ine Perron prtition P 1 o [0,1]. I q is positive integer stisying q 1 < δ 1, then P 2 := {( (k 1 1)q 1,[(k 1 1)q 1,k 1 q 1 ] ) : k 1 = 1,...,q } is δ 1 -ine Perron prtition o [0,1]. A contrdiction ollows: q 1 > S(,P 2 ) = (kq 1 (k 1)q 1 ) = 1. The ollowing theorem shows tht the one-dimensionl Henstock- Kurzweil integrl is useul or ormulting Fundmentl Theorem o Clculus. Theorem 1.2.5. Let : [,b] R nd let F C[,b]. I F is dierentible on (,b) nd F (x) = (x) or ll x (,b), then HK[,b] nd = F(b) F().

The one-dimensionl Henstock-Kurzweil integrl 7 Proo. Let ε > 0 be given. Since F is continuous on [, b], or ech x [,b] there exists δ 1 (x) > 0 such tht F(x) F(y) < ε 6 whenever y [,b] (x δ 1 (x),x+δ 1 (x)). Since F is dierentible on (,b), or ech x (,b) there exists δ 2 (x) > 0 such tht F ε(v u) (x)(v u) (F(v) F(u)) 3(b ) whenever x [u,v] (,b) (x δ 2 (x),x+δ 2 (x)). Deine guge δ on [,b] by the ormul } min {δ 1 (x),δ 2 (x), 12 (x ), 12 (b x) i < x < b, δ(x) = ε i x {,b}. 6( () + (b) +1) I P is δ-ine Perron prtition o [,b], then S(,P) (F(b) F()) = {(t)(v u) ( F(v) F(u) )} (t)(v u) (F(v) F(u)) t {,b} + <t<b t {,b} t {,b} (t)(v u) (F(v) F(u)) ( (t) (v u)+ F(v) F(u) ) + (t) (v u)+ t {,b} < ε( () + (b) ) 6( () + (b) +1) + 2ε 6 + <t<b <t<b F(v) F(u) + ε(v u) 3(b ) < ε. Since ε > 0 is rbitrry, we conclude tht HK[,b] nd = F(b) F(). ε(v u) 3(b ) <t<b ε(v u) 3(b )

8 HENSTOCK-KURZWEIL INTEGRATION ON EUCLIDEAN SPACES The ollowing exmples re specil cses o Theorem 1.2.5. Exmple 1.2.6. Let x 2 sin 1 x F(x) = 2 i 0 < x 1, 0 i x = 0. Then F is dierentible on [0,1]. In prticulr, F HK[0,1] nd 1 0 F = sin1. Exmple 1.2.7. Let 2 x (x) = 3 2cosx sin 3 i 0 < x π 2 x, 0 i x = 0. Then HK[0, π 2 ] nd π 2 = 2 0 3 4 π 2. The ollowing theorem is consequence o Theorem 1.2.5. Theorem 1.2.8. I : [,b] R is derivtive on [,b], then HK[,b]. We remrk tht the converse o Theorem 1.2.8 is not true. More detils will be given in Section 4.5. 1.3 Simple properties The im o this section is to prove some bsic properties o the Henstock- Kurzweil integrl vi Deinition 1.2.2. Theorem 1.3.1. I,g HK[,b], then +g HK[,b] nd ( +g) = + g.

The one-dimensionl Henstock-Kurzweil integrl 9 Proo. Let ε > 0 be given. Since HK[,b], there exists guge δ 1 on [,b] such tht S(,P 1) < ε 2 or ech δ 1 -ine Perron prtition P 1 o [,b]. Similrly, there exists guge δ 2 on [,b] such tht S(g,P 2) g < ε 2 or ech δ 2 -ine Perron prtition P 2 o [,b]. Deine guge δ on [,b] by setting δ(x) = min{δ 1 (x),δ 2 (x)}, (1.3.1) nd let P be δ-ine Perron prtition o [,b]. Since (1.3.1) implies tht the δ-ine Perron prtition P is both δ 1 -ine nd δ 2 -ine, the identity S( +g,p) = S(,P)+S(g,P) nd the tringle inequlity yield { S( +g,p) + g} S(,P) + S(g,P) < ε. Since ε > 0 is rbitrry, we conclude tht +g HK[,b] nd ( +g) = + Theorem 1.3.2. I HK[,b] nd c R, then c HK[,b] nd { } c = c. Proo. Let ε > 0 be given. Since HK[,b], there exists guge δ on [,b] such tht S(,P 1) < ε c +1 g. g

10 HENSTOCK-KURZWEIL INTEGRATION ON EUCLIDEAN SPACES or ech δ-ine Perron prtition P 1 o [,b]. I P is δ-ine Perron prtition o [,b], then { S(c,P) c } = c S(,P) < c ε c +1 < ε. Since ε > 0 is rbitrry, we conclude tht c HK[,b] nd { } c = c. Theorem 1.3.3. I,g HK[,b] nd (x) g(x) or ll x [,b], then Proo. Let ε > 0 be given. Since HK[,b], there exists guge δ 1 on [,b] such tht S(,P 1) < ε 2 or ech δ 1 -ine Perron prtition P 1 o [,b]. Similrly, there exists guge δ 2 on [,b] such tht S(g,P 2) g < ε 2 or ech δ 2 -ine Perron prtition P 2 o [,b]. Deine guge δ on [,b] by setting g. δ(x) = min{δ 1 (x),δ 2 (x)}, (1.3.2) nd we pply Cousin s Lemm to ix δ-ine Perron prtition P 0 o [,b]. Since (1.3.2) implies tht the δ-ine Perron prtition P 0 is both δ 1 -ine nd δ 2 -ine, it ollows rom the inequlity S(,P 0 ) S(g,P 0 ) tht < S(,P 0 )+ ε 2 S(g,P 0)+ ε 2 < g +ε, nd the desired inequlity ollows rom the rbitrriness o ε. The ollowing theorem gives useul necessry nd suicient condition or unction to be Henstock-Kurzweil integrble on [, b].

The one-dimensionl Henstock-Kurzweil integrl 11 Theorem 1.3.4 (Cuchy Criterion). A unction : [,b] R is Henstock-Kurzweil integrble on [,b] i nd only i given ε > 0 there exists guge δ on [,b] such tht S(,P) S(,Q) < ε (1.3.3) or ech pir o δ-ine Perron prtitions P nd Q o [,b]. Proo. (= ) Let ε > 0 be given. Since HK[,b], there exists guge δ on [,b] such tht S(,P 0) < ε (1.3.4) 2 or ech δ-ine Perronprtition P 0 o[,b]. I P nd Q re two δ-ine Perron prtitions o [, b], the tringle inequlity nd (1.3.4) yield S(,P) S(,Q) S(,P) + S(,Q) < ε. ( =) For ech n N we let δ n be guge on [,b] so tht S(,Q n ) S(,R n ) < 1 n or echpiroδ n -ine Perronprtitions Q n nd R n o[,b]. Next we deine guge n on [,b] by setting n (x) = min{δ 1 (x),...,δ n (x)}, nd pply Cousin s Lemm to ix n -ine Perron prtition P n o [,b]. We clim tht (S(,P n )) n=1 is Cuchy sequence o rel numbers. Let ε > 0 be given nd choose positive integer N so tht 1 N < ε. I n 1 nd n 2 re positive integers such tht min{n 1,n 2 } N, then P n1 nd P n2 re both min{n1,n 2}-ine Perron prtitions o [,b] nd so 1 S(,P n1 ) S(,P n2 ) < min{n 1,n 2 } 1 N < ε. Consequently, (S(,P n )) n=1 is Cuchy sequence o rel numbers. Since R is complete, the sequence (S(,P n )) n=1 converges to some rel number A. It remins to prove tht HK[,b] nd A =. Let P be N -ine Perron prtition o [,b]. Since our construction implies tht the sequence ( n ) n=1 o guges is non-incresing, we see tht the n -ine Perron prtition P n is N -ine or every integers n N. Thus S(,P) A = lim n S(,P) S(,P n) 1 N < ε. Sinceε > 0isrbitrry,weconcludetht HK[,b]ndA =.

12 HENSTOCK-KURZWEIL INTEGRATION ON EUCLIDEAN SPACES We re now redy to give n importnt clss o Henstock-Kurzweil integrble unctions. Theorem 1.3.5. I C[,b], then HK[,b]. Proo. Let ε > 0 be given. Since is continuous on [,b], or ech x [,b] there exists δ(x) > 0 such tht (y) (x) < whenever y [,b] (x δ(x),x+δ(x)). ε 2(b ) Clerly, the unction x δ(x) is guge on [,b]. Let P = {(t 1,[u 1,v 1 ]),...,(t p,[u p,v p ])} nd Q = {(w 1,[x 1,y 1 ]),...,(w q,[x q,y q ])} be two δ-ine Perron prtitions o [,b]. I [u j,v j ] [x k,y k ] is non-empty or some j {1,...,p} nd k {1,...,q}, we select nd ix point z j,k [u j,v j ] [x k,y k ]. On the other hnd, i [u r,v r ] [x s,y s ] is empty or some r {1,...,p} nd s {1,...,q}, we set z r,s =. Let µ 1 ( ) = 0 nd let µ 1 ([α,β]) = β α or ech pir o rel numbers α nd β stisying α β. By the tringle inequlity, S(,P) S(,Q) p q = (t j )(v j u j ) (w k )(y k x k ) j=1 p q q p = (t j )µ 1 ([u j,v j ] [x k,y k ]) (w k )µ 1 ([u j,v j ] [x k,y k ]) j=1 j=1 p q ( (tj ) (z j,k ) ) µ 1 ([u j,v j ] [x k,y k ]) j=1 p q ( + (wk ) (z j,k ) ) µ 1 ([u j,v j ] [x k,y k ]) < ε. j=1 An ppliction o Theorem 1.3.4 completes the proo. The ollowing theorem is lso consequence o Theorem 1.3.4. Theorem 1.3.6. I HK[,b], then HK[c,d] or ech intervl [c,d] [,b].

The one-dimensionl Henstock-Kurzweil integrl 13 Proo. Let [c,d] be proper subintervl o [,b]. For ech ε > 0 we use Theorem 1.3.4 to select guge δ on [,b] such tht S(,P) S(,Q) < ε or ech pir o δ-ine Perron prtitions P nd Q o [,b]. Since [c,d] is proper subintervl o [,b], there exists inite collection {[u 1,v 1 ],...,[u N,v N ]}opirwisenon-overlppingsubintervlso[,b]such tht [c,d] {[u 1,v 1 ],...,[u N,v N ]} nd N [,b] = [c,d] [u k,v k ]. For ech k {1,...,N} we ix δ-ine Perron prtition P k o [u k,v k ]. I P [c,d] nd Q [c,d] re δ-ine Perron prtitions o [c,d], then it is cler tht P [c,d] N P k nd Q [c,d] N P k re δ-ine Perron prtitions o [,b]. Thus S(,P[c,d] ) S(,Q [c,d] ) N N = S(,P [c,d])+ S(,P k ) S(,Q [c,d] ) S(,P k ) N N = S(,P [c,d] P k ) S(,Q [c,d] P k ) < ε. By Theorem 1.3.4, HK[c,d]. Remrk 1.3.7. I HK[,b] nd c [,b], we deine the Henstock- Kurzweil integrl o over {c} to be zero. Theorem 1.3.8. Let : [,b] R nd let < c < b. I HK[,c] HK[c,b], then HK[,b] nd = c + Proo. Given ε > 0 there exists guge δ on [,c] such tht c S(,P ) < ε 2 wheneverp isδ -ine Perronprtitiono[,c]. Asimilrrgumentshows tht there exists guge δ b on [c,b] such tht S(,P b) < ε 2 c c.

14 HENSTOCK-KURZWEIL INTEGRATION ON EUCLIDEAN SPACES or ech δ b -ine Perron prtition P b o [c,b]. Deine guge δ on [,b] by setting min{δ (x),c x} i x < c, δ(x) = min{δ (x),δ b (x)} i x = c, min{δ b (x),x c} i c < x b, nd let P = {(t 1,[u 1,v 1 ]),...,(t p,[u p,v p ])} be δ-ine Perron prtition o [,b]. Since our choice o δ implies tht c = u j = v k or some j,k {1,...,p}, we conclude tht P = P 1 P 2 or some δ-ine Perron prtitions P 1, P 2 o [,c] nd [c,b] respectively. Consequently, { S(,P) S(,P 1) < ε. c c + Since ε > 0 is rbitrry, the theorem ollows. Exercise 1.3.9. c } + S(,P 2) (i) Prove tht i,g R[,b], then +g R[,b] nd ( +g) = + (ii) Prove tht i R[,b] nd c R, then c R[,b] nd c = c (iii) Let : [,b] R nd let < c < b. Prove tht i R[,c] R[c,b], then R[,b] nd = c +. c. g. c 1.4 Sks-Henstock Lemm The im o this section is to estblish the importnt Sks-Henstock Lemm (Theorem 1.4.4) or the Henstock-Kurzweil integrl. As result, we deduce tht there re no improper Henstock-Kurzweil integrls (Theorems 1.4.6 nd 1.4.8). We begin with the ollowing deinitions.

The one-dimensionl Henstock-Kurzweil integrl 15 Deinition 1.4.1. A inite collection {(t 1,[u 1,v 1 ]),...,(t p,[u p,v p ])} o point-intervl pirs is sid to be Perron subprtition o [,b] i t i [u i,v i ] or i = 1,...,p, nd {[u 1,v 1 ],...,[u p,v p ]} is inite collection o nonoverlpping subintervls o [, b]. Deinition 1.4.2. Let {(t 1,[u 1,v 1 ]),...,(t p,[u p,v p ])} be Perron subprtition o [,b] nd let δ be guge on {t 1,...,t p }. The Perron subprtition {(t 1,[u 1,v 1 ]),...,(t p,[u p,v p ])} is sid to be δ-ine i [u i,v i ] (t i δ(t i ),t i +δ(t i )) or i = 1,...,p. By replcing δ-ine Perron prtitions by δ-ine Perron subprtitions in Deinition 1.2.2, we obtin the ollowing Lemm 1.4.3. Let HK[,b] nd let ε > 0. I δ is guge on [,b] such tht (x)(z y) < ε (x,[y,z]) Q or ech δ-ine Perron prtition Q o [,b], then { (t)(v u) or ech δ-ine Perron subprtition P o [, b]. v u } ε (1.4.1) Proo. Let P be δ-ine Perron subprtition o [, b]. I [u,v] = [,b], then (1.4.1) ollows rom Theorem 1.3.8. Henceorth we ssume tht [u,v] [,b]; in this cse, we choose nonoverlpping intervls [x 1,y 1 ],...,[x q,y q ] such tht q [,b] \ (u,v) = [x k,y k ]. For ech k {1,...,q} we iner rom Theorem 1.3.6 tht HK[x k,y k ]. Hence or ech η > 0 there exists guge δ k on [x k,y k ] such tht yk S(,P k) < η q x k or ech δ k -ine Perron prtition P k o [x k,y k ]. Since δ nd δ k re guges on [u k,v k ], we cn pply Cousin s Lemm to select nd ix min{δ,δ k }-ine Perron prtition Q k o [x k,y k ].

16 HENSTOCK-KURZWEIL INTEGRATION ON EUCLIDEAN SPACES Following the proo o Lemm 1.1.4, we conclude tht Q := P q Q k is δ-ine Perron prtition o [,b] such tht q S(,Q) = S(,P)+ S(,Q k ) nd = Consequently, { q = S(,Q) S(,Q) < ε+η. { (t)(v u) v v u } + u } { S(,Q k ) q q + S(,Q k) Since η > 0 is rbitrry, the lemm is proved. yk x k. q vk x k vk x k We re now redy to stte nd prove the ollowing crucil Sks-Henstock Lemm, which plys n importnt role or the rest o this chpter. Theorem 1.4.4 (Sks-Henstock). I HK[,b], then or ech ε > 0 there exists guge on δ on [,b] such tht v (t)(v u) < ε (1.4.2) or ech δ-ine Perron subprtition P o [, b]. Proo. Since is Henstock-Kurzweil integrble on [, b], it ollows rom Lemm 1.4.3 tht there exists guge δ on [,b] such tht { } y (z)(y x) x < ε (1.4.3) 2 (z,[x,y]) Q or ech δ-ine Perron subprtition Q o [,b]. Let P be δ-ine Perron subprtition o [, b], let { v } P + = (t,[u,v]) P : (t)(v u) 0 u u }

The one-dimensionl Henstock-Kurzweil integrl 17 nd let P = P\P +. Then P + P is δ-ine Perron subprtition o [,b] nd hence the desired result ollows rom (1.4.3): v (t)(v u) = < ε. + { (t)(v u) { (t)(v u) u v u v Theorem 1.4.5. Let HK[,b] nd let F(x) = x or ech x [,b]. Then F is continuous on [,b]. Proo. For ech ε > 0 we pply the Sks-Henstock Lemm to select guge δ on [,b] such tht v (t)(v u) < ε 2 or ech δ-ine Perron subprtition P o [,b]. By mking δ smller, we my ssume tht δ(t) or ll t [,b]. I x [,b], then ε 2(1+ (t) ) u u } } F(y) F(x) (x)(y x) (F(y) F(x)) + (x)(y x) < ε whenever y (x δ(x),x +δ(x)) [,b]. Since x [,b] is rbitrry, we conclude tht F is continuous on [, b]. Theorem 1.4.6 (Cuchy extension). A unction : [,b] R is Henstock-Kurzweil integrble on [,b] i nd only i or ech c (,b) the unction [,c] is Henstock-Kurzweil integrble on [,c] nd lim c b c In this cse, = lim c b c. Proo. (= ) This ollows rom Theorem 1.4.5. exists. (1.4.4) ( =) Let ε > 0 nd let (c n ) n=0 be strictly incresing sequence o rel numbers such tht c 0 = nd sup n N c n = b. Since (1.4.4) holds, there exists positive integer N such tht x c lim c b < ε (1.4.5) 4

18 HENSTOCK-KURZWEIL INTEGRATION ON EUCLIDEAN SPACES whenever x [c N,b). For ech k N we let δ k be guge on [c k 1,c k ] so tht the inequlity y (z)(y x) < ε 4(2 k (1.4.6) ) (z,[x,y]) Q k holds or ech δ k -ine Perron subprtition Q k o [c k 1,c k ]. Deine guge δ on [,b] by setting 1 2 (c 1 c 0 ) i x = c 0, min{δ k (c k ),δ k+1 (c k ), 1 2 (c k c k 1 ), 1 2 (c k+1 c k )} i x = c k or some k N, δ(x) = min{δ k (x), 1 2 (x c k 1), 1 2 (c k x)} x i x (c k 1,c k ) or some k N, ε min{b c N, 4( (b) +1) } i x = b, nd let P = {(t 1,[x 0,x 1 ]),...,(t p,[x p 1,x p ])} be δ-ine Perron prtition o [,b]. Ater suitble reordering, we my ssume tht = x 0 < x 1 < < x p = b. Since b [c k 1,c k ], our choice o δ implies tht t p = b nd x p 1 (c r,c r+1 ] or some unique positive integer r. We lso observe tht i k {1,...,r}, our choice o δ implies tht {(t,[u,v]) P : [u,v] [c k 1,c k ]} is δ k -ine Perron prtition o [c k 1,c k ]. Thus c S(,P) lim c b r { ck } (t)(v u) c k 1 [u,v] [c k 1,c k ] xp 1 + (t)(v u) + (b) (b x p 1 ) c r [u,v] [c r,x p 1] xp 1 c + lim c b < ε.

The one-dimensionl Henstock-Kurzweil integrl 19 Corollry 1.4.7. Let : [,b] R nd suppose tht [,c] is Riemnn c integrble on [,c] or every c (,b). I lim c b (t) dt exists, then HK[,b] nd lim c b c = Likewise, we hve the ollowing modiiction o Theorem 1.4.6. Theorem 1.4.8 (Cuchy extension). A unction : [,b] R is Henstock-Kurzweil integrble on [,b] i nd only i or ech c (,b) the unction [c,b] is Henstock-Kurzweil integrble on [c,b] nd lim c + c. exists. In this cse, = lim c + c. The ollowing corollry is n immedite consequence o Theorem 1.4.8. Corollry 1.4.9. Let : [,b] R nd suppose tht [c,b] is Riemnn integrble on [c,b] or every c (,b). I lim c + (t) dt exists, then c HK[,b] nd lim c b c = Exmple 1.4.10. Let 1 1 x i 0 x < 1, (x) = 0 i x = 0. Since is Henstock-Kurzweil integrble on [0,c] or ll c [0,1) nd c ( 1 lim dt = lim 2 2 ) 1 c = 2, c 1 1 t c 1 0 it ollows rom Theorem 1.4.6 tht HK[0,1] nd 1 0 = 2..

20 HENSTOCK-KURZWEIL INTEGRATION ON EUCLIDEAN SPACES Exmple 1.4.11. Let 1 x sin 1 2 x i 0 < x 1, h(x) = 0 i x = 0. It is cler tht h is Henstock-Kurzweil integrble on [c,1] or ll c (0,1). On the other hnd, since the limit 1 1 lim c 0 + x 2 sin 1 ( dx = lim cos1 cos 1 ) x c 0 + c c does not exist, n ppliction o Theorem 1.4.8 shows tht h is not Henstock-Kurzweil integrble on [0, 1]. Exercise 1.4.12. Show tht the Sks-Henstock Lemm remins true or the Riemnn integrl. Further pplictions o the Sks-Henstock Lemm will be given in the subsequent chpters. 1.5 Notes nd Remrks Cousin s Lemm hs been used by Gordon [46] to prove some clssicl results in nlysis. Theorem 1.1.6 is lso due to Gordon [46]. It is known tht R[, b] is liner spce. Further properties o the Riemnn integrl cn be ound in [88, Sections 1.3-1.5] or [6, Chpter 7]. 1n 1957, Kurzweil [71] gve slight but ingenious modiiction o the clssicl Riemnn integrl nd used it in his work on dierentil equtions. Lter, Henstock [55] discovered the integrl independently nd developed the theory urther. This integrl, which is now commonly known s the Henstock-Kurzweil integrl, is lso known s the Henstock integrl, the Kurzweil-Henstock integrl, or the generlized Riemnn integrl; see, or exmple, [4 6, 85, 88]. In dimension one, this integrl is equivlent to the Perron integrl in the ollowing sense: unction which is integrble in one sense is integrble in the other sense nd both integrls coincide; proo o this result is given in [44, Chpter 11]. A good overll view o the theory cn be ound in Bongiorno [10] nd Lee [86]. See lso [133].