Beberapa Sifat Integral Henstock Sekuensial

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1 JURNAL FOURER Otoer 2017, Vol. 6, No. 2, SSN X DO: /fourier E-SSN Beerp Sift ntegrl Hensto Seuensil Mlhyti Progrm Studi Mtemti Fults Sins dn Tenologi, UN Sunn Klijg, Jl. Mrsd Adisuipto No. 1 Yogyrt, ndonesi Korespondensi; Emil: mlhyti_01@yhoo.o.id Astr Peremngn teori integrl hususny integrl Hensto yng terru erhsil diemun oleh Lrmie Pxton, yitu integrl Hensto Seuensil. Pendefinisin ntegrl Hensto Seuensil hmpir sm seperti mendefinisin integrl Hensto, edny pd integrl Hensto Seuensil prtisiny melitn risn fungsi positif yng diseut prtisi tg δ n-fine. Lrmie memutin hw fungsi yng terintegrl Hensto jug terintegrl Hensto Seuensil egitupun seliny, sehingg sift-sift yng erlu pd integrl Hensto jug elu pd integrl Hensto Seuensil. Di dlm tulisn ini diutin eerp sift integrl Hensto Seuenisil dn teoremteorem dsr seperti yng erlu pd integrl Hensto. Kt Kuni ntegrl Hensto; ntegrl Hensto Seuensil; prtisi tg δ Astrt The development of integrl theories, in prtiulr the ltest Hensto integrl, ws sueeded y Lrmie Pxton, the Sequentil Hensto integrl. The definition of Sequentil Hensto ntegrl is lmost the sme s defining the integrl of Hensto, the differene in the Hensto integrl Sequentil prtition involves sequene of positive funtions lled the δ n-fine tg prtition. Lrmie proves tht Hensto's integrl funtion is lso integrl to Sequentil Hensto s well s vie vers, so tht the properties tht pply to the integrl Hensto lso pply to the Sequentil Hensto integrl. n this pper some of the ttriutes of Hensto Seuenisil integrls nd si theorems suh s those tht pply to the integrl Hensto. Keywords ntegrl Hensto; Sequentil Hensto ntegrl; δ tg prtition Pendhulun lmu pengethun merupn hl yng menglmi peremngn ser terus-menerus. Dintrny teori integrl yitu ilmu idng mtemti nlisis yng terus menglmi peremngn, dn memunginn untu terus diteliti dn diemngn. Pd thun 1854, teori integrl dengn penggunn prtisi segi dsr pengemngnny telh disusun oleh Riemnn. Teori ntegrl Riemnn merupn teori integrl yng mudh dipeljri dn dimengerti dlm mempeljriny. Nmun demiin, seiring jlnny wtu teori ntegrl Riemnn jug menglmi peremngn. Rlph Hensto (1957) seorng hli mtemtiwn, menermti d fungsi yng tid terintegrl Riemnn. Segimn diethui pendefinisin integrl yng dilun Riemnn hny memhs fungsi yng terts, nmun demiin tid semu fungsi yng terts terintegrln ser Riemnn, ontoh fungsi yng tid terintegrl Riemnn dlh fungsi Dirihlet. Dengn menggunn prtisi, Hensto menyusun teori integrl ru yng dienl dengn nm ntegrl Hensto. Bru-ru ini, tept di thun 2016 peremngn teori integrl yng terru erhsil diemun oleh Lrmie Pxton, yitu peremngn dri integrl Hensto menjdi integrl Hensto Seuensil mellui penggunn urutn umum tu is diseut risn, yitu, integrl Hensto didefinisin mellui pendetn risn. Leih lnjut, Lrmie memutin hw fungsi yng terintegrl Hensto jug terintegrl Hensto Seuensil egitupun seliny, sehingg sift-sift yng erlu pd integrl Hensto jug elu pd integrl Hensto Seuensil. Oleh ren itu, menri untu dipeljri sift-sift p sj yng erlu pd integrl Hensto Seuensil sert mengji leih dlm uti-uti yng tid dierin oleh Lrmie Pxton JURNAL FOURER Versi online vi

2 56 Mlhyti Lndsn Teori Pd gin ini, n dierin definisi prtisi, yng telh dienl eti mendefinisin integrl Riemnn mupun integrl Hensto. Nmun n dierin definisi non-overlpping terleih dhulu. Definisi 2.1. (Brtle, 2001:4) Du intervl teru pd R ditn tid tumpng tindih (nonoverlpping) pil irisn dri edu intervl terseut osong tu du intervl tertutup pd R ditn tid tumpng tindih (non-overlpping) pil irisn dri edu intervl terseut pling ny stu titi. Beriut n dierin ontoh gr leih mudh dlm memhmi definisi du intervl yng tid tumpng tindih (non-overlpping). Contoh 2.2. Dierin msing-msing du psng intervl pd R:. A = [0,2] dn B = [2,5] Du intervl terseut ditn tid tumpng tindih (non-overlpping), ren A B = {2}.. P = (2,4) dn Q = (5,7) ntervl P dn Q ditn tid tumpng tindih (non-overlpping), ren A B =.. R = [1,3] dn S = [2,4] Du intervl terseut ditn tumpng tindih (overlpping), ren R S = [2,3]. Definisi 2.3. (Brtle, 2000:145) Prtisi P pd sutu intervl = [, ] dlh olesi intervl-intervl tertutup yng non-overlpping sehingg = 1 n. Ditulis P = { 1,, n } dengn i = [x i 1, x i ] dimn = x 0 < < x i 1 < x i < < x n 1 < x n =. Titi-titi x i (i = 0,, n) diseut titi-titi prtisi P. Ji titi t i dipilih dri setip intervl i, i = 1,, n, m t i diseut tg dn himpunn psngnny diseut prtisi tg pd dpt ditulis Dengn x i 1 t i x i. P = {([x i 1, x i ], t i )} Beriut n dierin ontoh prtisi dn prtisi tg gr leih mudh dlm memhmi definisi prtisi di ts. Contoh 2.4. Dierin intervl A = [0,12] dientu P 1 = {[0,3], [3,6], [6,9], [9,12]} P 2 = {[0,2], [2,4], [4,6], [6,8], [8,10], [10,12]} m P 1 dn P 2 diseut prtisi. Dpt dipilih t i dri setip i, sehingg dpt dientu prtisi tg pd A: P 1 = {([0,3], 1), ([3,6], 5), ([6,9], 7), ([9,12], 11)} P 2 = {([0,2], 1), ([2,4], 3), ([4,6], 5), ([6,8], 7), ([8,10], 9), ([10,12], 11)} Selnjutny, n dierin definisi prtisi δ fine yng menjdi dsr terentuny integrl Riemnn. Definisi 2.5. (Pxton, 2016:3) Dierin δ > 0, prtisi tg P pd = [, ] ditn δ-fine pil setip suintervl [x i 1, x i ] [, ], memenuhi x i x i 1 < δ, i = 1, 2,,. JURNAL FOURER (2017)

3 Beerp Sift ntegrl Hensto Seuensil 57 Beriut n dierin ontoh prtisi δ finegr leih mudh dlm memhmi definisi prtisi δ fine. Contoh 2.6. Dierin δ = 3. Berdsrn Contoh 2.4. dpt disimpuln hw: P 1 un merupn prtisi δ fine ren d suintervl [0,3] A tetpi 3 0 > δ = 3 P 2 merupn prtisi δ fine. Definisi integrl Riemnn, memotivsi Hensto memodifisi prtisi yng digunn oleh Riemnn sehingg terentu integrl Hensto dengn definisi prtisi δ(x) fine eriut. Definisi 2.7. (Wells, 2011:10) Dierin = [, ] dn δ(x): R merupn fungsi positif sehingg δ(x) > 0, x. Himpunn P ditn prtisi δ(x)-fine pd pil memenuhi [x i 1, x i ] (t i δ(t i ), t i + δ(t i )) i = 1,, n, tu dengn t lin, prtisi tg P = {([x i 1, x i ], t i )} ditn δ(x)-fine pil setip suintervl [x i 1, x i ] memenuhi: dengn x i 1 t i x i. x i x i 1 < δ(t i ), i = 1, 2,,, Beriut n dierin ontoh prtisi δ(x) fine gr leih mudh dlm memhmi definisi prtisi δ(x) fine. Contoh 2.8. Dierin intervl = [0,1] dn didefinisin fungsi: dpt dientu P yng merupn prtisi δ(x): 1/4, x = 0 δ(x) = { 2x, 0 < x 1 P = {([0, 1 4 ], 1 4 ), ([1 4, 2 4 ], 1 4 ), ([2 4, 3 4 ], 2 4 ), ([3 4, 1], 7 8 )}. Definisi 2.9. (Pxton, 2016: 3) Dierin fungsi δ: R dengn δ(x) > 0, untu setip x, yng emudin δ(x) diseut guge pd. Definisi-definisi dits sngt ermnft dlm pendefinisin integrl Hensto seuensil sert pemutin eerp siftny. Lemm eriut ini n digunn dlm pemutin teorem-teorem pd integrl Hensto Seuensil. Lemm (Wells, 2011:14) Dierin fungsi positif δ(x) yng didefinisin pd [, ], dn intervl tg (t, [, ]). ntervl (t, [, ]) merupn prtisi tg δ(x)-fine pd [, ] ji dn hny ji {(t, [, t]), (t, [t, ])} merupn prtisi tg δ(x) fine pd [, ]. Selnjutny, f(t)( ) = f(t)(t ) + f(t)( t). JURNAL FOURER (2017)

4 58 Mlhyti Buti: (Syrt perlu) Diethui hw (t, [, ]) merupn prtisi tg δ(x) fine pd [, ] dengn [, t] [, ] dn [t, ] [, ] sert [, ] (t δ(t), t + δ(t)). Aitny, {(t, [, t]), (t, [t, ])} merupn prtisi tg δ(x) fine pd [, ]. (Syrt uup) Diethui hw {(t, [, t]), (t, [t, ])} merupn prtisi tg δ(x) fine pd [, ] errti [, t] (t δ(t), t + δ(t)) Dn Sehingg diperoleh dn [t, ] (t δ(t), t + δ(t)) [, t] [t, ] = [, ] [, ] (t δ(t), t + δ(t)) Oleh ren itu, (t, [, ]) merupn prtisi tg δ(x) fine pd [, ]. Selnjutny, f(t)( ) = f(t) f(t) + f(t)t f(t)t = f(t)(t ) + f(t)( t). Beriut ini dierin definisi integrl Hensto, dpt digunn dlm memhmi integrl Hensto Seuensil. Pendefinisin integrl Hensto menggunn prtisi tg δ(x)-fine. Definisi (Brtle, 2001:12) Fungsi f: = [, ] R ditn terintegrl Hensto pd pil terdpt ilngn positif A R dn untu setip ε > 0, terdpt δ(x) > 0 sehingg untu setip P = {([x i 1, x i ], t i )} yng merupn prtisi tg δ(x)-fine pd l erlu tu dpt ditulis f(t i )(x i x i 1 ) A < ε, S(f, P) A < ε. Bilngn A diseut nili integrl Hensto fungsi f pd ditulisa = f, emudin fungsif terintegrl Hensto pd dpt ditulisf R (). Beriut dierin ontoh fungsi terintegrl Hensto, gr leih mudh dlm memhmi definisi dits. Contoh Dierin fungsi Dirihlet, x [0,1] didefinisin f(x) = { 1, ji x Q 0, ji x Q 1 An diutin hw f(x) terintegrl Hensto pd [0,1] dn f(x) = 0 0 JURNAL FOURER (2017)

5 Beerp Sift ntegrl Hensto Seuensil 59 Penjelsn: Amil serng ε > 0, didefinisin E = {x i Q, x i [0,1]}, i = 1, 2, 3,, n, selnjutny didefinisin fungsi δ(x) pd [0,1] dengn ε δ(x i ) = { 2 i+1, ji x i E 0, ji x i E. Untu serng prtisi δ(x) fine pd [0,1] erlu f(t i )(x i x i 1 ) 0 = f(t i )(x i x i 1 ) = f(t i )(x i x i 1 ) + f(t i )(x i x i 1 ) x i E f(t i )(x i x i 1 ) + f(t i )(x i x i 1 ) x i E = 1 (x i x i 1 ) + 0 (x i x i 1 ) = (x i x i 1 ) + 0 = (x i x i 1 ) < δ(x i ) < = ε 1 2 i+1 ε 2 i+1 = 1 2 ε. 1 = 1 ε < ε. 2 x i E = 1 2 ε 1 2 i Dengn t lin, teruti hw f(x) terintegrl Hensto pd [0,1] dn f(x) = Hsil dn Pemhsn Beriut dierin definisi integrl Hensto Seuensil, gr mudh dlm memhminy dpt diliht emli definisi tentng prtisi yng telh dierin seelumny. Definisi 3.1 (Pxton, 2016:9) Fungsi f: = [, ] R ditn terintegrl Hensto Seuensil pd pil terdpt ilngn positif A dn sutu risn fungsi positif {δ n (x)} n=1 sehingg untu setip P n yng merupn prtisi tg δ n (x)-fine, erlu JURNAL FOURER (2017)

6 60 Mlhyti f(t i ) n (x i x i 1 ) n A < ε, tu dpt ditulis S(f, P n ) A < ε. Bilngn A diseut nili integrl Hensto Seuensil fungsi f pd dengn A = f, emudin fungsi f terintegrl Hensto Seuensil pd, dpt ditulis: f H (). Beriut n dierin ontoh fungsi terintegrl Hensto Seuensil, gr dpt memeri gmrn tentng definisi integrl Hensto Seuensil. Contoh 3.2. (Pxton, 2016:9) Dierin fungsi Dirihlet, x [0,1] didefinisin f(x) = { 1, x Q 0, x Q An diutin hw fungsi f terintegrl Hensto Seuensil pd [0,1] dn f(x) = 0. Penjelsn: Didefinisin elurg ilngn rsionl {q m } untu setip m N. Amil serng ilngn ε > 0, dn {δ n (x)} n=1 merupn fungsi guge yng turun pd [0,1] rtiny δ n+1 (x) < δ n (x), x itny, untu setip t i i, dengn i = [x i 1 x i ], i = 1, 2, 3,,, dn P n = {([x i 1 x i ], t i )} yng merupn prtisi δ n (x) fine jug turun, didefinisin i i [t i 1 2 δ n(t i ), t i δ n(t i )], i = 1, 2, 3,,. Ji t merupn titi tg, m it dpt memilih N yng uup esr dri risn fungsi guge tdi, dn didefinisin: ε δ N (t) = { 2 m, ji t = q m x, ji t Q. Dierin P n merupn prtisi δ n (x) fine pd [0,1] untu setip n N. Ji t i i dn t i Q, m f(t i ) = 0, itny jumlhn Riemnny ernili 0. Ji t i i dn t i Q, m f(t i ) = 1, perhtin du ondisi eriut terleih dhulu: 1. Ji q m merupn tg pd intervl i, m: i [q m 1 2 δ n(q m ), q m δ n(q m )] sehingg (x i x i 1 ) δ n (q m ) ε 2 m = δ N(t). 2. Ji q m merupn tg pd du intervl i yng erurutn, m: ren (x i x i 1 ) + (x i+1 x i ) ε 2 m, i [q m 1 2 δ n(q m ), q m ] x i x i δ n(q m ), 1 0 JURNAL FOURER (2017)

7 Beerp Sift ntegrl Hensto Seuensil 61 dn m, sehingg, i [q m, q m δ n(q m ), ] x i x i δ n(q m ) f(t i )(x i x i 1 ) ε 2 m S(f, P n ) 0 = f(t i )(x i x i 1 ) 0 = f(t i )(x i x i 1 ) ε 2 m = ε 1 2 m = ε. 1 = ε. Dengn t lin, teruti hw f(x) terintegrl Hensto Seuensil pd [0,1] dn f(x) = 0. Beriut n ditunjun hw integrl Hensto euivlen dengn integrl Hensto Seuensil. Teorem 3.3. (Pxton, 2016:11) Fungsi f: [, ] R terintegrl Hensto pd [, ] ji dn hny ji fungsi f terintegrl Hensto Seuensil pd [, ]. Buti: An diutin hw f R () ji dn hny ji f H (). Asumsin hw {δ n (x)} n=1 dlh risn fungsi guge yng turun, sehingg erlu (Syrt perlu) δ n+1 (x) < δ n (x), x Diethui f: R merupn fungsi yng terintegrl Hensto, rtiny untu setip ε > 0, terdpt δ(x) > 0 sehingg untu setip P yng merupn prtisi δ(x) fine, erlu S(f, P) A < ε. (3.1) Untu setip = 1,2,3, dierin ε n merupn ilngn rsionl ε sehingg 0 < ε < 1. Kren fungsi f terintegrl Hensto, m terdpt δ n (x) untu setip ε n yng memenuhi S(f, P) A < ε. Ji Q dlh himpunn ilngn rsionl yng terhitung, m {δ N (x)} n=1 dlh risn. Untu setip ε > 0, terdpt δ N (x) {δ n (x)} n=1 sehingg untu setip n N dn P n yng merupn risn prtisi δ n (x) fine pd, erlu Jdi, dengn t lin teruti hw f H (). S(f, P n ) A < ε JURNAL FOURER (2017)

8 62 Mlhyti (Syrt uup) Diethui f: R merupn fungsi yng terintegrl Hensto Seuensil, rtiny, untu setip ε > 0 terdpt risn fungsi guge {δ n (x)} n=1 sehingg untu setip P n yng merupn risn prtisi δ n (x) fine pd, erlu S(f, P n ) A < 1 n. Dierin λ > 0, dipilih δ N (x) {δ n (x)} n=1 sehingg untu setip δ(x) > 0, erlu δ(x) δ N (x) < λ x Ji dierin λ 0, m δ N (x) menjmin hw P N merupn prtisi δ n (x) finedn jug merupn prtisi δ(x) fine, sehingg untu P yng merupn prtisi δ(x) fine, it dpt mementu jumlhn Riemnn untu P &P N ser, sehingg S(f, P) S(f, P N ) < ε 2. Selnjutny, untu setip ε > 0, terdpt δ(x) sehingg dpt ditemun δ N (x) yng memenuhi δ(x) δ N (x) < λ x, dn risn {δ n (x)} n=1 merupn risn turun dimn n N, N N sehingg 1 < ε. untu N 2 setip P yng merupn prtisi δ(x) fine erlu S(f, P) A = S(f, P) S(f, P N ) + S(f, P N ) A S(f, P) S(f, P N ) + S(f, P N ) A < ε N < ε 2 + ε 2 = ε. Teorem dits ser tid lngsung mengtn hw semu sift-sift dn teorem yng erlu pd integrl Hensto erlu pul di integrl Hensto Seuensil. Beriut n diutin eerp sift yng telh erlu di integrl Hensto erlu pul di integrl Hensto Seuensil. Dintrny, pil dierin fungsi non negtive m nili integrl Hensto Seuensil dri fungsi terseut jug ernili non negtive. Selin itu pul nili integrl Hensto Seuensil dri perlin slr dn sutu fungsi sm dengn slr terseut dilin dengn nili dri integrl Hensto seuensil fungsi terseut. Teorem 3.4. (Pxton, 2016:15) Dierin f, g R merupn fungsi-fungsi yng terintegrl Hensto Seuensil pd = [, ] R, dn R. i) Ji f 0 pd m f 0. ii) f terintegrl Hensto Seuensil pd dn f = f iii) Ji f + g terintegrl Hensto Seuensil pd m (f + g) iv) Ji f(x) g(x), x, m f g.. = f + g. JURNAL FOURER (2017)

9 Beerp Sift ntegrl Hensto Seuensil 63 Buti: i) Dierin f 0 dn f H (), m untu setip ε > 0, terdpt δ N (x) {δ n (x)} n=1 sehingg untu setip n N dn P n yng merupn prtisi δ n (x) fine, erlu Artiny, sehingg, S(f, P n ) f ε. f(t i )(x i x i 1 ) f ε ε f(t i )(x i x i 1 ) f ε f ε f(t i )(x i x i 1 ) ε + f, (3.2) dengn f(t i ) 0, untu setip t i P n dn (x i x i 1 ) 0, m diperoleh 0 f(t i )(x i x i 1 ) = S(f, P n ), sehingg ji diominsin dengn (3.2) menjdi 0 S(f, P n ) f + ε n N. Jdi, dpt disimpuln hw untu setip ε > 0 erlu f 0. ii) Dierin f H (), rtiny untu setip ε > 0, terdpt δ N (x) {δ n (x)} n=1 sehingg untu setip n N dn P n yng merupn prtisi δ n (x) fine erlu Sehingg untu setip n N erlu S(f, P n ) f S(f, P n ) f ε, = f(t i )(x i x i 1 ) f = ( f(t i )(x i x i 1 ) f) = f(t i )(x i x i 1 ) f = S(f, P n ) f JURNAL FOURER (2017)

10 64 Mlhyti < ε = ε, sehingg f H () dn f = f. iii) Amil serng ε > 0, ren f H () m terdpt δ M (x) {δ m (x)} m=1 sehingg untu setip m M N dn P m yng merupn prtisi δ m (x) fine, erlu: S(f, P m ) f < ε 2, dn ren g H (), m terdpt δ K (x) {δ (x)} =1 sehingg untu setip M N dn P yng merupn prtisiδ (x) fine, erlu: S(g, P ) f < ε 2. Selnjutny, untu setip δ m (x) {δ m (x)} m=1 dn δ (x) {δ (x)} =1, dipilih δ n (x) = min{δ m (x), δ (x)} dengn n, m, = 1,2,3 x. Jels, δ n (x) merupn risn fungsi guge pd. Ji P n merupn prtisi δ n (x) fine, m P n merupn prtisi δ m (x) fine dn P n jug merupn prtisi δ (x) fine. Perhtin hw untu setip ε > 0, terdpt δ N (x) {δ n (x)} n=1 sehingg untu setip n N dn P n yng merupn prtisi δ n (x) fine P n, erlu S(f + g, P n ) ( f g) = (f + g)(t i )(x i x i 1 ) ( f g) = f(t i )(x i x i 1 ) f + g(t i )(x i x i 1 ) g f(t i )(x i x i 1 ) f + g(t i )(x i x i 1 ) g < ε 2 + ε 2 = ε, sehingg, f + g H () dn f + g = f + g. iv) Diethui f, g H () dn f(x) g(x) x. Dierin h = g f 0 erdsrn Teorem 3.3. gin iii) m erlu dn (g f) = g h = g f H () + f = g f JURNAL FOURER (2017)

11 Beerp Sift ntegrl Hensto Seuensil 65 Selnjutny, erdsrn Teorem 3.3. gin i), h 0, sehingg rtiny, sehingg, 0 h = (g f) = g 0 g f, f, f g. Beriut n dierin sift penmhn intervl yng menytn hw Apil sutu fungsi terintegrl Hensto Seuensil pd du suintervl tertutup yng dimut oleh sutu intervl tertutup, m fungsi terseut jug terintegrl Hensto Seuensil pd intervl tertutup terseut, dn nili integrl dri fungsi yng terintegrl Hensto Seuensil pd intervl tertutup terseut merupn penjumlhn dri nili integrl fungsi yng terintegrl Hensto Seuensil pd du suintervl terseut. Teorem 3.5. (Pxton, 2016:18) Dierin fungsi f: [, ] R dn (, ). Ji fungsi f merupn fungsi yng terintegrl Hensto Seuensil pd [, ] dn [, ], m fungsi f terintegrl Hensto Seuensil pd [, ] dn f = f + f. Buti: Amil serng ε > 0, ren f H ([, ]) m terdpt δ M (x) {δ m (x)} m=1 sehingg untu setip m M N dn P m yng merupn prtisi δ m (x) fine, erlu S(f, P m ) f < ε 2, dn jug ren f H ([, ]), m terdpt δ K (x) {δ (x)} =1 sehingg untu setip K N dn P yng merupn prtisiδ (x) fine, erlu S(f, P ) f < ε 2. Didefinisin guge δ N (x) {δ n (x)} n=1 sehingg untu setip n N N dn P n yng merupn prtisiδ n (x) fine, setip prtisi P n dpt dipehn menjdi: δ N (t) = min {δ m (t), 1 ( t)}, 2 min { δ m (), δ ()}, t [, ) t = untu m M dn K. { min {δ (t), 1 (t )}, 2 t (, ] JURNAL FOURER (2017)

12 66 Mlhyti Dierin P n merupn prtisi δ n (x)-fine, untu setip n N, ren P m merupn risn prtisi pd [, ] yng terdiri ts P n [, ], dn P merupn risn prtisi pd [, ] yng terdiri ts P n [, ] dengn n, m, = 1,2,3 m diperoleh S(f, P n ) = S(f, P m ) + S(f, P ). Untu setip ε > 0, terdpt δ N (x) {δ n (x)} n=1 sehingg untu setip n N erlu S(f, P n ) ( f + f j ) = f( p t i )(x i x i 1 ) n ( f + = f( t i )(x i x i 1 ) m f( = S(f, P m ) f + j i=p S(f, P ) f f) S(f, P m ) f + S(f, P ) f < ε 2 + ε 2 = ε. Sehingg, teruti hw f H ([, ]) dn f = f t i )(x i x i 1 ) ( f + + f. f) Beriut n dierin sift yng menytn hw Apil sutu fungsi terintegrl Hensto pd sutu intervl, m fungsi terseut jug terintegrl Hensto Seuensil pd setip suintervlny. Teorem 3.6. (Pxton, 2016:19) Dierin f: R R. Ji f H (), m f H ( i ) dengn i = 1,2,3, N Buti: An diutin hw f H ( i ) rtiny fungsi f terintegrl Hensto Seuensil pd setip suintervl i, i = 1,2,3, N dengn menggunn indusi mtemti. Pertm, n diutin enr untu = 2. Dierin ε > 0, m terdpt δ N (x) {δ n (x)} n=1 sehingg untu setip n N dn P n = {( i, t i )} yng merupn prtisi δ n (x) fine pd, erlu S(f, P n ) f < ε 2. Selnjutny, dierin intervl 1 dn 2 dengn 1 2 dn 1 2 =. Dierin pul δ 1n (x) dn δ 2n (x) erturut-turut merupn guge pd 1 dn 2, sehingg untu serng P 1n yng merupn prtisi δ 1n (x) fine dn P 2n yng merupn prtisi δ 2n (x) fine erlu 2 S(f, P n ) = f(t i )(x i x i 1 ) n = f(t 1 )(x 1 x 0 ) n + f(t 2 )(x 2 x 1 ) n = S(f, P 1n ) + S(f, P 2n ). JURNAL FOURER (2017)

13 Beerp Sift ntegrl Hensto Seuensil 67 Perhtin hw ji Q 16 merupn prtisi δ 1n (x) fine dn P 2n merupn prtisi δ 2n (x) fine m R n = Q 1n P 2n merupn prtisi δ n (x) fine, untu setip n = 1,2,3. Selnjutny dierin ε > 0, m terdpt δ N (x) {δ n (x)} n=1 sehingg untu setip n N dn P 1n, Q 1n merupn prtisi δ 1n (x) fine pd, erlu S(f, P 1n ) S(f, Q 1n ) = S(f, P 1n ) S(f, Q 1n ) + S(f, P 2n ) S(f, P 2n ) + f S(f, P 1n ) + S(f, P 2n ) f = S(f, P n ) f ε 2 + ε 2 = ε. + S(f, R n ) f f + S(f, Q 1n ) + S(f, P 2n ) f Selnjutny, disumsin enr untu = l, sehingg n diutin enr untu = l + 1. Dierin serng ε > 0, m terdpt δ N (x) {δ n (x)} n=1 sehingg untu setip n N dn P (l+1)n, Q (l+1)n yng merupn prtisi δ (l+1)n fine pd (l+1), erlu S(f, P (l+1)n ) S(f, Q (l+1)n ) S(f, P (l+1)n ) + [S(f, P 1n ) + S(f, P 2n ) + + S(f, P ln ] f = S(f, P n ) f ε 2 + ε 2 = ε, + S(f, Q (l+1)n + [S(f, P 1n ) + S(f, P 2n ) + + S(f, P ln )] f + S(f, R n ) f sehingg teruti hw fungsi f H ( i ) dengni = 1,2,3, N. Kesimpuln Pendefinisin ntegrl Hensto Seuensil hmpir sm seperti mendefinisin integrl Hensto, edny pd integrl Hensto Seuensil prtisiny melitn risn fungsi positif yng diseut prtisi tg δ δ. Sift-sift yng erlu pd integrl Hensto erlu pul pd integrl Hensto Seuensil, eerp dintrny eriut ini. Apil sutu fungsi terintegrl Hensto Seuensil pd du suintervl tertutup yng dimut oleh sutu intervl tertutup, m fungsi terseut jug terintegrl Hensto Seuensil pd intervl tertutup terseut, dn nili integrl dri fungsi yng terintegrl Hensto Seuensil pd intervl tertutup terseut merupn penjumlhn dri nili integrl fungsi yng terintegrl Hensto Seuensil pd du suintervl terseut. Selnjutny pil sutu fungsi terintegrl Hensto pd sutu intervl, m fungsi terseut jug terintegrl Hensto Seuensil pd setip suintervlny. Selin itu, pil dierin fungsi non negtive m nili integrl Hensto Seuensil dri fungsi terseut jug ernili non negtive. Sift linny mengtn hw nili integrl Hensto Seuensil dri perlin slr dn sutu fungsi sm dengn slr terseut dilin dengn nili dri integrl Hensto seuensil fungsi terseut. JURNAL FOURER (2017)

14 68 Mlhyti Referensi [1] Brtle Roert G., 2000, ntrdution to Rel Anlysis, Third Edition, John Wiley nd Sons, n, USA. [2] Brtle Roert G., 2000, A Modern Theory of ntegrtion, Grdute Studies in Mthemtis, Vol. 32. Amerin Mthemtil Soiety, Providene, R. [3] Lee Tuo Yeong, 2006, Multipliers for Generlized Riemnn ntegrls in the Rel Line, Mthemti Bohemi, Singpore [4] Lee Tuo Yeong, 2011, Hensto-Kurzweil ntegrtion on Euliden Spes, Series in Rel Anlysis, Vol.12, World Sientifi. [5] Pxton Lrmie, 2016, A Sequentil Approh to the Hensto ntegrl, Wshington Stte University. [6] Wells Jonthn, 2011, Generliztions of the Riemnn ntegrl:an nvestigtion of the Hensto ntegrl, dises tnggl 16 Agustus JURNAL FOURER (2017)