Zoma Da Lebegis Integrali LeqciaTa Kursi

Teimuraz axobaZe

zoma da lebegis integrali

(saleqcio kursi)

2

1. lebegis zoma namdvil ricxvTa simravleze

lebegis integralis SemoRebis sxvadasxva meTodi arsebobs. Cven kla-

sikur gzas avirCevT, anu SemoviRebT lebegis zomas, SeviswavliT mis Tvi-

sebebs da Semdgom am ukanasknelze dayrdnobiT avagebT lebegis integ-

rals. simartivisaTvis lebegis integralis Teorias ganvixilavT namdvil

ricxvTa simravleze. am ukanasknelis ganzogadeba SeiZleba mravalganzo-

milebiani evkliduri sivrceebisaTvis. maT aq mxolod gakvriT SevexebiT.

1.1 Ria da Caketili simravleebi

namdvil ricxvTa I qvesimravles (I⊂ ) ewodeba intervali, Tu mas aqvs

Tviseba: pirobidan x1, x2∈I gamomdinareobs, rom nebismieri namdvili x ric-xvi, romlisTvisac min{x1, x2}≤x≤max{ x1, x2}, gvaqvs x∈I. Zneli Sesamowmebeli

ar aris, rom intervalis rolSi SeiZleba ganvixiloT TiToeuli Semdegi

simravleTagan: (a,b), (a,b], [a,b), [a,b), sadac a SeiZleba iyos -∞, xolo b - +∞. piriqiT, Tu I intervalia (anu akmayofilebs intervalis gansazRvrebas),

maSin is daemTxveva erT-erT CamoTvlil simravles. marTlac, SemoviRoT

aRniSvna: infI =a, supI =b. vTqvaT a<b. ganvixiloT nebismieri namdvili x ric-xvi, romlisTvisac a<x<b. simravlis supremumisa da infimumis gansaz-

Rvris Tanaxmad iarsebebs am simravlidan iseTi ori x1 da x2 elementi, rom

x1<x<x2. amrigad, (a,b)⊂I . Tu supI∈I da infI∉I, an infI∈I da supI∉I, an infI∈I da supI∈I, maSin Sesabamisad gveqneba: (a,b]⊂I, [a,b)⊂I, [a,b]⊂I. meore mxriv, rad-gan I simravlis yoveli x elementisaTvis a≤x≤b, vaskvniT, rom Camoyalibe-

buli winadadeba WeSmaritia.

momavalSi (a,b) intervals vuwodebT Ria intervals.

I intervalis sigrZe |I| vuwodoT sxvaobas b-a, amasTan im SemTxvevaSi,

rodesac a da b ricxvTagan erT-erTi usasrulobaa, vgulisxmobT, rom

|I|=∞. ra Tqma unda, ∞ ar aris namdvili ricxvi, magram momavalSi mas xSi-

rad gamoviyenebT, davuSvebT ra:

a+∞=∞, ∀a∈ ,

a-∞=-∞, ∀a∈ , 0×∞=0, a×∞=∞, ∀a∈ , da a >0,

a<∞, ∀a∈ ,

a>-∞, ∀a∈ .

vTqvaT x0∈ . x0 wertilis midamo ewodeba nebismier Ria (a,b)⊂ inter-

vals, romelic Seicavs x0–s (a<x0<b). x0∈ wertils ewodeba G⊂ simrav-

lis Siga wertili, Tu arsebobs am wertilis iseTi (a,b) midamo, romlis

yoveli wertili G⊂ simravlidanaa ((a,b)⊂G).

3

G⊂ simravles ewodeba Ria simravle, Tu am simravlis yoveli werti-

li amave simravlis Siga wertilia.

F⊂ simravles ewodeba Caketili simravle, Tu am simravlis yoveli

krebadi mimdevrobis zRvari ekuTvnis amave simravles.

x0∈ wertils ewodeba E⊂ simravlis zRvruli (dagrovebis) wertili,

Tu am wertilis nebismieri midamo Seicavs x0 wertilisagan gansxvavebul

E simravlis erT wertils mainc. es cneba sxvagvaradac SeiZleba Camoya-

libdes. vityviT, rom x0∈ wertili aris E⊂ simravlis zRvruli werti-

li, Tu arsebobs E simravlis wertilTa mimdevroba, romelic krebadia

x0-ken da romlis wevrebic wyvil-wyvilad gansxvavebulia; an ase ganvmar-

toT: x0 wertils ewodeba E simravlis zRvruli wertili, Tu am werti-

lis nebismieri midamo Seicavs E simravlis usasrulod bevr wertils.

advili saCvenebelia, rom ukanaskneli sami ganmarteba erTmaneTis tolfa-

sia (aCveneT). simravlis zRvruli wertili SesaZlebelia ar ekuTvnodes

am simravles. magaliTisaTvis SeiZleba ganvixiloT (0,1) intervali, rom-

elsac ar miekuTneba misi zRvruli 0 da 1 wertilebi.

vTqvaT E⊂ . vityviT, rom x0∈ wertili aris E simravlis izolirebu-

li wertili, Tu arsebobs x0 wertilis iseTi midamo, romelic x0-gan gan-

sxvavebul E simravlis arcerT wertils ar Seicavs.

Teorema 1.1.1. imisaTvis, rom F⊂ simravle iyos Caketili, aucilebe-

li da sakmarisia, rom misi damateba G≡ \ F iyos Ria simravle.

damtkiceba. vTqvaT F⊂ Caketili simravlea. vaCvenoT, rom misi damate-

biTi G≡ \ F simravle aris Ria. davuSvaT es ase ar aris, e.i. G simravlis

yoveli wertili ar aris Siga wertili. aviRoT erT-erTi aseTi x0 werti-

li (x0∈G). am wertilis yovel midamoSi moiZebneba erTi mainc iseTi wer-

tili, romelic ar ekuTvnis G simravles, e.i. ekuTvnis F-s. yoveli natu-

raluri n-Tvis ganvixiloT x0 wertilis 1/n-midamo. am midamoSi moTavse-

buli F simravlis erT-erTi wertili aRvniSnoT xn-iT. cxadia, |xn-x0|<1/n. amrigad, lim nn

x→∞ 0 ,x= e.i. x0∈F. es ki winaaRmdegobaa.

piriqiT, vTqvaT G Ria simravlea, maSin F≡ \G iqneba Caketili simrav-

le. marTlac, vTqvaT F ar aris Caketili, e.i. F simravleSi arsebobs ise-

Ti krebadi (xn) mimdevroba, romlis zRvari x0≡ lim nnx

→∞∉F, anu x0∈G. amrigad,

x0 aris G simravlis Siga wertili. es ki niSnavs, rom arsebobs am werti-

lis iseTi midamo, romelSic ar iqneba moTavsebuli F simravlis arcer-

Ti wertili, anu (xn) mimdevroba ar SeiZleba iyos krebadi x0–ken.

Caketili simravlis magaliTia [a,b] daxuruli Sualedi (a,b∈ ), xolo

(a,b) Ria intervali Ria simravlea.

namdvil ricxvTa simravle erTdroulad Caketilic aris da Riac.

amitom Teorema 1.1.1-is ZaliT carieli ∅ simravle aseTive Tvisebisaa.

Teorema 1.1.2. a) Caketil simravleTa nebismieri sistemis TanakveTa

Caketili simravlea. b) Caketil simravleTa sasruli sistemis gaerTiane-

ba Caketilia.

4

damtkiceba. a) vTqvaT gvaqvs namdvil ricxvTa simravlis Caketil

qvesimravleTa {Fα}α∈A sistema. vaCvenoT, rom F≡ A Fα α∈∩ aris Caketili sim-

ravle. ganvixiloT F simravlis raime krebadi (xn) mimdevroba. vTqvaT

lim nnx

→∞= x0. vaCvenoT, rom x0∈F. ra Tqma unda, yoveli α∈A indeqsisaTvis mim-

devroba (xn) aris Fα simravlis mimdevroba. radgan Fα Caketili simravlea,

amitom x0∈Fα. amrigad, x0∈ A Fα α∈∩ = F. b) vTqvaT F1, F2,…, Fn Caketili simravleebia, maSin F≡ 1

nk kF=∪ aris aseve

Caketili simravle. ukanasknelis dasamtkiceblad ganvixiloT F simrav-lis nebismieri krebadi (xn) mimdevroba. vTqvaT lim nn

x→∞

= x0. am mimdevrobis

usasrulod bevri wevri ekuTvnis erT-erT Fk, k=1,2,…,n, simravles mainc.

zogadobis SeuzRudavad vigulisxmoT, rom (xn) mimdevrobis raRac ( )knx

qvemimdevrobis yoveli wevri ekuTvnis 0kF simravles (k0∈{1,2,…, n}). cxadia,

0lim .knk

x x→∞

= radgan 0kF Caketili simravlea, amitom x0∈

0kF ⊂ 1nk kF=∪ = F.

simravleTa {Bα}α∈A sistemisTvis gaviTvaliswinebT, ra morganis formu-

lebs (oradobis princips)

A

Bαα∈

=∪ \ ( \ ),A

Bαα∈∩ R

( )\ \ ,A A

B Bα αα α∈ ∈

=∩ ∪R R

martivad davaskvniT, rom samarTliania

Teorema 1.1.3. a) Ria simravleTa nebismieri sistemis gaerTianeba Ria

simravlea. b) Caketil simravleTa sasruli sistemis gaerTianeba Caketi-

li simravlea.

sazogadod, Ria simravleTa usasrulo TanakveTa (magaliTad, simravle

1(0,1 1/ )n n∞= +∩ ) ar aris Ria. aseve, Caketil simravleTa usasrulo gaerTia-

neba (magaliTad, 1[0,1 1/ ]n n∞= −∪ ) sazogadod ar aris Caketili simravle.

Teorema 1.1.4. namdvil ricxvTa simravleSi yoveli Ria G simravle

warmoidgineba TanaukveT, Ria intervalTa araumetes Tvladi gaerTianebis

saxiT.

damtkiceba. aviRoT nebismieri wertili τ∈G. vTqvaT aτ=inf{x: (x,τ)⊂G}, bτ= sup{x:(τ,x)⊂G}. maSin (aτ,bτ)⊂G, amasTan aτ,bτ∉G. maSasadame, G simravlis yovel

τ elements Seesabameba (aτ,bτ)⊂G intervali. Tu intervalebs ( )1 1,a bτ τ da

( )2 2,a bτ τ aqvT saerTo wertili, maSin isini emTxveva erTmaneTs. ganvixi-

loT yvela gansxvavebuli (aτ,bτ) intervali. radgan isini wyvil-wyvilad

ar TanaikveTebian, amitom maTi raodenoba Tvlads ar aRemateba. amis gar-

da, yvela aseTi TanaukveTi intervalis gaerTianeba emTxveva G-s.

Tu F simravle Caketilia, maSin 1.1.1 Teoremis ZaliT misi damatebiTi

simravle Riaa. amitom es ukanaskneli 1.1.4 Teoremis safuZvelze warmoid-

gineba araumetes Tvladi, TanaukveTi Ria intervalebis gaerTianebis sa-

xiT. TiToeul aseT intervals ewodeba F simravlis mosazRvre inter-

vali.

E simravlis zRvrul wertilTa simravles ewodeba E simravlis war-

moebuli simravle da aRiniSneba simboloTi E′.

5

simravles ewodeba srulyofili simravle, Tu is emTxveva missave war-

moebul simravles. [a,b] segmenti srulyofili simravlis erT-erTi maga-

liTia.

Teorema 1.1.5. warmoebuli E′ simravle Caketili simravlea.

damtkiceba. Tu E′ simravle carielia, maSin Teoremis mtkiceba cxadia.

vTqvaT E′≠∅ da x0 aris E′ simravlis zRvruli wertili. ganvixiloT x0 wer-

tilis nebismieri (a,b) midamo. masSi moiZebneba x0-gan gansxvavebuli E′ sim-ravlis erTi x∗

wertili mainc. aviRoT x∗ wertilis (c,d)⊂(a,b) midamo. es mi-

damo (da, maSasadame, (a,b) intervalic) zRvruli wertilis ganmartebis Za-

liT Seicavs E simravlis usasrulod bevr wertils. amrigad, x0∈E. Teorema 1.1.6. vTqvaT G aris aracarieli Ria simravle da (a,b)⊂G, ma-

Sin G Ria simravlis warmomdgen intervalTa Soris arsebobs iseTi,

romelic Seicavs (a,b) intervals.

damtkiceba. vTqvaT x0∈(a,b) da G= ( , )i i iα β∪ ; maSin arsebobs i0∈N, rom

x0∈( )0 0,i iα β ; vTqvaT

0iβ <b. es niSnavs, rom

0iβ ∈(a,b)⊂G. meore mxriv,

0iβ ∉ G. es

ki SeuZlebelia; e.i. b≤0i

β . analogiurad vaCvenebT, rom 0i

α ≤ a.

1.1.5 Teoremis ZaliT srulyofili simravle aris Caketili simravle.

magram yoveli Caketili simravle ar aris srulyofili simravle. maga-

liTad, nebismieri sasruli aracarieli simravle Caketilia, magram ar

aris srulyofili simravle. aseve {0,[1,2]} aris Caketili arasrulyofili

simravlis magaliTi.

bunebrivad ismeba kiTxva: ra damatebiT pirobas unda akmayofilebdes

simravle, rom is iyos srulyofili simravle? am kiTxvaze pasuxis gasa-

cemad winaswar davamtkicoT

Teorema 1.1.7. vTqvaT Faris aracarieli Caketili simravle, xolo

[a,b] aris umciresi segmenti, romelic F simravles Seicavs (anu a=infF, xo-lo b=supF). maSin

1. Tu x0 aris F simravlis ori mosazRvre intervalis saerTo werti-

li, maSin is aris F simravlis izolirebuli wertili.

2. Tu a wertili (an b wertili) aris F simravlis erT-erTi mosazRvre

intervalis bolo wertili, maSin igi aris F simravlis izolirebuli

wertili.

3. F simravles, garda 1 da 2 punqtebSi CamoTvlili wertilebisa, sxva

izolirebuli wertili ar aqvs.

damtkiceba. 1 da 2 punqtebis WeSmariteba cxadia. davamtkicoT punqti

3. vTqvaT x0 aris F simravlis izolirebuli wertili. jer vigulisxmoT,

rom a<x0<b. izolirebuli wertilis ganmartebis Tanaxmad arsebobs am

wertilis iseTi (α,β) midamo, romelic ar Seicavs x0–gan gansxvavebul F simravlis sxva wertils. cxadia, (α,β)⊂[a,b]. maSin intervali (x0, β) ar Sei-cavs F simravlis wertilebs, e.i. (x0, β)⊂[a,b]\ F. radgan [a,b]\ F aris Ria sim-ravle, amitom 1.1.6 Teoremis ZaliT arsebobs F simravlis mosazRvre in-

tervali (r,s), romelic Seicavs (x0, β)-s. Tu r<x0, maSin x0 ar SeiZleba ekuT-

vnodes F simravles. maSasadame, x0≤ r. radgan (x0, β)⊂(r,s), amitom r≤x0; anu x0= r. maSasadame, x0 aris F simravlis erT-erTi mosazRvre intervalis kidura

wertili.

analogiurad vaCvenebT, rom x0 iqneba erT-erTi mosazRvre intervalis

marjvena bolo.

6

sabolood, Tu x0=a an x0 =b, maSin x0 iqneba erT-erTi mosazRvre interva-

lis bolo wertili. aqedan miiReba

Teorema 1.1.8. yoveli SemosazRvruli srulyofili P simravle aris

segmenti an miiReba raRac segmentidan sasruli an Tvladi Ria interva-

lebis amoyriT, romelTac ar aqvT arc erTmaneTTan da arc segmentTan

saerTo bolo wertili. piriqiT, yoveli simravle, romelic miiReba am

gziT, srulyofilia.

ganvixiloT srulyofili simravlis mniSvnelovani magaliTi –kanto-

ris simravle (kantoris diskontinuumi).

vTqvaT E0=[0,1], E1=[0,1]\(1/3,2/3)=[0,1/3]∪ [2/3,1]. miRebuli TiToeuli segmen-

tidan amovagdoT Sua mesamedi sigrZis segmenti. vTqvaT E2=[0,1/9]∪[2/9,3/9]∪ [6/9,7/9]∪[8/9,1]. Tu ase gavagrZelebT miviRebT En simravleTa mimdevrobas,

romelTaTvisac: a) E1⊃ E2⊃...⊃ En⊃...; b) En aris gaerTianeba 2n cali segment-

isa, romelTagan TiToeulis sigrZe iqneba 3 n−. P≡ 1n nE∞

=∩ simravles ewode-

ba kantoris simravle. TiToeuli En aris Caketil simravleTa sasruli

gaerTianeba, amitom Caketilia. 1.1.2 Teoremis ZaliT TanakveTa P= 1n nE∞=∩

iqneba Caketili. cxadia, es simravle SemosazRvrulicaa. P simravlis age-

bidan davaskvniT, rom arcerTi 3 1 3 2,

3 3m m

k k+ +⎛ ⎞⎜ ⎟⎝ ⎠

saxis intervali, sadac m da

k nebismieri naturaluri ricxvebia, ar TanaikveTebian P simravlesTan. ma-

Sasadame, P simravle ar Seicavs arcerT intervals, anu, rogorc ityvian,

aris arsad mkvrivi. radgan P simravle miiReba [0,1] segmentidan iseTi

Tvladi raodenoba TanaukveTi intervalebis amoyriT, romelTac ar aqvT

erTmaneTTan saerTo wertili da ar Seicaven [0,1] segmentis boloebs

(rogorc zemoT iyo aRniSnuli, TiToeul aseT intervals ewodeba P sim-ravlis mosazRvre intervali), amitom 1.1.8 Teoremis ZaliT P simravle

aris srulyofili simravle.

kantoris simravles aqvs kontinuumis simZlavre. amis saCveneblad gan-

vixiloT kantoris P simravlis agebisas pirveli amogdebuli (1/3,2/3) in-tervali. am intervalis yoveli x wertili samwiladad gaSlisas warmoid-

gineba saxiT x=0,a1a2a3... (ak=0,1,2), amasTan a1=1. rac Seexeba am intervalis

kidura wertilebs, TiToeuli maTgani SeiZleba Caiweros ori sxvadasxva

saxiT:

0,100000...1 ;0,022222...3

⎧= ⎨

⎩

0,12222...2 .0, 20000...3

⎧= ⎨

⎩

[0,1] segmentis yvela sxva wertilis samobiTi gaSlisas mZimis Semdeg

pirvel adgilze ar SeiZleba idges cifri 1. amrigad, agebis pirvel etap-

ze [0,1] segmentidan iyreba is da mxolod is wertilebi, romelTa pirveli

samobiTi niSani aucileblad aris 1. analogiurad davaskvniT, rom agebis

meore etapze iyreba is da mxolod is wertilebi, romelTa samobiT gaS-

laSi meore niSani aucileblad aris 1, da a. S. maSasadame, P simravle

mxolod iseTi wertilebisagan Sedgeba, romelTa samobiT gaSla SesaZle-

belia saxiT x=0,a1a2a3..., sadac TiToeuli niSani ak=0 an ak=2. calsaxobisa-

Tvis ganvixiloT P simravlis wertilebis mxolod aseTi warmodgena.

meore mxriv, [0,1] segmentis yoveli x wertili gavSaloT orobiT siste-

maSi: x=0,b1b2b3..., sadac TiToeuli niSani bk=0 an bk=1. amasTan warmodgenis

7

calsaxobisaTvis ar davuSvaT 0-gan gansxvavebuli ricxvis orobiTi gaS-

la usasrulo perioduli orwiladiT, periodiT 0.

axla ar gagviWirdeba [0,1] segmentsa da P simravles Soris bieqciuri

Tanadobis damyareba. [0,1] segmentis 0,b1b2b3... orwilads SevuTanadoT P sim-ravlis 0,a1a2a3... samwiladi Semdegnairad: Tu bk=0, maSin vigulisxmoT, rom

ak=0, xolo Tu bk=1, maSin davuSvaT ak=2. vTqvaT τ∈ da E⊂ . manZili x wertilidan E simravlemde ase ganimar-

teba: ρ(τ, E)=inf{|τ-x|: y∈E}. Tu E simravle Caketilia da τ∉E, maSin ρ(τ, E)>0. marTlac, winaaRmdeg

SemTxvevaSi, infimumis ganmartebis Tanaxmad, yoveli naturaluri n-Tvis

arsebobs iseTi τn wertili, rom |τ-τn|<1/n, anu lim ;nnτ τ

→∞= es ki, E simravlis

Caketilobis gamo, niSnavs: τ∈E. namdvil ricxvTa or A da B qvesimravles Soris manZili ganimarteba

Semdegnairad:

ρ(A, B)=inf{|x-y|: x∈A, y∈B}. cxadia, ρ(A, B)≥0. Tu A da B Caketili simravleebia, erT-erTi Semosaz-

Rvrulia da A∩B=∅, maSin ρ(A, B)>0; amasTan moiZebneba iseTi x0∈A da y0∈B wertilebi, rom ρ(A, B)=|x0-y0|. marTlac or simravles Soris manZilis gan-

martebis Tanaxmad arsebobs iseTi (xn) da (yn) mimdevrobebi, saTanadod A da B simravleebidan, rom lim | | ( , ).n nn

x y A Bρ→∞

− = garkveulobisaTvis vigulis-

xmoT, rom A aris SemosazRvruli simravle, maSin bolcanosa da vaier-

Strasis Teoremis Tanaxmad (xn) mimdevrobidan gamoiyofa krebadi ( )knx qve-

mimdevroba. vTqvaT 0limknk

x x A→∞

= ∈ (A simravlis Caketilobis gamo). amrigad,

lim | | ( , ).k kn nk

x y A Bρ→∞

− = davuSvaT ( )ikn aris ( )kn mimdevrobis is qvemimdevroba,

romlisTvisac k kn nx y− niSans inarCunebs (vTqvaT arauaryofiTia). maSin

( )limk ki in ni

x y→∞

− = ( , ).A Bρ maSasadame, arsebobs 0lim ;kini

y y B→∞

= ∈ e.i ( , )A Bρ = |x0-y0| >0.

aRsaniSnavia, rom moTxovna erT-erTi simravlis SemosazRvrulobis Se-

saxeb arsebiTia. vTqvaT A={1, 2, 3, … , n, …}, B={1+1/2, 2+1/3, 3, … , n+1/(n+1), …}. cxadia, orive simravle Caketilia da ( , )A Bρ =0.

Teorema 1.1.9. vTqvaT f: → aris uwyveti funqcia. amasTan vigulis-

xmoT, rom A⊂ . a) Tu A Caketili simravlea, maSin f -1(A) Caketilia. b) Tu A Ria simravlea, maSin f -1(A) Riaa.

damtkiceba. a) vTqvaT A Caketili simravlea da f -1(A)=F. vaCvenoT, rom F simravle Caketilia. aviRoT am simravlidan nebismieri krebadi (xn) mimdev-

roba. garkveulobisaTvis vigulisxmoT, rom lim nnx

→∞= x0. radgan f funqcia

uwyvetia, amitom lim ( )nnf x

→∞= f(x0). amis garda, yoveli naturaluri n-Tvis f(xn)

∈A da A Caketili simravlea, amitom f(x0)∈A. es ki niSnavs, rom x0∈F. maSasa-dame, F Caketili simravlea.

am debulebis b) nawilis mtkiceba gamomdinareobs Teorema 1.1.1 da f -1( \ A)= \ f -1(A)

tolobidan.

8

1.2 gare zoma

gansazRvreba 1.2.1. namdvil ricxvTa simravlis nebismieri A qve-simravlisaTvis SemoviRoT gare zomis (m∗( A)) cneba Semdegnairad

m∗( A)=inf ⎧ ⎫⎨ ⎬⎩ ⎭∑

1 1

| | : ∪ ii i

I A I∞∞

= =

⊂ .

am gansazRvrebaSi A simravlis yoveli dafarva Sedgeba Tvladi rao-

denoba intervalebisagan. es ar gamoricxavs nebismieri sasruli raodeno-

ba intervalebiT Sedgenili sistemiT A simravlis dafarvas, radgan am da-

farvaSi SeiZleba intervalebi gameordes. SesaZlebelia isic moxdes, rom

A simravlis Ii intervalebiT yoveli dafarvisas mwkrivi 1 ii

I∞

=∑ iyos gan-

Sladi. am SemTxvevaSi vigulisxmebT, rom m∗(A)=∞. yvela sxva SemTxvevaSi

m∗(A) aris arauaryofiTi namdvili ricxvi.

aRsaniSnavia, rom A⊂ simravlis gare zomis ganmartebisas Cven SeiZ-

leba davkmayofildeT mxolod Ria intervalTa Tvladi sistemebiT da-

farvebiT. marTlac, davasaxeloT ε>0. infimumis ganmartebis ZaliT moi-

Zebneba { In} intervalTa Tvladi sistema, romelTaTvisac

1n

nI

∞

=∑ < m∗(A)+ ε /2.

meore mxriv, am sistemis TiToeuli In intervalTvis arsebobs iseTi Ria

nI intervali, rom

1/ 2nn nI I ε +< + .

aqedan

1 1/ 2.n n

n nI I ε

∞ ∞

= =

< +∑ ∑

amrigad, mxolod Ria simravleTa Tvladi sistemiT gansazRvruli gare

( )m A∗ zomisTvis gveqneba

( )m A∗<

1/ 2n

nI ε

∞

=

+∑ ,

e.i.

( )m A∗< ( )m A∗

+ ε. ukanaskneli ki niSnavs, rom ( )m A∗

= ( )m A∗.

axla SevniSnoT, rom gare zomis ganmartebaSi SeiZleba ganvixiloT

mxolod is intervalebi, romelTa sigrZeebi ar aRematebian winaswar da-

saxelebul δ>0 ricxvs. marTlac, Tu A simravlis damfaravi sistemis ro-

melime Ik intervalis sigrZe aRemateba δ ricxvs, maSin am intervals dav-

yofT sasrul raodenoba intervalebad, romelTa sigrZec naklebi iqneba

δ–ze. am procesis Sedegad miRebuli intervalTa sistema, cxadia, Tvla-

dia.

Cven SevudgebiT gare zomis Tvisebebis Seswavlas. daviwyoT monotonu-

robis TvisebiT.

Teorema 1.2.2. Tu A⊂ B, maSin m∗(A)≤ m∗(B).

9

damtkiceba. am debulebis mtkiceba uSualod gamomdinareobs im faqti-

dan, rom B simravlis intervalebiT yoveli dafarva warmoadgens A sim-ravlis dafarvasac.

Semdeg debulebas ewodeba gare zomis Tvladad naxevrad adiciurobis

Tviseba.

Teorema 1.2.3. vTqvaT A1, A2,..., An,... namdvil ricxvTa simravlis qve-

simravleTa Tvladi sistemaa. maSin

( )⎛ ⎞⎜ ⎟⎝ ⎠

∑11

.∪ i iii

m A m A∞ ∞

∗ ∗

==

≤

damtkiceba. am debulebis mtkiceba cxadia im SemTxvevaSi, roca raime

i-Tvis (i∈N) m∗(Ai)=∞. amitom Cven vigulisxmebT, rom yoveli i naturaluri

ricxvisTvis (i∈N) m∗(Ai)<∞. dasaxelebuli ε>0-Tvis da nebismieri i∈N ric-

xvisTvis moiZebneba intervalTa iseTi {Ii,j}j∈N sistema, romelic faravs Ai

simravles da

,1| |i j

jI

∞

=

≤∑ m∗(Ai)+ ε /2i.

amitom

,1 1 1

i i ji i j

A I∞ ∞ ∞

= = =

⊂∪ ∪∪ da ,1 1 1

| | ( ) .i j ii j i

I m A ε∞ ∞ ∞

∗

= = =

≤ +∑∑ ∑

maSasadame,

( )11

.i iii

m A m A ε∞ ∞

∗ ∗

==

⎛ ⎞≤ +⎜ ⎟

⎝ ⎠∑∪

ε−is nebismierobis gamo Teorema 1.2.3 damtkicebulia.

cxadia, ukanaskneli debuleba samarTliania namdvil ricxvTa simrav-lis qvesimravleTa nebismieri sasruli sistemisTvisac.

simravles, romlis gare zoma 0-is tolia, vuwodoT nulsimravle. cxa-

dia, nulsimravleTa nebismieri Tvladi gaerTianeba nulsimravlea. es wi-

nadadeba uSualo Sedegia Teorema 1.2.3-isa. Tu raime winadadeba sruldeba A simravlis raRac B qvesimravleze da

amasTan A \ B aris nulsimravle, maSin vambobT, rom es winadadeba srulde-

ba A simravleze TiTqmis yvelgan (T.y).

Teorema 1.2.4. nebismieri I intervalisaTvis m∗(I)=|I|. damtkiceba. radgan I intervali faravs Tavis Tavs, amitom m∗(I)≤|I|. ax-

la vaCvenoT, rom m∗(I)≥|I|. jer ganvixiloT SemTxveva, roca I aris sasruli Caketili [a,b] Suale-

di. vTqvaT (Ii) aris Ria intervalTa mimdevroba, romlis wevrebis gaerTi-

aneba faravs [a,b] segments. am sistemidan gamovyoT (a1, b1), (a2, b2),..., interva-

lebi Semdegnairad. vTqvaT (a1, b1) aris erT-erTi Ik intervali umciresi in-

deqsiT, romelic Seicavs a wertils. Tu b1>b, maSin es sistema agebulia.

Tu b1≤b, maSin SevarCevT iseT umciresindeqsian Ik intervals, romelic Se-

icavs b1 wertils. vTqvaT SerCeuli intervalia (a2, b2). Tu n-1 nabijze bn-1 ≤ b, maSin am process gavagrZelebT. vTqvaT (an, bn) aris umciresindeqsiani

intervali, romelic Seicavs bn-1–s. es procesi Sewydeba sasruli nabijis

Semdeg. winaaRmdeg SemTxvevaSi gveqneba wertilTa usasrulod zrdadi

(bn) mimdevroba, romelic krebadi iqneba raRac b0≤b wertilisken. radgan

10

{Ii} sistema faravs [a,b] Sualeds, amitom b0∈[a,b] wertilisaTvis iarsebebs

iseTi k0∈N, rom b0∈0kI da bn∈

0kI , roca n> k0. (an, bn) intervalTa agebis Za-

liT es niSnavs, rom yoveli (an, bn) intervali, romelis indeqsi aWarbebs

k0-s, win uswrebs 0kI -s, rac SeuZlebelia. amitom arsebobs iseTi n0∈N, rom

0.nb b> maSin gvaqvs

(b1- a1)+ (b2- a2)+...+0 0n nb a−

>(b1- a)+ (b2- b1)+...+0 0 1( )n nb b −−

=0nb a− > b- a.

maSasadame, 1| | .iiI b a∞

=> −∑ amrigad, m∗(I)≥b-a.

vTqvaT axla I SemosazRvrulia, magram ar aris Caketili intervali.

vigulisxmoT, rom misi bolo wertilebia a da b. am SemTxvevaSi [a+ε/2, b-ε/2], 0<ε<b-a, segmentisTvis Cven gamoviyenebT ukve Catarebul msjelobas.

maSin gveqneba

m∗(I)≥(b-ε/2)-(a+ε/2)=(b-a)-ε. radgan ε nebismieri dadebiTi ricxvia, amitom amjeradac miviRebT:

m∗(I)≥b-a. Tu I ar aris SemosazRvruli, maSin nebismieri M>0 ricxvisaTvis mov-

ZebnoT I qveintervals, romlis sigrZea M. gare zomis monotonurobis

gamo (ix. Teorema 1.2.2)

m∗(I)≥m∗( I )=M. amrigad, m∗(I)=∞.

Teorema 1.2.5. Tu F1, F2, ..., Fn wyvil-wyvilad TanaukveTi, SemosazRvru-

li, Caketili simravleebia, maSin

( ).⎛ ⎞⎜ ⎟⎝ ⎠

∑11

∪n n

i iii

m F m F∗ ∗

==

=

damtkiceba. sakmarisia ganvixiloT n=2 SemTxveva. zogadi SemTxveva mar-

tivad miiReba n-is mimarT ariTmetikuli induqciis meTodis gamoyenebiT.

radgan F1 da F2 SemosazRvruli, Caketili, TanaukveTi simravleebia, ami-

tom (ix. punqti 1.2) arsebobs iseTi δ>0 ricxvi, rom |u-v|>δ yoveli u∈F1 da

u∈F2. nebismieri ε>0 ricxvisaTvis arsebobs intervalTa iseTi Tvladi {Ii}

sistema, rom TiToeuli Ii intervalis diametri naklebia δ /2ricxvze, amas-Tan F1∪F2⊂∪Ii da Σ| Ii|≤ m∗( F1∪F2)+ε. aRvniSnoT Ik(F1) (Ij(F2)) simboloTi iseTi

Ii intervalebi, romlebic Seicaven F1 (F2) simravlis wertilebs. radgan

TiToeuli Ii intervali SeiZleba Seicavdes maqsimum erTi (F1 an F2) simrav-

lis wertilebs, amitom gveqneba

m∗(F1)+m∗(F2)≤ 1 2| ( ) | | ( ) |k jk j

I F I F+∑ ∑

≤1

ii

I∞

=∑ ≤ m∗(F1∪ F2)+ε.

radgan ε nebismieri dadebiTi ricxvia, amitom

m∗(F1)+m∗(F2)≤ m∗( F1∪ F2). sapirispiro utoloba gamomdinareobs Teorema 1.2.3-dan.

Teorema 1.2.6. Tu A da B namdvil ricxvTa simravlis qvesimravle-

ebia da ρ(A,B)≡δ>0, maSin

11

m∗(A∪ B)= m∗(A)+ m∗( B). ukanaskneli debuleba iseve mtkicdeba, rogorc Teorema 1.2.5. Teorema 1.2.7. Tu A aris TanaukveT Ii intervalebis Tvladi gaerTi-

aneba, maSin

m∗(A)= ∑1

ii

I∞

=

.

damtkiceba. radgan 1i iI∞=∪ intervalTa gaerTianeba faravs Tavissave

Tavs, amitom gare zomis ganmartebis Tanaxmad

m∗(A)≤ 1

ii

I∞

=∑ .

jer vigulisxmoT, rom Ii Caketili intervalebia. maSin 1.2.5 Teoremis

ZaliT yoveli naturaluri n-Tvis gveqneba

m∗(A)≥ 11

( ).n n

i iii

m I m I∗ ∗

==

⎛ ⎞=⎜ ⎟

⎝ ⎠∑∪

magram 1.2.4 Teoremis gaTvaliswinebiT

m∗(Ii)=|Ii|, e.i.

m∗(A)≥ 1

.n

ii

I=∑

n-is nebismierobis gamo

m∗(A)≥ 1

.ii

I∞

=∑

zogad SemTxvevaSi davasaxeloT ε>0 ricxvi. yovel Ii intervalSi “Cav-

xazoT” Caketili Ji intervali ise, rom

|Ji|>|Ii|-ε/2ι.

maSin ukve damtkicebuli 1.2.5 Teoremis Tanaxmad

m∗(A)≥ 11

| |,n n

i iii

m J J∗

==

⎛ ⎞=⎜ ⎟

⎝ ⎠∑∪

e.i.

m∗(A)≥ 1| |i

iJ

∞

=

>∑1

.ii

I ε∞

=

−∑

ε−is nebismierobis gamo Teorema damtkicebulia.

1.3 simravlis lebegis zoma.

zomadi simravleebis ZiriTadi Tvisebebi

gare zoma marTalia gansazRvrulia simravlis nebismier qvesimrav-

leze, magram mas ar aqvs iseTi Tvisebebi, romelTa arseboba aucilebelia

analizis Semdgomi ganviTarebisaTvis. magaliTad, gare zomisTvis, sazo-

gadod, ar aris samarTliani e.w. Tvladad adiciurobis Tviseba:

12

11

( ),i iii

m A m A∞ ∞

∗ ∗

==

⎛ ⎞=⎜ ⎟

⎝ ⎠∑∪

sadac Ai, i=1,2,..., wyvil-wyvilad TanaukveTi simravleebia. gamosavali arse-

bobs, Tu SevzRudavT gare zomis gansazRvris ares.

gansazRvreba 1.3.1. A⊂ simravle aris zomadi lebegis azriT, Tu

yoveli ε>0 ricxvisaTvis arsebobs iseTi F Caketili da G Ria simravle-

ebi, rom F⊂A⊂G da m∗(G \ F)<ε. am SemTxvevaSi vambobT, rom m∗(A) aris A simravlis lebegis zoma da aRvniSnavT mas simboloTi m(A).

Teorema 1.3.1. a) nulsimravle aris zomadi simravle; b) nebismieri

intervali aris zomadi simravle.

damtkiceba. nulsimravlis SemTxvevaSi nebismieri dadebiTi ε ricxvisa-Tvis Caketili simravlis rolSi aviRoT F=∅, xolo G Ria simravlis

rolSi vigulisxmoT Ria intervalTa {Ii} sistemas, romelic faravs nul-

simravles da Σ|Ii|<ε. maSin 1.2.3 da 1.2.4 Teoremebis ZaliT

m∗(G \ F)=m∗G≤1

( )iim I∞ ∗

==∑ 1

| | .iiI ε∞

=<∑

maSasadame, nulsimravle aris zomadi simravle.

axla vaCvenoT, rom I intervali, romlis boloebia a da b zomadia.

vTqvaT mocemulia ε>0 ricxvi. Tu a=-∞, b=∞, maSin vigulisxmebT F=G= . Tu

a>-∞, b=∞, maSin vigulisxmebT G=(a-ε/3,∞), F=[a+ε/3,∞). 1.2.4 Teoremis safuZ-

velze

m∗(G \ F)=m∗(a-ε/3, a+ε/3)=2ε/3<ε. analogiurad, I intervali iqneba zomadi, Tu b=-∞, a<∞. axla ganvixiloT

SemTxveva, roca a da b sasruli ricxvebia. am SemTxvevaSi davuSaT ε′= min{ε/3, (b-a)/2}, da

G=(a-ε′, b+ε′), F=[a+ε′, b-ε′]. maSin kvlav 1.2.3 da 1.2.4 Teoremebis gamoyenebiT miviRebT

m∗(G \ F)= m∗((a-ε′, a+ε′)∪(b-ε′, b+ε′)) ≤ m∗(a-ε′, a+ε′)+ m∗(b-ε′, b+ε′) =|(a-ε′, a+ε′)|+|(b-ε′, b+ε′)|=4ε′<ε. maSasadame, I aris zomadi.

Teorema 1.3.2. Tu A aris zomadi, maSin aseve zomadia misi damatebac Ac= \A.

damtkiceba. gansazRvris Tanaxmad nebismieri ε ricxvisaTvis arsebobs

iseTi F Caketili da G Ria simravle, rom F⊂A⊂G da m∗(G \ F)<ε. cxadia, Gc⊂Ac⊂Fc

,

sadac Gc aris Caketili, xolo Fc

aris Ria simravle, amasTan

m∗(Fc \ Gc)=m∗(G \ F)<ε. es ki niSnavs, rom Ac

aris zomadi.

Teorema 1.3.3. Tu A da B aris zomadi, maSin maTi TanakveTac A∩B zomadi simravlea. damtkiceba. vTqvaT ε>0 mocemuli ricxvia. radgan A da B zomadi simrav-leebia, amitom miZebneba iseTi F1, F2 Caketili da G1, G2 Ria simravle, rom

F1⊂A⊂G1, F2⊂B⊂G2, da

m*(G1\ F1)<ε/2 , m*(G2\ F2)<ε/2.

13

cxadia, F≡F1∩ F2 Caketili simravlea, xolo G≡G∩G2 – Ria. amave dros F⊂ ⊂A∩ B⊂G. amis garda, advili Sesamowmebelia, rom

G \ F= G∩ F c

=(G1∩G2)∩ (F1∩ F2)c

=(G1∩G2)∩ ( 1cF ∪ 2

cF ) =(G1∩G2∩ 1

cF )∪ (G1∩G2∩ 2cF )

⊂(G1∩ 1cF )∪(G2∩ 2

cF ) =(G1\ F1)∪ (G2\ F2).

maSasadame,

m∗(G \ F)≤m∗((G1\ F1)∪(G2\ F2)) ≤m∗(G1\ F1)+m∗(G2\ F2) <ε.

lema 1.3.1. vTqvaT G SemosazRvruli Ria simravlea, xolo F aris G

simravlis Caketili qvesimravle. maSin

m∗(G \ F)=m∗(G) - m∗(F). damtkiceba. rogorc cnobilia, namdvil ricxvTa simravlis yoveli

Ria qvesimravle warmoidgineba TanaukveT Ii intervalTa araumetes Tvla-

di gaerTianebis saxiT (ix. Teorema 1.1.4). amitom G \ F=∪ Ii, sadac Ii=(ai, bi) aris SemosazRvruli Ria simravleebi. maSasadame, 1.2.2-1.2.4 Teoremebis Za-

liT

m∗(G \ F)≤ ( )i ii

b a−∑ <∞.

dasaxelebuli ε>0 ricxvisaTvis avirCioT iseTi naturaluri k ricxvi, rom

m∗(G \ F)≤ ( )i ii

b a−∑ ≤1

( )k

i ii

b a=

−∑ +ε.

yovel (ai, bi) intervalSi SevarCioT iseTi daxuruli Fi≡[ci, di] Sualedi, rom

bi-ai<di-ci+ε/2i. maSin 1.2.2-1.2.5 Teoremebis safuZvelze davaskvniT

m∗(G)≤m∗(F)+m∗(G \ F)

≤m∗(F)+ 1

( )k

i ii

b a=

−∑ +ε

<m∗(F)+ 1

( )k

i ii

d c=

−∑ +2ε

= m∗(F)+ 1

( )k

ii

m F∗

=∑ +2ε

= m∗(F∪ F1∪ F2∪ ...∪ Fk)+2ε ≤m∗(G) +2ε. amrigad,

m∗(G)≤m∗(F)+m∗(G \ F)<m∗(G) +2ε. ε−is nebismierobis gamo

m∗(F)+m∗(G \ F)=m∗(G). Teorema 1.3.4. SemosazRvruli A simravle zomadia maSin da mxolod

maSin, roca yoveli ε>0 ricxvisaTvis arsebobs A simravlis iseTi Cake-

tili F qvesimravle, rom m∗(F)>m∗(A) - ε.

14

damtkiceba. (aucilebloba) Tu A aris zomadi simravle, maSin (ix. gan-

sazRvreba 1.3.1) arsebobs iseTi Caketili F da Ria G simravle, rom F⊂A⊂ G da

m∗(G \ F)<ε. maSin 1.2.3 da 1.2.3 Teoremis ZaliT

m∗(F)>m∗(A)-m∗(A\F) ≥ m∗(A)-m∗(G\F)> m∗(A)-ε.

(sakmarisoba) vigulisxmoT, rom dasaxelebuli ε>0 ricxvisaTvis arse-

bobs A simravlis iseTi Caketili F qvesimravle, rom m∗(A) - m∗(F)<ε. rad-gan A aris SemosazRvruli, amitom arsebobs misi damfaravi iseTi Ria Ii

intervalTa sistema, rom |Ii|<1 da

| |ii

I∑ -ε/2< m∗(A)<∞.

G-Ti aRvniSnoT im Ii intervalTa gaerTianeba, romelTac A simravlesTan

aqvT erTi saerTo wertili mainc. cxadia, G simravle SemosazRvrulia

da F⊂A⊂ G. 1.3.5 lemis Tanaxmad

m∗(G \ F)=m∗(G) - m∗(F)< | |ii

I∑ - m∗(F)< m∗(A)+ε/2 - m∗(F)<ε.

lebegis zomas aqvs Tvladad adiciurobis Tviseba, anu samarTliania

Teorema 1.3.5. Tu (Ai) zomad TanaukveT simravleTa mimdevrobaa da A= i iA∪ , maSin A aris zomadi da

m(A)= ( ).ii

m A∑

damtkiceba. jer ganvixilavT SemTxvevas, roca A aris SemosazRvruli.

1.3.6 Teoremis ZaliT nebismieri ε>0 ricxvisaTvis arsebobs iseTi Caketi-

li Fi⊂Ai simravle, rom

m∗(Fi)>m∗(Ai) - ε/2i+1.

magram (ix. Teorema 1.2.3) ( ) ( ).i

im A m A∗ ∗≤ ∑

cxadia, arsebobs iseTi k naturaluri ricxvi, rom

1( )

k

ii

m A∗

=∑ > m∗(A) - ε/2.

vTqvaT F≡ 1 .ki iF=∪ maSin

m∗(F)=1

( )k

ii

m F∗

=∑ >

1( )

k

ii

m A∗

=∑ - ε/2> m∗(A) - ε.

maSasadame, 1.3.4 Teoremis ZaliT A aris zomadi simravle. nebismieri na-

turaluri n-Tvis gvaqvs

1( )

n

ii

m A∗

=∑ <

1( )

n

ii

m F∗

=∑ +ε/2=

1

n

ii

m F∗

=

⎛ ⎞⎜ ⎟⎝ ⎠∪ +ε/2≤m∗(A) +ε/2.

es ki niSnavs, rom

1( )i

im A

∞∗

=∑ ≤ m∗(A).

aqedan 1.2.3 Teoremis gaTvaliswinebiT miviRebT tolobas

1( )i

im A

∞

=∑ =

1( )i

im A

∞∗

=∑ =m∗(A)=m(A).

15

axla ganvixiloT zogadi SemTxveva. vTqvaT Ij=(-j,j+1], j∈Z. maSin = j jI∈∪ Z da

A= ( )i iA∈∪ N ( )j jI∈∩ ∪ Z = ( ),i j i jA I∈ ∈∪ ∩N Z = ( ).j i i jA I∈ ∈∪ ∪ ∩Z N

1.3.3 Teoremis ZaliT Ai∩Aj simravle zomadia, amitom Teorema 1.3.1-sa Teorema 1.3.5-is damtkicebis pirveli nawilis Tanaxmad

Bj= i ji

A I∈

∩∪N

aris zomadi da

m∗(Bj)= ( )1

.i ji

m A I∞

∗

=∑ ∩

avirCioT Caketili Fi da GJ Ria simravle ise, rom Fj⊂Bj⊂Gj da

m∗(Gj \ Fj)<ε/2|j|+2. vTqvaT F=∪Fj da G=∪Gj. maSin G aris Ria. vaCvenoT, rom F aris Caketili

simravle. marTlac, F simravlidan ganvixiloT raime krebadi mimdevro-

ba. radgan es mimdevroba SemosazRvrulicaa, amitom misi elementebi SeiZ-

leba ekuTvnodes mxolod sasrul raodenoba Fj simravleebs. amasTan yo-

veli Fj aris Caketili simravle, amitom maTi sasruli gaerTianeba Caketi-

li simravlea, e.i. ganxiluli krebadi mimdevrobis zRvari ekuTvnis aR-

niSnul gaerTianebas da, maSasadame, is aris F simravlis wertili. rad-

gan A=∪Bj, amitom F⊂A⊂G. amis garda, G \ F ⊂ ∪(Gj \ Fj), e.i. m∗(Gj \ Fj)≤ ( \ )j j

im G F∗∑ < | | 2/ 2 .j

iε ε+ <∑

amrigad, A aris zomadi simravle.

axla vaCvenoT, rom lebegis zomas aqvs Tvladad adiciurobis Tviseba.

yoveli n∈N ricxvisaTvis gvaqvs

| | | | | |

( ) ( ) ( \ )j j j jj n j n j n

m B m F m B F∗ ∗ ∗

≤ ≤ ≤

≤ +∑ ∑ ∑

| | | |

( ) ( \ )j j jj n j n

m F m G F∗ ∗

≤ ≤

≤ +∑ ∑

| |

( ) .jj n

m F m Aε ε∗ ∗

≤

⎛ ⎞≤ + ≤ +⎜ ⎟

⎝ ⎠∪

maSasadame,

( ) ( .)jj

m B m A∗ ∗

∈

≤∑Z

ukanasknelidan miviRebT

,( ) ( ) ( ) ( );j i j j

i i j j

m A m A I m B m A∗ ∗ ∗ ∗≤ = ≤∑ ∑ ∑∩

anu

( ) ( ).ji

m A m A∗ ∗≤∑

sapirispiro utoloba 1.2.3 Teoremis uSualo Sedegia.

1.3.5 TeoremaSi gadmocemul Tvisebas lebegis zomis Tvladad adiciu-

robis Tviseba ewodeba. Teorema 1.3.6. Tu A1, A2,... zomad simravleTa mimdevrobaa, maSin a) ∪ Ai

da b) ∩ Ai zomadi simravleebia. damtkiceba. jer erTi, ori zomadi D da E simravlisTvis D\E= D∩ Ec. am-

itom 1.3.2 da 1.3.3 Teoremis ZaliT D\E aris zomadi simravle. ori zomadi

16

D da E simravlis gaerTianebac zomadia. ukanaskneli martivad gamomdina-

reobs D∪ E=(D\E)∪ (E\D)∪ (E∩D) warmodgenidan da Teorema 1.3.5-dan. ariT-

metikuli induqciis meTodis gamoyenebiT martivad davaskvniT, rom zomad

simravleTa sasruli gaerTianeba zomadia.

axla davubrundeT dasamtkicebel a) winadadebas. advili Sesamowmebe-

lia, rom

A=D1∪D2∪ ..., sadac

D1=A1, Dn=An\ 11 ,n

i iA−=∪ n=2,3,....

Tu gaviTvaliswinebT, rom A wyvil-wyvilad TanaukveT zomad Dn simravle-

Ta Tvladi gaerTianebaa, maSin 1.3.5 Teoremis safuZvelze davaskvniT, rom

Teorema 1.3.6-is a) nawili damtkicebulia. rac Seexeba Teorema 1.3.6-is b) nawilis mtkicebas, igi martivad gamomdinareobs morganis formulis (ix.

gv. 4) gamoyenebiT.

Teorema 1.3.7. yoveli Ria da Caketili simravle zomadia. damtkiceba. Ria simravle zomadobis dasasabuTeblad sakmarisia Sev-

niSnoT, rom yoveli intervali zomadi simravlea, nebismieri Ria simrav-

le warmoidgineba intervalTa TanaukveTi gaerTianebis saxiT da gamoviye-

noT 1.3.5 Teorema. radgan nebismieri Caketili simravle raRac Ria sim-

ravlis damatebaa da adgili aqvs 1.3.2 Teoremas, amitom Sedegi 1.3.7 dam-tkicebulia.

rogorc zemoT iqna naCvenebi (ix. Sesabamisi msjeloba Teorema

1.1.8-is Semdeg), kantoris diskontinuumi P aris srulyofili, kontinuumis

simZlavris simravle, romelic ar Seicavs arcerT intervals. igi Caketi-

li simravlea da, maSasadame, aris zomadi. Zneli ar aris imis Cveneba,

rom P simravlis [0,1] Sualedze moTavsebuli mosazRvre intervalebis si-

grZeTa jamia 11 2 4 2... ... 1.

3 9 27 3

n

n

−

+ + + + + =

amitom, Tu mosazRvre intervalTa gaerTianebas aRvniSnavT G simboloTi,

maSin 1.2.4 Teoremis ZaliT m(G)=1. amis garda, [0,1]= P∪G. aqedan davas-

kvniT, rom m([0,1])= m(P)+m(G), e.i. 1=m(P)+1. amrigad, m(P)=0. 1.3.1 gansazRvrebiT zemoT SemoRebul iqna zomadi simravlis cneba. aR-

saniSnavia, rom es cneba SeiZleba sxvagvaradac iyos mocemuli. amis saS-

ualebas gvaZlevs Semdegi

Teorema 1.3.8 (karaTeodori) vTqvaT A⊂ . A simravle aris zomadi

maSin da mxolod maSin, roca yoveli E⊂ simravlisaTvis

m∗(E)= m∗(A∩ E)+m∗(Ac∩ E). damtkiceba. (aucilebloba) sakmarisia vaCvenoT, rom m∗(E)≥m∗(A∩ E)+

m∗(Ac∩ E). ukanasknelis samarTlianoba cxadia, roca m∗(E)=∞. sxva SemTxve-

vaSi movZebniT Ria G⊃F simravles, rom m∗(G)<m∗(E)+ε. maSin 1.3.1 lemis Za-

liT

m∗(A∩ E)+m∗(Ac∩ E)≤m∗(A∩G)+m∗(Ac∩G)=m∗(G)<m∗(E)+ε. radgan ε nebismieri ricxvia, amitom ukanasknelidan miiReba saWiro uto-

loba.

17

(sakmarisoba) vTqvaT In=(-n,n), An=A∩ In, n∈N. radgan A= 1i nA∞=∪ , amitom

sakmarisia vaCvenoT An–is zomadoba. movZebnoT Ria simravleTa iseTi (Gk)

mimdevroba, rom An⊂Gk da m∗(Gk)<m∗(An)+1/k, k∈N. maSin m∗(An)+m∗( c

nA ∩Gk∩ In)=m∗(A∩Gk∩ In)+m∗(Ac∩Gk∩ In) = m∗(Gk∩ In)≤ m∗(Gk) <m∗(An)+1/k.

maSasadame, m∗( cnA ∩Gk∩ In)<1/k. es ki niSnavs, rom

( )( )cn k k nm A G I∗ ∩∩ ∩ =0.

meore mxriv,

An= ( )n k k nA G I∩∩ ∩ = ( )k k nG I∩ ∩ \ ( )cn k k nA G I∩∩ ∩ .

amrigad, An= ( )k k nG I∩ ∩ \ N, sadac N aris nulsimravle. aqedan gamomdinare-

obs An simravlis zomadoba.

Teorema 1.3.9. simravle A⊂ aris zomadi maSin da mxolod maSin,

roca arsebobs iseTi Caketil simravleTa (Fi) mimdevroba da N nulsimrav-

le, rom A=N∪(∪Fi). damtkiceba. (sakmarisoba) Tu gaviTvaliswinebT, rom nulsimravle da

nebismieri Caketili simravle zomadia, gamoviyenebT Teorema 1.3.6-s, davas-kvniT A simravlis zomadobas.

(aucilebloba) vTqvaT A zomadi simravlea. 1.3.1 gansazRvrebis Tanax-

mad yoveli ε>0 da i∈N ricxvisaTvis arsebobs iseTi Caketili Fi⊂A simrav-

le, rom m(A \ Fi)<ε/2ι. aqedan miviRebT m(A \ (∪Fi))<ε/2ι, anu m(A \ (∪Fi))=0. vTqvaT

1\ .i iA F N∞= ≡∪

maSin m(N)=0 da A= 1 .i iN F∞=∪∪

Teorema 1.3.10. A⊂ simravle aris zomadi maSin da mxolod maSin,

Tu moiZebneba Ria simravleTa iseTi (Gi) mimdevroba da N nulsimravle,

rom A=∩Gi \ N. damtkiceba. (sakmarisoba) radgan Ria simravle da N simravle zomadia;

amasTan zomad simravleTa Tvladi gaerTianeba zomadi simravlea (ix. Te-

orema 1.3.6), da bolos, zomad simravleTa sxvaoba zomadia (ix. Teorema

1.3.6-is damtkiceba), amitom debulebis sakmarisoba damtkicebulia.

(aucilebloba) vTqvaT A aris zomadi simravle. 1.3.1 gansazRvrebis Ta-

naxmad nebismieri ε>0 da i∈N ricxvisaTvis arsebobs Ria (Gi) simravleTa

iseTi mimdevroba, rom A⊂Gi, m(Gi \ A)<ε/2ι. amitom m((∩Gi) \ A) <ε/2ι, e.i. m((∩Gi) \ A) =0. debulebis dasamtkiceblad sakmarisia SemoviRoT aRniSvna

(∩Gi) \ A ≡N. maSin m(N)=0 da A=∩Gi \ N.

SemoviRoT Gδ da Fσ simravleTa tipis cneba. vityviT, rom A aris Gδ tipis simravle, Tu is warmoidgineba, rogorc Ria simravleTa Tvladi

TanakveTis saxiT. vityviT, rom A simravle aris Fσ tipis, Tu igi SeiZle-

ba warmoidgenil iqnes, rogorc Caketil simravleTa Tvladi gaerTianeba.

am ganmartebebis safuZvelze Teorema 1.3.9 da Teorema 1.3.10 SeiZleba asec

Camoyalibdes.

A simravlis zomadobisaTvis aucilebelia da sakmarisi, rom moiZeb-

nos iseTi Fσ tipis B simravle, rom A= B∪N, sadac N aris raRac nul-

simravle.

18

A simravlis zomadobisaTvis aucilebelia da sakmarisi, rom moiZeb-

nos A simravlis Semcveli iseTi Gδ tipis C simravle, rom A= C ∪ N, sad-ac N raime nulsimravlea.

Teorema 1.3.11. a) Tu A1, A2,... zomad simravleTa mimdevrobaa, romelTa-

Tvisac yoveli naturaluri i ricxvisTvis Ai⊂Ai+1, maSin simravle A=∪ Ai

aris zomadi da m(A)= .lim ( )iim A

→∞

b) vTqvaT A1, A2,... zomad simravleTa iseTi mimdevrobaa, rom Ai+1⊂ Ai (i= 1,2,...) da raRac i0 naturaluri ricxvisTvis ( )0i

m A < ∞ . maSin A=∩ Ai zoma-

dia da m(A)= .lim ( )iim A

→∞

damtkiceba. a) Tu raime i0-Tvis ( )0,im A = ∞ maSin Teoremis a) nawilis sa-

marTlianoba cxadia. amitom davuSvaT, rom yoveli i∈N-Tvis m(Ai)<∞. advi-li Sesamowmebelia, rom

A= ( )1 1 1 1 \ .i i n n nA A A A∞ ∞= = +=∪ ∪∪

An+1\ An simravleebi wyvil-wyvilad TanaukveTi zomadi simravleebia, ami-

tom 1.3.5 Teoremis ZaliT

( )( )1 1 1( ) \n n nm A m A A A∞= += ∪∪

( ) ( )1 11

( ) i ii

m A m A m A∞

+=

= + −⎡ ⎤⎣ ⎦∑

( ) ( )1

1 11

( ) limn

i in i

m A m A m A−

+→∞=

= + −⎡ ⎤⎣ ⎦∑

( ) ( )1 1( ) lim .n nnm A m A m A+→∞

= + −⎡ ⎤⎣ ⎦

( )lim .nnm A

→∞=

b) es SemTxveva martivad daiyvaneba wina SemTxvevaze. marTlac, zogado-

bis SeuzRudavad SegviZlia vigulisxmoT, rom m(A1)<∞. ganvixiloT nebis-

mieri Ria G simravle, romelic Seicavs A1simravles da m(G)<∞. cxadia, G \ A1⊂ G \ A2⊂ ... G \ An⊂...

da

( )1 1\ \ .i i i iG A G A∞ ∞= ==∪ ∪

1.3.11 Teoremis a) punqtis ZaliT

( ) ( )1\ lim \ .i i nnm G A m G A∞

= →∞=∩

aqedan

( ) ( )1 lim .i i nnmG m A mG m A∞

= →∞− = −∩

SevniSnoT, rom Tu ukanaskneli Teoremis b) nawilSi yoveli i–Tvis

m(Ai)=∞, maSin debuleba, sazogadod, ar aris samarTliani. marTlac,

vTqvaT An=(n,∞), n=1,2,... . maSin m(An)=∞ (n=1,2,...). amitom ( )lim .nnm A

→∞= ∞ meore

mxriv, ∩ An=∅, e.i. m(∩ An)=0. A⊂ simravlis Siga zoma aRiniSneba simboloTi m∗ (A) da ganimarteba

Semdegnairad: m∗(A)=sup{m∗(F): F⊂A, F aris Caketili simravle}.

19

radgan nebismieri Caketili simravle zomadia, amitom ukanasknel gan-

martebaSi m∗(F) SeiZleba SevcvaloT simboloTi m(A). Teorema 1.3.12. vTqvaT A⊂ da m∗(A)<∞. A simravle aris zomadi maS-

in da mxolod maSin, roca

m∗(A)=m∗(A). damtkiceba. (aucilebloba) m∗(A)≤m∗(A). marTlac, ganvixiloT nebismieri

Caketili F⊂A, maSin m∗(F)≤m∗(A). amitom m∗(A)=sup{m∗(F): F⊂A}≤m∗(A). sapiris-piro utolobis dasamtkiceblad gamoviyenoT Teorema 1.3.4. maSin yoveli

ε>0 ricxvisaTvis iarsebebs A simravlis iseTi Caketili F qvesimravle,

rom

m∗(F)> m∗(A)-ε. aqedan

m∗(A)≥ m∗(F)> m∗(A)-ε. ε-is nebismierobis gamo

m∗(A)≥ m∗(A). (sakmarisoba) vTqvaT axla daculia toloba m∗(A)=m∗(A). davasaxeloT

ε>0 ricxvi. maSin qveda zomis ganmartebis Tanaxmad iarsebebs iseTi Cake-

tili F⊂A simravle, rom

m∗(F)>m∗(A)-ε=m∗(A)-ε. es ki Teorema 1.3.4-is ZaliT niSnavs A simravlis zomadobas.

ukanaskneli debulebis aucilebeli nawili samarTliania nebismieri

zomadi A⊂ simravlisaTvis. jer erTi, cxadia, rom m∗(A)≤m∗(A). sapiris-piro utoloba SeiZleba damtkicdes 1.3.4 Teoremis gamoyenebis gareSec.

marTlac, radgan A simravle aris zomadi, amitom yoveli ε>0 ricxvisa-Tvis arsebebs iseTi F Caketili da Ria G simravle, rom F⊂A⊂G da m∗(G\F) <ε. G Ria simravlisaTvis moiZebneba Ria simravleTa iseTi Tvladi {Ii}

sistema, rom G⊂∪ Ii da m∗(G)>1| | .iiI ε∞

=−∑ gvaqvs

m∗(G)≤ m∗(F)+ m∗(G\F)<m∗(F)+ε. aqedan

m∗(F)> m∗(G)-ε> 1| | 2iiI ε∞

=−∑ ≥m∗(A)-2ε,

anu m∗(F)>m∗(A)-2ε, saidanac ε–is nebismierobis gamo davaskvniT, rom m∗(A) =m∗(A).

qvemoT naCvenebi iqneba, rom m∗(A)=∞ SemTxvevaSi tolobidan m∗(A)= m∗(A), sazogadod, ar gamomdinareobs A simravlis zomadoba.

1905 wels vitalis mier agebul iqna arazomadi V simravlis magaliTi.

ganvixiloT igi.

vTqvaT A⊂ da x∈ . maSin simravles x+A={a+x: a∈A} ewodeba A simravlis

Zvra x–iT. gare zomis ganmartebis Tanaxmad, A simravle aris zomadi maSin

da mxolod maSin, roca x+A aris zomadi raime x∈ –Tvis, amasTan m∗(A)= m∗(x+A).

(0,1] intervalze ganvixiloT “∼” mimarTeba: vityviT, rom (0,1] Sualedis

a da b ricxvTa Soris aris a∼b mimarTeba, Tu a-b∈Q. advilia imis Cveneba,

rom “∼” aris eqvivalentobis mimarTeba (anu 1) a∼a, 2) (a∼b) ⇒ (b∼a), 3) (a∼b, b∼c) ⇒ (a∼c)). am mimarTebiT (0,1] Sualedi daiyofa TanaukveT qvesimravle-

ebad; kerZod, (0,1]=∪ Aα, sadac Aα simravleebi aris wyvil-wyvilad Tanau-

20

kveTi da TiToeuli maTgani Seicavs “∼” mimarTebiT eqvivalentur ele-

mentebs (e.i. raime x-Tvis Aα=(x+Q)∩ (0,1]).

axla yoveli Aα simravlidan avirCioT TiTo elementi da SevadginoT

simravle V. (aq Cven gamoviyeneT amorCevis aqsioma). vaCvenoT, rom V sim-ravle ar aris zomadi. vTqvaT r1, r2,... aris (0,1] intervalis yvela racio-

naluri ricxvisagan Sedgenili mimdevroba. SemoviRoT simravle

Vk=(rk+V∩ [0,1- rk ])∪ (( rk -1)+ V∩ (1- rk,1]), ∀k∈N.

SevniSnoT, rom m∗(Vk)=m∗(V), amasTan Vi ∩ Vj ≠∅, i≠j, da (0,1]=∪ Vk. davuSvaT

V simravle aris zomadi. maSin Tvladad adiciurobis Tvisebis ZaliT

gveqneba:

1=m∗((0,1])=1 1

( ) ( ).kk k

m V m V∞ ∞

∗ ∗

= =

=∑ ∑

es ki SeuZlebelia, radgan ukanaskneli gamosaxulebis marjvena mxare Se-

iZleba iyos 0 an ∞.

es magaliTi saSualebas gvaZlevs pasuxi gavceT zemoT dasmul kiTx-

vas: arsebobs arazomadi simravle A⊂ , romlisTvisac m∗(A)=m∗(A). vTqvaT

A=V∪(1,∞), sadac V aris vitalis arazomadi simravle (0,1] Sualedidan. cxa-

dia, m∗(A) ≤m∗(A), amasTan m∗(A)=∞. amitom m∗(A)=m∗(A)=∞. A simravle ar aris

zomadi, radgan winaaRmdeg SemTxvevaSi zomadi iqneba simravle V= A∩ [0,1]. Cven vnaxeT, rom arsebobs arazomadi SemosazRvruli simravle. buneb-

rivad ismis kiTxva: xom ar SeiZleba lebegis mier SemoTavazebuli meTo-

dis ise ganzogadeba, rom simravlis nebismieri qvesimravle aRniSnuli

ganzogadebis azriT iyos zomadi? ufro zustad, dgas Semdegi amocana:

yovel SemosazRvrul E simravles unda SevusabamoT iseTi arauaryofiTi

µ(E) ricxvi (vuwodoT mas E simravlis zoma), rom sruldebodes Semdegi:

1. Tu A=[0,1], maSin µ(A)∈(0,∞);

2. Tu A da B simravleebi kongruentulia, maSin µ(A)=µ(B); 3. Tu A aris Tvladi gaerTianeba TanaukveTi Ak (k=1, 2,...) simravleebi-

sa, maSin µ(A)= ( ).kkAµ∑

am kiTxvaze pasuxs iZleva

Teorema 1.3.13. µ Sesabamisoba, romelic akmayofilebs CamoTvlil 1, 2

da 3 pirobebs, ar arsebobs.

damtkiceba. lebegis azriT arazomadi simravlis ganxilvisas agebul

iqna kongruentul TanaukveTi Vk simravleebi, romelTaTvisac (0,1]= ∪ Vk. Tu zogadobis SeuzRudavad vigulisxmebT, rom 1∈V1, maSin gveqneba [0,1]=

∪ Vk. davuSvaT, rom aRniSnuli µ funqcia arsebobs. es imas niSnavs, rom

1( ) ([0,1]).k

kVµ µ

∞

=

=∑

Vk simravleTa kongruentulobis gamo nebismieri k∈N ricxvisaTvis µ(Vk)

mniSvneloba ar iqneba damokidebuli k–ze. radgan µ([0,1]) sasruli dade-

biTi ricxvia, amitom miviRebT winaaRmdegobas.

21

2. zomadi funqciebi

am TavSi Cven SeviswavliT zomad funqciebs.

2.1 zomadi funqciebis ZiriTadi Tvisebebi

gansazRvreba 2.1.1. f : → funqcias ewodeba zomadi, Tu nebismieri

V⊂ Ria simravlisaTvis f -1(V) aris zomadi.

aRniSvnis simartivisaTvis {x∈ | f(x)≤a} simravlisaTvis Cven gamoviye-

nebT Canawers { f≤a}. aseve analogiuri azriT gavigebT gamosaxulebebs:

{ f<a}, { f>a}, { f≥a},.... Teorema 2.1.2. vTqvaT f : → . Semdegi winadadebebi aris tolfasi:

(1) f aris zomadi funqcia; (2) { f<a} aris zomadi simravle yoveli a∈ ricxvisTvis;

(3) { f≤a} aris zomadi simravle yoveli a∈ ricxvisTvis;

(4) { f>a} aris zomadi simravle yoveli a∈ ricxvisTvis;

(5) { f≥a} aris zomadi simravle yoveli a∈ ricxvisTvis;

damtkiceba. radgan { f<a}= f -1((-∞,a)), amitom (1) ⇒ (2).

(2) ⇒ (3), radgan (-∞,a]= ( , 1/ )n

a n−∞ +∩ da f -1(∩ Ai)= ∩ f -1(Ai).

(3) ⇒ (4), radgan (a,∞)=(- ∞,a]c, f -1(Ac)= f -1(A)c. (4) ⇒ (5), radgan [a,∞)= ∩ ( a-1/n,∞) da f -1(∩ Ai)=∩ f -1(Ai). (5) ⇒ (1). marTlac, Catarebuli msjelobis analogiurad SeiZleba vaC-

venoT, rom { f>a} da { f<b} zomadi simravlebia, amitom { a< f <b}={ f>a}∩ { f<b} simravlec zomadia. axla Tu gaviTvaliswinebT, im faqts, rom namdvil

ricxvTa simravleSi yoveli Ria simravle warmoidgineba Ria intervalTa

araumetes Tvladi gaerTianebis saxiT da gamoviyenebT tolobas f -1(∪ Ai)= ∪ f -1(Ai), davaskvniT, rom Teorema damtkicebulia.

Teorema 2.1.3. vTqvaT (fn) aris zomad funqciaTa mimdevroba., maSin

sup fn, inf fn, lim sup fn da lim inf fn zomadi funqciebia.

damtkiceba. radgan {sup fn ≤ a}=∩ n{ fn ≤ a} da samarTliania Teorema 2.1.2, amitom sup fn aris zomadi. analogiurad vaCvenebT, rom inf fn zomadi funq-

ciaa.

lim sup fn da lim inf fn funqciebis zomadoba Sedegia Semdegi warmodgenebi-

sa:

limsup fn=infksup{ fn | n≥k}, liminf fn=supkinf{ fn | n≥k}. Teorema 2.1.4. Tu (fn) aris zomadi funqciebis mimdevroba, maSin

simravleebi {sup fn=∞} da {inffn=-∞} zomadi funqciebia.

damtkiceba. Teorema 2.1.3-is ZaliT sup fn da inf fn zomadi funqciebia,

amitom zomadia (ix Teorema 1.3.8) simravleebic:

22

{sup fn=∞}=∩ k{sup fn>k} da {inf fn=-∞}=∩ k{inf fn<-k}.

dasaxelebuli f funqciisaTvis ganvsazRvroT:

f +(x)=max{0, f(x)}=(| f(x)|+f(x))/2, f -(x)=max{0,-f(x)}=(| f(x)|- f(x))/2.

cxadia, f(x)= f +(x) - f -(x) da |f(x)|= f +(x)+ f -(x). ukanaskneli debulebidan gamomdi-

nareobs, rom f aris zomadi funqcia, maSin f + da f –

funqciebic zomadia.

Teorema 2.1.5. Tu f, g: → zomadi funqciebia, maSin aseTivea cf, f + g, fg, f / g funqciebic, sadac c aris nebismieri namdvili ricxvi.

damtkiceba. jer vaCvenoT, rom cf aris zomadi funqcia. roca c=0, maSin es ukanaskneli cxadia. Tu c>0, maSin {cf<a}={f<a/c}; Tu c<0, maSin {cf<a}= {f>a/c}. maSasadame, 2.1.2 Teoremis Tanaxmad cf aris zomadi.

f + g funqciis zomadobis dasamtkiceblad Cven gamoviyenebT tolobas

{ f + g <a}= ( ){ } { } .r Q f a r g r∈ < − <∪ ∩

jer vaCvenoT ukanaskneli tolobis samarTlianoba. Tu erTdroulad f(x)< a-r da g(x)<r, maSin, cxadia, f (x)+g(x)<a. piriqiT, Tu f (x)+g(x)<a da, g(x)=b, ma-Sin arsebobs iseTi r racionaluri ricxvi, rom g(x)<r da f(x)+r <a; anu f(x)< a-r. amrigad, dasamtkicebeli toloba WeSmaritia.

Tu axla gaviTvaliswinebT Teorema 1.3.8-s, davaskvniT, rom f + g funqcia zomadia.

imisaTvis, rom vaCvenoT fg funqciis zomadoba, SevniSnoT, rom

f (x)+g(x)=(f +(x) - f -(x)) (g+(x) – g-(x)) =f +(x)g+(x)- f +(x)g-(x)- f -(x)g+(x)+ f -(x)g-(x).

amitom sakmarisia fg funqciis zomadoba davamtkicoT arauaryofiTi f da g funqciebisaTvis. aseTi funqciebisTvis

{ f g <a}= ( )(0, ) { / } { } ;r Q f a r g r∈ ∞ < <∩∪ ∩

ukanasknelis damtkiceba SeiZleba wina tolobis analogiurad.

imisaTvis, rom vaCvenoT f / g funqciis zomadoba, sakmarisia davamtki-coT 1/g funqciis zomadoba, romelic gamomdinareobs tolobidan:

{1/g <a}={1/ , 0} {1/ , 0}, 0},{ 0}, 0.

a g ag a g ag ag a

< > > < ≠⎧⎨ < =⎩

∩

Teorema 2.1.6. Tu f da g funqciebi namdvili ricxvTa simravleze mo-

cemuli zomadi funqciebia, maSin { f = g}, { f < g}, { f ≤ g} zomadi simrav-leebia.

damtkiceba. cxadia,

{ f < g}={ f - g <0}, { f ≤ g}={ f - g ≤0},

{ f =g}={ f≤g}∩ {g≤f }. debulebis dasamtkiceblad sakmarisia gamoviyenoT Teoremebi 2.1.5, 1.3.8.

Teorema 2.1.7. Tu (fn) aris zomadi funqciebis mimdevroba, maSin yvela

im x wertilTa simravle, romelTaTvisac (fn(x)) krebadia, aris zomadi. damtkiceba. debulebis mtkiceba uSualod gamomdinareobs limsup fn–isa

da liminf fn funqciebis zomadobidan (ix. Teorema 2.1.3), Teorema 2.1.6-dan da

im faqtidan, rom gansaxilveli simravle warmoidgineba saxiT:

{ liminf fn =limsup fn}. Teorema 2.1.8. f funqcia zomadia maSin da mxolod maSin, roca zoma-

dia f + da f -. amis garda, f –is zomadoba iwvevs | f| funqciis zomadobas.

23

damtkiceba. radgan f –is zomadobidan gamomdinareobs f + da f – funqcieb-

is zomadoba (ix. Teorema 2.1.3), amitom |f |= f + + f - funqcia iqneba zomadi. Tu

funqciebi f + da f –

zomadia, maSin f = f +- f - funqciac iqneba zomadi (ix. Teo-

rema 2.1.4). Teorema 2.1.9. Tu f : → zomadi funqciaa, xolo g : → aris uwyveti, maSin g f aris zomadi.

damtkiceba. radgan g funqcia uwyvetia, amitom nebismieri namdvili a ricxvisTvis Ga≡{g>a} Ria simravlea. vTqvaT Ga= 1 ,n nI∞

=∪ sadac {In} aris Ria

TanaukveT intervalTa Tvladi sistema (ix. Teorema 1.1.4). garkveulobisa-

Tvis vigulisxmoT, rom In=(an,bn), n=1,2,.... cxadia, {x|f(x)∈In}={x| an<f(x)<bn}={x| f(x)>an}∩ {x| f(x)<bn}.

es niSnavs, rom simravle {x|f(x)∈In} aris zomadi. SemoviRoT aRniSvna ϕ=g f. maSin gveqneba

{x| ϕ(x)>a}={x| g(f(x))>a}= {x| f(x)∈Ga}= 1{ | ( ) }.n nx f x I∞

= ∈∪ maSasadame, yoveli namdvili a ricxvisTvis simravle {x| ϕ(x)>a} warmoidgi-neba rogorc zomadi simravleebis Tvladi gaerTianeba.

debulebis dasamtkiceblad sakmarisia gamoviyenoT Teorema 1.3.8. mocemuli A simravlisaTvis ganvixiloT R–ze gansazRvruli funqcia

χA(x)=1, ,0, .

x Ax A

∈⎧⎨ ∉⎩

Tu

Tu

χA funqcias A simravlis maxasiaTebeli funqcia ewodeba.

Teorema 2.1.10. vTqvaT a1, a2,..., an gansxvavebul namdvil ricxvTa siste-

maa, xolo A1, A2,..., An namdvil ricxvTa simravlis raRac TanaukveTi qve-

simravleebia. funqcia 1 21 2 n...

nA A Af a a a= + + +χ χ χ zomadia maSin da mxolod

maSin, roca simravleebi A1, A2,..., An zomadia.

damtkiceba. SeiZleba vigulisxmoT, rom a1< a2 <...< an. es daSveba zogado-bas ar zRudavs.

davuSvaT A1, A2,..., An zomadi simravleebia.

ganvixiloT sxvadasxva SemTxveva:

1. vTqvaT a uaryofiTi ricxvia da a<a1, maSin { f >a}= ;

2. Tu 0≤a<a1, maSin { f >a}= 1 ;nk kA=∪

3. Tu a≥0 da ak≤a<ak+1, k=1,2,...,n-1, maSin { f >a}= 1 ;ni k iA= +∪

4. Tu a uaryofiTi ricxvia, romlisTvisac ak≤a<ak+1, k=1,2,...,n-1, maSin { f >a}= \ 1 ;k

i iA=∪

5. Tu a uaryofiTi ricxvia da an≤a<∞, maSin { f >a}= \ 1 ;ni iA=∪

6. Tu a≥0 da an≤a<∞, maSin { f >a}=∅; eqvsive SemTxvevaSi { f >a} simravle, misi warmodgenis sxvadasxva for-

mis miuxedavad, zomadia.

axla davuSvaT, rom f zomadi funqciaa. maSin { x| f(x)=ai}=Ai, i=1,2,...,n, Tu

ai≠0. Tu ai=0, maSin Ai –is rolSi SegviZlia vigulisxmoT simravle

Ai= \ { | ( ) }.j i jx f x a≠ =∪ ukanaskneli warmodgenis gamo Ai iqneba zomadi simravle.

24

2.2 mniSvnelovani debulebebi zomadi funqciebis Sesaxeb

am paragrafSi Cven SeviswavliT zomadi funqciebis Semdgom Tvisebebs.

am sakiTxebis kvleva dakavSirebulia lebegis, egorovis, luzinis, risisa

da sxvaTa saxelebTan.

upirveles yovlisa davadgenT kavSirs zomad da uwyvet funqciebs So-

ris. rogorc cnobilia, funqcia f uwyvetia maSin da mxolod maSin, roca

yoveli Ria G simravlisTvis f -1(G) aris Ria. kerZod, Tu G–s rolSi avi-

RebT intervals (a,∞), sadac a nebismieri ricxvia, maSin simravle f -1((a,∞)) ={ f>a} iqneba Ria, da, maSasadame, warmoadgens zomad simravles. amrigad,

yoveli uwyveti funqcia aris zomadi. magram yoveli zomadi funqcia ar

aris uwyveti. magaliTisaTvis SeiZleba ganvixiloT funqcia

sgn(x)=1, (0, ),0, 0,1, ( ,0).

xx

x

∈ ∞⎧⎪ =⎨⎪− ∈ −∞⎩

Tu

Tu

Tu

miuxedavad amisa, zomadi funqciebi arc ise Sors dganan uwyveti funq-

ciebisagan. ufro zustad, adgili aqvs Semdeg debulebas.

Teorema 2.2.1 (luzini) f : → funqcia zomadia maSin da mxolod maS-

in, roca yoveli ε>0 ricxvisaTvis moiZebneba iseTi zomadi E simravle,

rom m(E)<ε, da f funqciis SezRudva \ E simravleze aris uwyveti.

damtkiceba. aviRoT racionalur ricxvTa Q={r1, r2,..., rn,...} simravle da

ganvixiloT intervalTa (rk-1/n, rk+1/n)k,n∈N sistema. advili saCvenebelia, rom

es sistema aris Tvladi. intervalebi am sistemidan aRvniSnoT sim-

boloebiT U1, U2,.... vTqvaT f aris zomadi funqcia. maSin yoveli i∈N ricxvisaTvis arsebobs

iseTi Caketili Fi da Ria Gi simravle, rom Fi⊂ f -1(Ui)⊂Gi da m(Gi \ Fi)<ε/2i.

vTqvaT E=∪ (Gi \ Fi). cxadia, m(E)<ε. vTqvaT g aris f funqciis SezRudva \E

simravleze (g=f | \E). maSin g -1(Ui)= f -1(Ui) \ E= Gi ∩ ( \E). maSasadame, g -1(Ui) aris

Ria \E simravlis mimarT (anu g -1(Ui) simravle aris Ria simravlisa da \E simravlis TanakveTa. cxadia, yoveli Ria U simravle warmoidgineba Ui sim-

ravleTa raRac qvesistemis ∪Ui gaerTianebiT, amitom g -1(U)=g -1(∪Ui)= (∪Gi ) ∩ ( \E) aris Ria simravle \E simravlis mimarT. es ki niSnavs, rom g fun-

qcia uwyvetia \E simravleze.

piriqiT, vTqvaT (Ei) aris zomad simravleTa iseTi mimdevroba, romlis-

Tvisac m(Ei)<1/i da fi = \ iEf R aris uwyveti \Ei simravleze (i=1,2,...). aqedan gamomdinareobs, rom yoveli Ria U simravlisaTvis arsebobs iseTi Ria Gi

simravle, rom 1( )if U−

=Gi \ Ei (i∈N). vTqvaT E=∩ Ei. cxadia, m(E)=0. amrigad,

f -1(U)∩ E aris zomadi. gvaqvs f -1(U)=[ f -1(U)∩ E]∪ [ f -1(U)∩ Ec] =[ f -1(U)∩ E]∪ [ f -1(U)∩ (∪i

ciE )]

25

=[ f -1(U)∩ E]∪ [∪ i(f -1(U)∩ ciE )]

=[ f -1(U)∩ E]∪ [∪ i1( )if U− ]

=[ f -1(U)∩ E]∪ [∪ i(Gi \ Ei)]. maSasadame, f -1(U) aris zomadi simravle. es niSnavs, rom f aris zomadi funqcia.

qvemoT Cven ganvixilavT zomad funqciaTa mimdevrobis krebadobas

sxvadasxva azriT, da maT Soris damokidebulebas.

Teorema 2.2.2 (egorovi) vTqvaT A⊂ aris sasruli zomis simravle. vi-

gulisxmoT, rom zomad funqciaTa (fn) mimdevroba krebadia TiTqmis yvel-

gan (T.Yy.) A simravleze. maSin yoveli ε>0 ricxvisaTvis arsebobs iseTi

zomadi E⊂A simravle, rom m(E)<ε da (fn) mimdevroba Tanabrad krebadia f funqciisken A\ E simravleze.

damtkiceba. nebismieri n,k∈N ricxvebisaTvis davuSvaT

, { | | ( ) ( ) | 1/ }.n k ii n

A x A f x f x k∞

=

= ∈ − ≥∪

cxadia, A1,k ⊃ A2,k ⊃ ... da yoveli k∈N ricxvebisaTvis ( )1 , 0.n n km A∞= =∩ radgan

m(A1,k)≤m(A)<∞, amitom 1.3.13 Teoremis b) nawilis ZaliT ,lim ( ) 0n knm A

→∞= yoveli

k∈N-Tvis. dasaxelebuli ε>0 ricxvisTvis da k∈N-Tvis movZebnoT iseTi na-

turaluri n(k) ricxvi, rom m(An(k),k)<ε/2k. aviRoT E= ( ),1

.n k kkA∞

=∪ maSin m(E)<ε.

yoveli k∈N-Tvis gvaqvs A\E⊂A\ An(k),k. maSasadame, yoveli i≥n(k)-sa da nebismi-

eri x∈A\E-Tvis |fi(x)- f(x)|<1/k. amrigad, funqciaTa fn mimdevroba A\E simravle-

ze Tanabrad krebadia f funqciisken.

gansazRvreba 2.2.3 vTqvaT A⊂ zomadi simravlea, xolo (fn) aris A

simravleze zomad funqciaTa mimdevroba. vityviT, rom (fn) mimdevroba A

simravleze aris zomiT krebadi, Tu yoveli δ>0 ricxvisTvis

{ }( )−lim | | 0.nnm f f A

→∞≥ =δ ∩

funqciaTa mimdevrobis zomiT krebadobidan, sazogadod, ar gamomdina-

reobs amave mimdevrobis wertilovnad krebadoba. ganvixiloT Semdegi ma-

galiTi. davasaxeloT nebismieri naturaluri n ricxvi da ganvixiloT

warmodgena n=2k+r, sadac 0≤r<2k. ganvsazRvroT fn : [0,1]→ funqcia Semdegna-

irad

1, [ / 2 , ( 1) / 2 ],( )

0, \ [ / 2 , ( 1) / 2 ].

k k

n k k

x r rf x

x r r⎧ ∈ +⎨

∈ +⎩ RTu

Tu

advili Sesamowmebelia, rom (fn) mimdevroba [0,1] segmentze zomiT krebadia

igivurad 0-is toli funqciisken. maSin, roca (fn) mimdevroba ar aris 0-ken

krebadi [0,1] segmentis arcerT wertilSi.

Teorema 2.2.4 (f.risi) vTqvaT A⊂ zomadi simravlea da (fn) mimdevro-

ba A simravleze aris zomiT krebadi f funqciisken. maSin arsebobs (fn)

mimdevrobis iseTi ( )knf qvemimdevroba, romelic T.Yy. krebadia A-ze.

26

damtkiceba. avarCioT naturalur ricxvTa iseTi n1< n2<... mimdevroba, rom m({|f -

knf |≥1/k}∩A)<1/2k. maSin yoveli k∈N-Tvis davuSvaT

Ai= { }| | 1/knk i

f f k A∞

=− ≥ ∩∪ .

cxadia, gvaqvs A1 ⊃ A2 ⊃ ... . vTqvaT B= .i iA∈N∩ radgan m(A1)<Σk1/2k<∞, amitom

1.3.13 Teoremis ZaliT m(B)=limi m(Ai)< limiΣk≥i1/2k=0. maSasadame, B aris nulsim-

ravle, amasTan A \ B simravleze ( )knf qvemimdevroba krebadia f funqciisken.

marTlac, Tu x∈A \ B, maSin raRac i∈N-Tvis x∈ ciA ∩A; e.i. x∈∩ k≥i{|f -

knf |<1/k}∩A,

saidanac gamomdinareobs, rom ( )knf x → f(x), roca k→∞.

ukanaskneli debulebidan gamomdinareobs

Teorema 2.2.5 Tu (fn) funqciaTa mimdevroba A simravleze zomiT kre-

badia rogorc f, ise g funqciisken, maSin T.y. A simravleze f=g. damtkiceba. risis 2.2.4 Teoremis Tanaxmad SeiZleba movZebnoT ( )knf qve-

mimdevroba, romelic A \ A1 simravleze krebadia f funqciisken, sadac A1

raRac nulsimravlea. cxadia, ( )knf zomiT krebadia g funqciisken. amitom

kvlav risis Teoremis Tanaxmad am qvemimdevrobidan gamoiyofa ( )k pnf qve-

mimdevroba, romelic krebadi iqneba A \ A2 simravleze g funqciisken, sadac

A2 aris nulsimravle. aqedan gamomdinare davaskvniT, rom ( )k pnf qvemimdev-

roba A \ (A1∪ A2) simravleze krebadia rogorc f, aseve g funqciisken. es ki imas niSnavs, rom A \ (A1∪ A2) simravleze f=g. debulebis dasamtkiceblad

sakmarisia SevniSnoT, rom A1∪ A2 aris nulsimravle.

Teorema 2.2.6 (lebegi) vTqvaT A⊂ aris sasruli zomis simravle, xo-

lo (fn) funqciaTa mimdevroba krebadia A simravleze TiTqmis yvelgan. ma-

Sin (fn) mimdevroba iqneba zomiT krebadi A simravleze.

damtkiceba. am Teoremis damtkiceba egorovis 2.2.2 Teoremis Sedegia.

27

3. lebegis integrali

am TavSi ganvixilavT lebegis integralis Teorias. igi warmoadgens

rimanis sakuTrivi integralis ganzogadebas da moqmedebs funqciaTa ga-

cilebiT farTo klasze, vidre rimanis azriT integrebad funqciaTa kla-

sia. lebegis integrals rimanis integralTan SedarebiT aqvs sxvadasxva

RirsSesaniSnavi Tviseba da warmoadgens kvlevis umniSvnelovanes iaraRs

Tanamedrove maTematikur analizSi.

3.1 ganmartebebi da ZiriTadi Tvisebebi

Cven SevecdebiT ganvsazRvroT lebegis integrali.

gansazRvreba 3.1.1 funqcias ewodeba martivi, Tu igi aris sasruli

raodenoba maxasiaTebeli funqciis (ix. paragrafi 2.2) wrfivi kombinacia.

debuleba 3.1.2 vTqvaT Ai, Bj aris zomadi simravleebi da ai, bj≥0 (1≤i≤k, 1≤j≤l). vTqvaT ϕ1=∑ 1 i

ki Ai

a=

χ da ϕ2= .∑ 1 j

lBj

bjχ=

Tu ϕ1≤ϕ2, maSin

∑1

( )k

i ii=

a m A ≤ .∑1

( )l

j jj=

b m B

amis garda, Tu ϕ1=ϕ2, maSin ukanasknel formulaSi adgili aqvs tolobas.

damtkiceba. jer vaCvenoT, rom arsebobs iseTi TanaukveTi 1 2, ,..., kA A A

simravleebi da arauaryofiTi 1 2, ,..., ka a a ricxvebi, rom

1i

k

i Ai

a χ=

=∑1

,i

k

i Ai

a χ=∑

1( )

k

i ii

a m A=

=∑1

( ).k

i ii

a m A=∑

cxadia, ukanaskneli tolobebi samarTliania, roca k=1. davuSvaT am to-

lobebs adgili aqvs k=n-Tvis. maSin gveqneba

1

1i

n

i Ai

a χ+

=

=∑ 111

ni

n

i n AAi

a aχ χ++

=

+∑

=1\

1i n

n

i A Ai

a χ+

=∑ +

111

( )i n

n

i n A Ai

a a χ++

=

+∑ ∩ +1 1

1 \ nn j

n A Aja χ

+ =+ ∪

da

11

( \ )n

i i ni

a m A A +=∑ + 1 1

1( ) ( )

n

i n i ni

a a m A A+ +=

+∑ ∩ + 1 1 1( \ )nn n j ja m A A+ + =∪

= 1

( )n

i ii

a m A=∑ + 1 1( )n na m A+ + =

1

1

( ).n

i ii

a m A+

=∑

maSasadame, tolobebi sruldeba k=n+1-Tvisac. es imas niSnavs, rom tolo-

bebi samarTliania nebismieri k∈N-Tvis.

axla davubrundeT uSualod debulebis mtkicebas. radgan zemoTCata-

rebuli msjeloba SeiZleba gamoviyenoT ϕ2 funqciisaTvisac, amitom sakmarisia vaCvenoT debulebis samarTlianoba im SemTxvevaSi, rodesac {Ai},

{Bj} qmnis -is danawilebas. am SemTxvevaSi ai≤bj, Tu Ai∩Bj≠∅ da, maSasadame,

28

1( )

k

i ii

a m A=∑ =

1 1

( )k l

i i ji j

a m A B= =∑∑ ∩ ≤

1 1

( )l k

j i jj i

b m A B= =

∑∑ ∩ =1

( ).l

j jj

b m B=

∑

debulebis bolo winadadebis samarTlianoba gamomdinareobs im faqti-

dan, rom toloba ϕ1=ϕ2 miiReba ϕ1≤ϕ2 da ϕ1≥ϕ2 Sefasebebidan.

gansazRvreba 3.1.3 vTqvaT f aris arauaryofiTi zomadi funqcia, xo-

lo Sf aris iseT martiv ϕ funqciaTa simravle, rom 0≤ϕ≤ f. lebegis f∫R

integrali f funqciidan ganimarteba Semdegnairad

∫ fR

==∑

∑1

1sup ( ).k

i A fii

k

i iia S

a m A=

=∈ϕ χ

3.1.2 debulebidan arauaryofiTi martivi ϕ=1 i

ki Ai

a χ=∑ funqciisaTvis mivi-

RebT

ϕ∫R =1

( ).k

i ii

a m A=∑

radgan Sf⊂Sg, amitom pirobidan f ≤g gamomdinareobs f∫R ≤ .g∫R

gansazRvreba 3.1.4 zomadi f funqciisaTvis ganvsazRvroT lebegis in-

tegrali

f∫R = f +∫R - f −∫R ,

amasTan igulisxmeba, rom ukanaskneli tolobis marjvena mxareSi moTav-

sebul SesakrebTagan erTi mainc sasruli ricxvia. lebegis integrali

zomad E simravleze ganimarteba ase

∫Ef = .∫ ER

f χ⋅

vityviT, rom f aris lebegis azriT integrebadi E simravleze, Tu ∫Ef

≠ .±∞ E simravleze lebegis azriT integrebad funqciaTa simravle aRi-

niSneba simboloTi L1(E)≡L(E). Teorema 3.1.5 lebegis integrals aqvs Semdegi fundamenturi Tvise-

bebi:

(a) yoveli c∈R ricxvisaTvis ∫ ∫E Ecf c f= ;

(b) martivi ϕ1 da ϕ2 funqciebisaTvis +∫ ∫ ∫1 2 1 2(E E E

+ ) =ϕ ϕ ϕ ϕ . kerZod, zo-

madi Ai simravleebisTvis da ai∈R (1≤i≤k) ricxvebisaTvis

( )∑ ∑∫ =1 =1( );

i

k ki i ii iAE

a a m A E=χ ∩

(g) Tu E1∩ E2=∅, maSin +∫ ∫ ∫

1 2 1 2

;∪E E E E

f f f=

(d) Tu f≤g, maSin

∫ ∫ ;E E

f g≤

(e) Tu E1⊂E2 da f≥0, maSin

∫ ∫1 2

;E E

f f≤

(v) f aris integrebadi E simravleze maSin da mxolod maSin, roca

amave simravleze integrebadia | f |, da amis garda

29

.∫ ∫E Ef f≤

damtkiceba.

(a) 3.1.4 ganmartebidan gamomdinare, sakmarisia vigulisxmoT, rom f≥0. Tu c≥0, aseTi funqciebisaTvis (integralis ganmartebis Tanaxmad) dasam-

tkicebeli toloba trivialuria. vTqvaT –c>0, maSin kvlav integralis

gansazRvrebis ZaliT

( ) ( )E E EE

cf cf cf cfχ χ χ+ −= = − =∫ ∫ ∫ ∫R R R

- ( )Ecf χ −

∫R

= ( ) ( )/ 2E E Ecf cf cfχ χ χ− − = − −∫ ∫R R

= ( ) .E EE

c f c f c fχ χ− − = =∫ ∫ ∫R R

(b) upirveles yovlisa SevniSnoT, rom, rogorc 3.1.3 gansazRvrebis

Semdeg iyo aRniSnuli, nebismieri zomadi arauaryofiTi martivi ϕ=

1 i

ki Ai

a χ=∑ funqciisaTvis

ϕ∫R

=1

( )k

i ii

a m A=∑ .

ganvixiloT nebismieri zomadi martivi ϕ=1 i

ki Ai

a χ=∑ funqcia. aRvniSnoT M-

iT (analogiurad, - N-iT) yvela im i indeqsTa simravle (i=1,2,...,k), romel-

TaTvisac ai≥0 (ai<0). amitom ϕ ϕ ϕ+ −= −∫ ∫ ∫

R R R

= ( ) ( ) ( )i i i ii M i N

a m A a m A∈ ∈

− −∑ ∑

1

( )k

i ii

a m A=

= ∑ .

Tu ϕ-s rolSi ganvixilavT martivi ϕ⋅χE funqcias, maSin misTvis adgili

eqneba warmodgenas ϕ⋅χE= 1 i

ki A Ei

a χ=∑ ∩ . amitom ukve damtkicebulis gamo

E

ϕ =∫1

( ).k

i ii

a m A E=∑ ∩

vTqvaT axla ϕ1=1

(1)(1)

1 i

ki Ai

a χ=∑ da ϕ2=

2( 2)

(2)1

.j

kj Aj

a χ=∑ cxadia,

ϕ1+ϕ2= ( )1 2

(1) ( 2)(1) (2)

1 1

.i j

k k

i j A Ai j

a a χ= =

+∑∑ ∩

ukve damtkicebulis Tanaxmad

( )1 2ϕ ϕ+ =∫R

( )1 2

(1) (2) (1) (2)

1 1

( )k k

i j i ji j

a a m A A= =

+∑∑ ∩

=1 2

(1) (1) (2)

1 1

( )k k

i i ji j

a m A A= =∑∑ ∩ +

2 1(2) (1) (2)

1 1

( )k k

j i jj i

a m A A= =

∑∑ ∩

=1 2

(1) (1) (2)

1 1

( )k k

i i ji j

a m A A= =∑ ∑ ∩ +

2 1(2) (1) (2)

1 1

( )k k

j i jj i

a m A A= =

∑ ∑ ∩

=1

(1) (1)

1( )

k

i ii

a m A=∑ +

2(2) (2)

1

( )k

j jj

a m A=

∑ = 1 2.ϕ ϕ+∫ ∫R R

aqedan davaskvniT, rom

30

( )1 2E

ϕ ϕ+ =∫ ( )1 2 Eϕ ϕ χ+∫R

= ( )1 2E Eϕ χ ϕ χ+∫R

= 1 Eϕ χ∫R

+ 2 Eϕ χ∫R

= 1E

ϕ +∫ 2.E

ϕ∫

(g) integralis gansazRvris Tanaxmad sakmarisia ganvixiloT mxo-

lod arauaryofiTi f funqcia. davuSvaT ϕ=1 2( )1 i

ki A E Ei

a χ=∑ ∩ ∩ . maSin gveqneba

1 2E E

f =∫∪

1 2E Ef χ∫ ∪R

=

1 2

1 21

sup ( ( ))f E E

k

i iS i

a m A E Eχϕ ⋅∈ =

∑∪

∩ ∪

≤1

11

sup ( )f E

k

i iS i

a m A Eχϕ ⋅∈ =

∑ ∩

+

2

21

sup ( )f E

k

i iS i


∑ ∩

=1Ef χ∫

R

+2Ef χ∫

R

=1 2

.E E

f f+∫ ∫

amis garda, ganvixiloT martivi 1ϕ =1

11 i

ki A Ei

a χ=∑ ∩ da 2ϕ =

2

21

kj Bj Ej

b χ=∑ ∩ funqciebi.

vTqvaT 1ϕ ∈1EfS χ⋅ da 2ϕ ∈

2EfS χ⋅ . maSin cxadia, 1ϕ + 2ϕ ∈1 2

.E EfS χ⋅ ∪

amitom

1 2

1 21 1

( ) ( )k k

i i j ji j

a m A E b m B E= =

+∑ ∑∩ ∩

= ( )1 2

1 21 1

( ) ( )k k

i j i ji j

a b m A B E E= =

+∑∑ ∩ ∩ ∪

≤1 2

1 2

.E E

E Ef fχ =∫ ∫∪

∪R

maSasadame,

1E

f =∫1

1

11

sup ( )f E

k

i iS i


∑ ∩

≤2

1 2

21

( ),E E

k

j jj

f b m B E=

− ∑∫∪

∩

saidanac miviRebT

1E

f∫ +

2E

f ≤∫1 2E E

f∫∪

.

(d) dasamtkicebeli utoloba arauaryofiTi f funqciisaTvis uSua-

lod gamomdinareobs aseTi funqciebisaTvis integralis ganmartebidan.

zogad SemTxvevaSi sakmarisia SevniSnoT, rom x wertilSi pirobidan f (x)≤ g(x) gamomdinareobs f +(x)≤ g+(x) da f -(x)≥ g -(x).

(e) dasamtkicebeli utoloba (d) punqtis uSualo Sedegia. marTlac,

radgan E1⊂E2, amitom 1 2E Ef fχ χ≤ . maSasadame,

1E

f∫ =1 2

2

.E EE

f f fχ χ≤ =∫ ∫ ∫R R

31

(v) f funqciis integrebadobidan gamomdinareobs, rom E

f +∫ da E

f −∫ sas-

ruli ricxvebia. vTqvaT E1={x | x∈E, f(x)≥0}, E2={x | x∈E, f(x)<0}. maSin E1∩ E2=∅.

amitom (g) punqtis Tanaxmad

1 2 1 2E E E E E

f f f f= = +∫ ∫ ∫ ∫∪

=

1 2

.E E E E

f f f f+ − + −+ = + < ∞∫ ∫ ∫ ∫

Tu f zomadia da | f | integrebadia E simravleze, maSin (d) punqtis ZaliT

E

f +∫ da E

f −∫ sasruli ricxvebia. es ki niSnavs f funqciis integrebadobas E

simravleze. radgan f ≤| f | da - f ≤| f |, amitom

E

f∫ ≤E

f∫ da -E

f∫ ≤E

f∫ .

aqedan vaskvniT, rom (v) punqtis mtkiceba dasrulebulia.

Teorema 3.1.6 vTqvaT f aris E simravleze arauaryofiTi zomadi fun-

qcia. ∫ = 0E

f maSin da mxolod maSin, roca f=0 T.Yy. E simravleze.

damtkiceba. vTqvaT An={f >1/n}∩ E, n∈N. maSin

m(An)=n (1/ ) 0.nA

E E

n n fχ ≤ =∫ ∫

maSasadame, An aris nulsimravle. aqedan gamomdinareobs, rom simravle

{f >0}=∪ nAn aris nulsimravle.

lebegis integrals gaaCnia umniSvnelovanesi Tviseba. es aris integra-

lis Tvladad adiciuroba. ufro zustad, adgili aqvs Semdeg dabulebas.

Teorema 3.1.7 a) vTqvaT f zomadi, arauaryofiTi funqciaa zomad E⊂R

simravleze da E= 1 ,i iE∞=∪ sadac Ei simravleebi wyvil-wyvilad TanaukveTi

zomadi simravleebia. maSin

.∑∫ ∫iE E

ιf f

∞

=1

=

b) igive winadadeba aris samarTliani im SemTxvevaSi, rodesac f aris E simravleze integrebadi (ara aucileblad arauaryofiTi) funqcia.

damtkiceba. Teoremis b) punqtis mtkiceba martivad gamomdinareobs ama-

ve Teoremis a) punqtidan. amisaTvis sakmarisia ganvixiloT warmodgena f = f + - f – da TiToeuli SesakrebisaTvis gamoviyenoT a) punqti.

radgan a) punqtSi moiTxoveba f funqciis arauaryofiToba, amitom 3.1.5 Teoremis (g) da (e) punqtidan miviRebT

1

.i i

n

E E Ei

f f f=

= ≤∑∫ ∫ ∫∪

aqedan davaskvniT, rom

1.

iE Ei

f f∞

=

≤∑∫ ∫

axla vaCvenoT sapirispiro utolobis samarTlianoba. vTqvaT s∈Sf. vi-

gulisxmoT, rom s=1 i

ki Ai

a χ=∑ . maSin Teorema 3.1.5-is (b) punqtidan da lebe-

gis zomis Tvladad adiciurobis gamo (ix. Teorema 1.3.7)

32

( )( )11 1

( )k k

i i i i jjEi i

s a m A E a m A E∞

== =

= =∑ ∑∫ ∩ ∩ ∪

=1 1 1 1

( ) .j j

k

i i j E Ej i j j

a m A E s f∞ ∞ ∞

= = = =

= ≤∑∑ ∑ ∑∫ ∫∩

aqedan arauaryofiTi funqciidan integralis ganmartebis ZaliT

1

sup .jf

E E Es S jf s f

∞

∈ =

= ≤∑∫ ∫ ∫

3.2 zRvarze gadasvla lebegis integralis

niSnis QqveS

Teorema monotonur funqciaTa mimdevrobis Sesaxeb aris umniSvnelova-

nesi debuleba lebegis integrebis TeoriaSi. am paragrafSi Cven ganvixi-

lavT am Teoremas da mis Sedgebs. Teorema 3.2.1 vTqvaT (fn) aris arauaryofiT zomad funqciaTa zrdadi,

f funqciisaken krebadi mimdevroba zomad E simravleze. maSin

lim .n→∞∫ ∫= nE E

f f

damtkiceba. 3.1.5 Teoremis (d) punqtis Tanaxmad ricxviTi nEf∫ mimdev-

roba aris zrdadi. amave dros, es mimdevroba “zemodan SemosazRvrulia”

Ef∫ integraliT. maSasadame, arsebobs zRvari lim nEn

f→∞ ∫ da

lim nEnf

→∞ ∫ ≤E

f∫ .

Cveni mizania vaCvenoT, rom

Ef∫ ≤ lim nEn

f→∞ ∫ .

ϕ=1 i

ki Ai

a χ=∑ ∈Sf funqciisaTvis da dasaxele-buli t∈(0,1) wertilisaTvis

davuSvaT En(t)={ fn-tϕ≥0}. maSin E1(t)⊂ E2(t) ⊂... . Tu x∈E wertilisTvis ϕ(x)=0, maSin x∈ En(t) yoveli n-Tvis. Tu ϕ(x)>0, maSin f(x)≥ ϕ(x)>tϕ(x). radgan

lim ( ) ( ),nnf x f x

→∞= amitom arsebobs iseTi n, rom fn(x)>tϕ(x). ma-Sasadame, aseTi n-

Tvis x∈ En(t). amrigad, E=∪nEn(t). 3.1.7 Teoremis Tanaxmad

E

tϕ =∫1

( )k

i ii

t a m A E=∑ ∩

= ( )1

( )k

i n i ni

t a m A E t=∑ ∪ ∩

= ( )1

lim ( )k

i i nnit a m A E t

→∞=∑ ∩

= ( )1

lim ( )k

i i nn it a m A E t

→∞=∑ ∩

33

=

( )

limn

nE t

tϕ→∞ ∫ ≤

( )

limn

nnE t

f→∞ ∫

≤( )

limn

nnE t

f→∞ ∫ ≤ lim nn

E

f→∞ ∫ .

radgan t aris (0,1) intervalis nebismieri wertili, amitom miviRebT

lim .nnE E

fϕ→∞

≤∫ ∫

maSasadame,

lim .nnE E

f f→∞

≤∫ ∫

aRsaniSnavia, rom es Teorema ZalaSia maSinac, roca erT-erTi E

f∫ an

nEf∫ integralidan aris ∞.

axla Cven ganvixilavT 3.2.1 Teoremis ramdenime Sedegs. manamde davad-

genT lebegis integralis wrfivobas. ukanasknelis misaRwevad davamtki-

cebT erT mniSvnelovan debulebas.

Teorema 3.2.2 vTqvaT f aris arauaryofiTi zomadi funqcia E simrav-leze. maSin arsebobs funqciaTa zrdadi (ϕn) mimdevroba Sf klasidan, rom-

elic krebadia f –ken. amis garda, nebismier simravleze, romelzedac f aris SemosazRvruli, krebadoba Tanabaria.

damtkiceba. davuSvaT

ϕn= ( ),

2 11 1

,0

1, , , [ , ) .2 2 2

n

n i n

n

E F n i nn n ni

i i in E f F f nχ χ−

− −

=

⎛ + ⎞⎡ ⎞+ = = ∞⎟⎜ ⎟⎢⎣ ⎠⎝ ⎠∑

yoveli n-Tvis ϕn aris martivi funqcia. amasTan nebismieri n-Tvis da yo-

veli x-Tvis ϕn(x)≤ϕn+1(x) f funqciis gansazRvris simravlidan. marTlac,

vTqvaT f(x)∈ 0 0 1,2 2n n

i i +⎡ ⎞⎟⎢⎣ ⎠, sadac i0 raRac arauaryofiTi mTeli ricxvia, rome-

lic ar aRemateba n2n-1-s. am SemTxvevaSi f(x)∉Fn, e.i. ϕn(x)=i0/2n; amave dros

0 0 0 0 0 01 1 1 1 1 1

2 2 2 2 2 1 2 1 2 2( ) , , , .2 2 2 2 2 2n n n n n n

i i i i i if x + + + + + +

+ + + +⎡ ⎞ ⎡ ⎞ ⎡ ⎞∈ =⎟ ⎟ ⎟⎢ ⎢ ⎢⎣ ⎠ ⎣ ⎠ ⎣ ⎠∪

maSasadame, ϕn+1(x)=2i0/2n+1=i0/2n an ϕn+1(x)=(2i0+1)/2n+1

. amrigad, ganxilul SemT-

xvevaSi saWiro utoloba damtkicebulia. Tu f(x)∈Fn, maSin f(x)≥n da ϕn(x)=n. Tu f(x)≥n+1, maSin ϕn+1(x)=n+1, e.i. ϕn+1(x)>ϕn(x). Tu f(x)∈[n, n+1), maSin f(x) miekuT-vneba [n, n+1) intervalis 2n+1

cal tol qveintervalad danawilebis rome-

liRac 1,

2 2n n

i i +⎡ ⎞⎟⎢⎣ ⎠ intervals (i=n2n+1

, ... ,(n+1)). amitom ϕn+1(x) SeiZleba iyos to-

li n, n+1/2n+1, ... , n +(2n+1-1) /2n+1 ricxvebidan erT-erTis.

Catarebuli msjelobidan gamomdinareobs, rom dasaxelebuli f(x)–Tvis

da sakmarisad didi yoveli n–Tvis | f(x)-ϕn(x)|<1/2n, anu funqciaTa (ϕn) mim-

devroba wertilovnad krebadia f funqciisken, roca n→∞.

Tu funqcia f Semosazrvrulia, maSin (ϕn) funqciaTa mimdevrobis Tana-

bari krebadoba f–ken martivad gamomdinareobs im faqtidan, rom Fn=∅ sak-

marisad didi yoveli n–Tvis.

Teorema 3.2.3 Tu zomadi f da g funqciebi gansazRvrulia zomad E simravleze da erT-erTi aris integrebadi, maSin

34

( )E

f + g =∫ Ef∫ +

Eg∫ .

damtkiceba. zogadobis SeuzRudavad vigulisxmoT, rom g aris integre-badi. jer ganvixiloT SemTxveva, roca f da g arauaryofiTia. 3.2.2 Teore-

mis Tanaxmad arsebobs E simravleze, Sesabamisad, f da g funqciisaken krebadi arauaryofiT, zrdad, martiv funqciaTa (ϕn) da (ψn) mimdevrobebi.

cxadia, (ϕn+ψn) aris zrdadi mimdevroba, romelic krebadia f+g funqciisa-ken. 3.1.5 Teoremis (a) punqtisa da Teorema 3.2.1-is ZaliT

( ) lim ( ) lim lim .n n n nn n nE E E E E E

f g f gϕ ψ ϕ ψ→∞ →∞ →∞

+ = + = + = +∫ ∫ ∫ ∫ ∫ ∫

Semdeg, vaCvenoT, rom arauaryofiTi f da g funqciebisaTvis

( ) .E E E

f g f g− == −∫ ∫ ∫ (3.1)

davuSvaT E1={ f ≥g}∩ E, E2={ f <g}∩ E da h = f - g. damtkicebis pirveli na-

wilis Tanaxmad

1 1 1

,E E E

h g f+ =∫ ∫ ∫

2 2 2

( ) .E E E

h f g− + =∫ ∫ ∫

radgan

1E

g∫ sasrulia, amitom pirveli tolobidan miviRebT

1 1 1

( ) ;E E E

f g f g− = −∫ ∫ ∫

xolo g funqciis integrebadobidan gamomdinareobs 2

( )E

h−∫ da

2E

f∫ –is sas-

ruloba. amrigad,

2 2 2

( ) .E E E

g f g f− = −∫ ∫ ∫

amitom lebegis integralis gansazRvrisa da 3.1.5 Teoremis (g) punqtis Za-

liT

( ) ( ) ( )E E E

f g f g f g+ −− = − − −∫ ∫ ∫

=1

( )E

f g−∫ -2

( )E

g f−∫

=1E

f∫ -1E

g∫ -2E

g∫ +2E

f∫

=1 2E E

f f⎛ ⎞

+⎜ ⎟⎜ ⎟⎝ ⎠∫ ∫ -

1 2E E

g g⎛ ⎞

+⎜ ⎟⎜ ⎟⎝ ⎠∫ ∫

= .E E

f g−∫ ∫

maSasadame, sruldeba (3.1) toloba.

zogad SemTxvevaSi (3.1)-is gamoyenebiT miviRebT

( ) ( )E E

f g f f g g+ − + −+ = − + −∫ ∫

= ( ) ( )E

f g f g+ + − −⎡ ⎤+ − +⎣ ⎦∫

= ( ) ( )E E

f g f g+ + − −+ − +∫ ∫

35

=E E E E

f g f g+ + − −+ − −∫ ∫ ∫ ∫

=E E E E

f f g g+ − + −⎛ ⎞ ⎛ ⎞− + −⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠∫ ∫ ∫ ∫

= .E E

f g+∫ ∫

Cven vagrZelebT lebegis 3.2.1 Teoremis Sedegebis ganxilvas.

Teorema 3.2.4 (levi) vTqvaT (fn) aris zomad E simravleze gansazRvru-

li arauaryofiT zomad funqciaTa mimdevroba. maSin

.∑ ∑∫ ∫1 1

=i iE Ei= i=

f f∞ ∞

damtkiceba. SemoviRoT aRniSvna Sn= 1

nii

f=∑ . cxadia, funqciaTa (Sn) mim-

devroba aris E simravleze zomadi da zrdadi. aRvniSnoT misi zRvariTi

funqcia S–iT. rogorc cnobilia (ix. Teorema 2.1.3), S zomadi, arauaryo-fiTi funqciaa. amitom 3.2.1 Teoremis ZaliT

1lim .n iE E En i

S S f∞

→∞=

= = ∑∫ ∫ ∫

magram 3.2.3 Teoremis Tanaxmad 1

.nn iiE E

S f=

= ∑∫ ∫ . amitom

1 1

lim .n

i iE En i if f

∞

→∞= =

=∑ ∑∫ ∫

Teorema 3.2.5 (fatu) vTqvaT zomad E simravleze mocemulia arauar-

yofiT, zomad funqciaTa (fn) mimdevroba. maSin

.∫ ∫liminf liminfn nE En nf f

→∞ →∞≤

damtkiceba. vTqvaT gn=inf{ }kk nf

≥. maSin (gn) aris zrdad funqciaTa mimdev-

roba, romelic krebadia liminf nnf

→∞–ken, amasTan gn≤fn. amitom yoveli n–Tvis

.n nE Eg f≤∫ ∫ maSasadame,

liminf lim lim liminf liminf .n n n n nE E E E En n n n nf g g g f

→∞ →∞ →∞ →∞ →∞= = = ≤∫ ∫ ∫ ∫ ∫

Teorema 3.2.6 (lebegis Teorema SemosazRvrulad krebadobis Sesaxeb)

davuSvaT zomad funqciaTa (fn) mimdevroba wertilovnad krebadia f funq-ciisken zomad E simravleze. vigulisxmoT, rom g funqcia integrebadia E simravleze da yoveli naturaluri n-Tvis am simravleze | fn|≤. g maSin fn

integrebadia E simravleze da

∫lim 0.nEnf f

→∞− =

damtkiceba. f zomadi funqciaa da E simravleze |f |≤g. amitom g-s integ-rebadobidan gamomdinareobs fn da f funqciis integrebadoba. meore mxriv, radgan | fn - f |≤| fn |+| f |≤2g, amitom 2g-| fn - f | aris zomadi da arauaryofiTi. fa-

tus 3.2.5 Teoremis ZaliT

2 lim(2 | |)nE E ng g f f

→∞= − −∫ ∫

= liminf (2 | |)nE ng f f

→∞− −∫

≤ liminfn→∞

(2 | |)nEg f f− −∫

36

= ( )liminf 2 | |nE Eng f f

→∞− −∫ ∫

= 2E

g∫ - limsup | |.nEnf f

→∞−∫

maSasadame, 2E

g∫ integralis sasrulobis gamo limsup | | 0nEnf f

→∞− ≤∫ . aqedan

gamomdinareobs, rom lim | | 0.nEnf f

→∞− =∫

Sedegi 3.2.7 lebegis 3.2.6 Teoremis pirobebSi adgili aqvs tolobas

∫ ∫lim nE Enf f

→∞= .

ukanaskneli martivad gamomdinareobs Sefasebidan (ix. Teorema 3.1.5,

punqti (v))

nE Ef f− =∫ ∫ [ ]nE

f f− ≤∫ nEf f−∫ .

SeniSvna 3.2.8 adgili aqvs 3.2.6 Teoremaze ufro zogad debulebas. ker-

Zod, am TeoremaSi fn mimdevrobis wertilovnad krebadobis pirobis Sec-

vla zomiT krebadobiT ar iwvevs am debulebis daskvnis cvlilebas.

3.3 lebegisa da rimanis integralebis SedarebaQ

qvemoT Cven vaCvenebT, rom segmentze rimanis azriT integrebadi yove-

li funqcia integrebadia lebegis azriTac. magram segmentze lebegis az-

riT integrebadi nebismieri funqcia SeiZleba ar iyos rimanis azriT in-

tegrebadi imave segmentze. lebegis Teoria uzrunvelyofs funqciaTa ga-

cilebiT farTo klasis integrebadobas, vidre rimanis azriT integrebad

funqciaTa klasia. Tumca ukanaskneli SeniSvna exeba rimanis mxolod sa-

kuTriv integrals. marTlac, maTematikuri analizis kursidan kargadaa

cnobili, rom rimanis arasakuTrivi integrali ( )1

sin /x x dx∞

∫

1

0

1 1sin dxx x

⎛ ⎞⎜ ⎟⎝ ⎠∫

krebadia, magram krebadia ara absoluturad. es ki niSnavs (ix. Teorema

3.1.5, punqti (v)), rom ukanaskneli integralis integralqveSa funqcia ver

iqneba lebegis azriT integrebadi.

lebegis integralis yvelaze mniSvnelovani upiratesoba rimanis integ-

ralTan SedarebiT mdgomareobs lebegis integralSi zRvarze gadasvlis

operaciis ”Tavisuflad” ganxorcielebaSi.

erT-erTi siZnele, romelic rimanis integralis TeoriaSi gvxvdeba is

aris, rom rimanis azriT integrebad funqciaTa (ufro metic, uwyvet fun-

qciaTa) mimdevrobis zRvruli funqcia SeiZleba ar iyos rimanis azriT

integrebadi. magaliTisaTvis ganvixiloT funqciaTa (fn) mimdevroba [0,1]

segmentze. gadavnomroT am segmentis racionaluri ricxvebi, anu davamya-

roT bieqcia [0,1]∩Q={rk} simravlesa da naturalur ricxvTa N simravles

Soris. yoveli x∈[0,1] wertilisTvis da nebismieri n∈N-Tvis davuSvaT

37

1 2

1

1, , ,..., ,( )

0, [0,1] \ .n

n ni i

x r r rf x

x r=

=⎧= ⎨ ∈⎩ ∪

Tu

Tu

advili saCvenebelia, rom fn funqcia wyvetilia mxolod r1, r2,... ,rn werti-

lebSi. amitom yoveli fn funqcia aris rimanis azriT integrebadi [0,1] seg-

mentze. amasTan

1, [0,1],( ) lim ( )

0,nn

xf x f x

→∞

∈⎧≡ = ⎨

⎩

∩QTu

sxva SemTxvevaSi.

maTematikuri analizis kursidan kargadaa cnobili, rom zRvruli f fun-qcia ar aris rimanis azriT integrebadi [0,1] segmentze.

lebegis integralis TeoriaSi analogiuri siZnele TiTqmis gamoric-

xulia (radgan zomad funqciaTa mimdevrobis zRvari zomadi funqciaa (ix.

Teorema 2.1.3)). Tu funqcia integrebadia [a,b] segmentze lebegis azriT, maSin Sesaba-

mis integrals aRvniSnavT simboloTi [ , ]

( ) ( )b

a a bf x dx f x dx≡∫ ∫ . imisaTvis, rom

ganvasxvavoT erTmaneTisagan lebegis da rimanis integralebi, rimanis in-

tegrals CavwerT saxiT ℜ ( ) .b

af x dx∫

Teorema 3.3.1 a) Tu f funqcia integrebadia [a,b] segmentze rimanis az-riT, maSin es funqcia iqneba imave segmentze integrebadi lebegis azri-

Tac da

∫ ( )b

af x dx =ℜ ∫ ( )

b

af x dx . (3.2)

b) vTqvaT funqcia SemosazRvrulia [a,b] segmentze. imisaTvis, rom f fun-qcia [a,b] segmentze iyos rimanis azriT integrebadi, aucilebelia da sak-

marisi es funqcia iyos uwyveti T.Yy. [a,b] segmentze. damtkiceba. a) vTqvaT f funqcia aris SemosazRvruli [a,b] segmentze. yo-

veli naturaluri k–Tvis ganvixiloT [a,b] segmentis Pk danawileba: a= ( )0

kx < ( )

1kx < ( )

2kx ...< ( )k

nx =b. vigulisxmoT, rom Pk⊂Pk+1 da lim ( ) 0,kkPλ

→∞= sadac λ(Pk) aR-

niSnavs Pk danawilebis parametrs, anu λ(Pk)= ( )( ) ( )11

max .k ki ii n

x x −≤ ≤− vTqvaT

( ) ( )1 1

( ) sup ( ),k k

i i

ki

x x xM f x

− −≤ ≤=

( ) ( )1 1

( ) inf ( ),k k

i i

ki

x x xm f x

− −≤ ≤= i=1,2,...,n.

aRvniSnoT darbus zeda da qveda jamebi, Sesabamisad, S(Pk, f) da s(Pk, f) sim-boloebiT, anu

S(Pk, f)= ( )( )

( ) ( ) ( )1

1,

n kk k k

i i ii

M x x −=

−∑

s(Pk, f)= ( )( )

( ) ( ) ( )1

1.

n kk k k

i i ii

m x x −=

−∑

vTqvaT M(k)(x)= ( )kiM da m(k)(x)= ( )k

im , roca x∈[ ( )1

kix − , ( )k

ix ], i=1,2,...,n. cxadia,

S(Pk, f)= ( ) ( )b k

aM x dx∫ da s(Pk, f)= ( ) ( )

b k

am x dx∫ . (3.3)

radgan Pk+1 aris Pk danawilebis gagrZeleba (anu Pk+1⊂Pk), amitom yoveli

x∈[a,b] wertilisaTvis (1) (2) (2) (1)( ) ( ) ... ( ) ... ( ) ( ).M x M x f x m x m x≥ ≥ ≥ ≥ ≥ ≥

davuSvaT ( )( ) lim ( )k

kM x M x

→∞= da

( )( ) lim ( ).k

km x m x

→∞=

38

axla vigulisxmoT, rom f aris [a,b] segmentze integrebadi funqcia ri-manis azriT. maSin, rogorc maTematikuri analizis kursidan kargadaa

cnobili, roca λ(Pk)→0, k→∞, maSin

lim ( , )kkS P f

→∞= ℜ ( )

b

af x dx∫ da lim ( , )kk

s P f→∞

= ℜ ( )b

af x dx∫ .

meore mxriv, M(k) da m(k) funqciebi zomadi, erTobliv SemosazRvruli funq-

ciebia; aseTivea M da m funqciebi. maSasadame, lebegis Teoremis ZaliT

monotonurad krebadobis Sesaxeb (ix. Teorema 3.2.1, SesaZlebelia gamo-

viyenoT SeniSvna 3.2.7) davaskvnaT

( )lim ( ) ( )b bk

a akM x dx M x dx

→∞=∫ ∫ da

( )lim ( ) ( )b bk

a akm x dx m x dx

→∞=∫ ∫ . (3.4)

aqedan gamomdinare, gveqneba

( )b

aM x dx∫ = ( )

b

am x dx∫ =ℜ ( )

b

af x dx∫ . (3.5)

radgan [a,b] segmentze m(x)≤f(x)≤M(x), amitom M(x)-m(x)≥0, x∈[a,b] da

[ ( ) ( )] 0.b

aM x m x dx− =∫

ukanasknelidan Teorema 3.1.6–is safuZvelze davaskvniT, rom T.Yy. [a,b] seg-mentze M(x)=m(x)=f(x). radgan M da m

zomadi funqciebia, amitom aseTivea f funqciac. radgan f aris SemosazRvruli, amitom is integrebadic iqneba

lebegis azriT da

( )b

af x dx∫ = ( )

b

aM x dx∫ = ( )

b

am x dx∫ .

ukanaskneli niSnavs, rom adgili eqneba (3.2) tolobas, anu Teoremis a)

punqti damtkicebulia.

axla vaCvenoT b) punqtis samarTlianoba. upirveles yovlisa davadgi-

noT erTi saintereso faqti.

yoveli Pk danawileba (k=1,2,...) Seicavs sasrul raodenoba wertilebs,

amitom k kP∪ simravle Tvladia. maSasadame, is aris nulsimravle. aviRoT

nebismieri x∈[a,b] wertili, romelic ar miekuTvneba k kP∪ simravles. advi-

li saCvenebelia, rom f funqcia uwyvetia x wertilze maSin da mxolod

maSin, roca M(x)=m(x). marTlac, vTqvaT f funqcia uwyvetia x0∈[a,b] \ k kP∪

wertilze. es niSnavs, rom nebismieri ε>0 ricxvisaTvis arsebobs δ=δ(ε)>0, rom roca x∈(x0-δ, x0+δ), maSin f (x0)-ε< f (x)< f (x0)+ε. radgan lim ( ) 0,kk

Pλ→∞

= amitom

arsebobs iseTi N(ε), rom roca k≥ N, maSin Pk danawilebis erT-erTi ∆k seg-

-mentisTvis gvaqvs x0∈∆k⊂(x0-δ, x0+δ). es niSnavs, rom aseTi k–Tvis f (x0)-ε ≤ m(k)(x0)≤ f (x)≤M(k)(x0)≤ f (x0)+ε. ukanaskneli utoloba niSnavs, rom f (x0)-ε≤m(x0)≤ f (x0)≤M(x0)≤ f (x0)+ε. ε–is nebismierobis gamo

m(x0)= M(x0)=f (x0). (3.6) vTqvaT axla m(x0)=M(x0). Tu gaviTvaliswinebT m(x0)–isa da M(x0)–is gan-

martebas, maSin yoveli ε>0 ricxvisaTvis iarsebebs k0=k0(ε), rom M(k)(x0)- m(k)(x0)< ε. ganvixiloT

0kP danawilebis is 0k∆ qvesegmenti, romlis Siga

wertilia x0. maSin am qvesegmentis yoveli x wertilisTvis 0( )0( )km x ≤ ( )f x ≤

0( )0( ).kM x kerZod, aseTi x–ebisTvis | f (x)-f (x0)|<ε. es f funqciis uwyvetobas

niSnavs x0 wertilSi.

davubrundeT b) punqtis mtkicebas. vTqvaT f aris rimanis azriT integ-

rebadi [a,b] segmentze. maSin damtkicebuli a) punqtis ZaliT adgili aqvs

39

(3.6) tolobas, rac Tavis mxriv [a,b] segmentis TiTqmis yvela x0 wertilis-

Tvis niSnavs f funqciis uwyvetobas. piriqiT, Tu f uwyvetia T.y. [a,b] seg-mentze, maSin sruldeba (3.6). maSasadame, adgili aqvs (3.5) tolobaTagan

pirvels. axla (3.4) da (3.3)-dan davaskvniT, rom yoveli ε–Tvis moiZebneba

iseTi k, rom S(Pk, f)-s(Pk, f)<ε.

ukanasknelidan darbus kriteriumis gamoyenebiT davaskvniT, rom f aris rimanis azriT integrebadi [a,b] segmentze.

40

4. sasruli variaciis funqciebi

stiltiesis integrali

am TavSi ganvixiluli iqneba funqciaTa mniSvnelovani klasi. amasTan

Camoyalibebuli debulebebis mxolod mcire nawili iqneba damtkicebuli.

4.1 monotonuri funqciebi

maTematikuri analizis kursidan kargadaa cnobili, rom intervalze

gansazRvrul monotonur funqcias am intervalis yovel Siga wertilze

gaaCnia sasruli calmxrivi zRvrebi. aseve cnobilia, rom monotonuri

funqciis wyvetis wertilTa simravle Tvlads ar aRemateba; e.i. monoto-

nuri funqcia SeiZleba wyvetili iyos mxolod nulsimravleze. gacile-

biT Rrma Sinaarsis Semcvelia Semdegi debuleba.

Teorema 4.1.1 (lebegi) vTqvaT f aris zrdadi (ara aucileblad uwyve-

ti) funqcia [a,b] segmentze. maSin [a,b] segmentis T.y. wertilSi arsebobs

sasruli warmoebuli f ′. samarTliania ufro Zlieri debuleba.

Teorema 4.1.2 (lebegi) vTqvaT f zrdadi funqciaa [a,b] segmentze, maSin f ′ integrebadia da

′∫ ( ) ( ) - ( ).b

af x dx f b f a≤

Teorema 4.1.1 gauZlierebadia, radgan samarTliania

Teorema 4.1.3 rogori E⊂[a,b] nulsimravlec ar unda davasaxeloT,

arsebobs iseTi uwyveti, zrdadi f0 funqcia [a,b] segmentze, rom E simrav-lis nebismier x wertilze gveqneba ′0 ( ) =f x ∞.

aRsaniSnavia, rom zogad SemTxvevaSi Teorema 4.1.2-Si mocemuli utolo-

ba sazogadod ar SeiZleba Seicvalos tolobiT. marTlac, risis mier

agebul iqna [0,1] segmentze iseTi mkacrad zrdadi, uwyveti F funqciis ma-

galiTi, rom F(0)=0, F(1)=1; amasTan F funqciis warmoebuli TiTqmis yvel-

gan 0–is tolia. es niSnavs, rom

( ) 0 1 (1) (0).b

aF x dx F F′ = < = −∫ (4.1)

vaCvenoT, rom aseTi F funqcia arsebobs. Teorema 4.1.4 (risi) arsebobs [0,1] intervalze iseTi uwyveti, mkac-

rad zrdadi funqcia, romlis warmoebuli T.Yy. 0-is tolia.

damtkiceba. davafiqsiroT nebismieri t (0<t<1) da avagoT funqciaTa (Fn) mimdevroba induqciiT: F0(x)=x, x∈[0,1]. davuSvaT Fn gansazRvrulia, uwyvetia

da wrfivia TiToeul (α,β) saxis intervalze, sadac α=k2-n, β=(k+1)2-n. Fn+1

funqcias ganvsazRvravT ise, rom Fn+1(x) toli iyos Fn(x)–is x=α da x=β wer-tilebSi, xolo am intervalebis Sua x=(α+β)/2 wertilebSi ganisazRvreba

ase:

1 2nF α β+

+⎛ ⎞⎜ ⎟⎝ ⎠

=1 1( ) ( ),2 2n n

t tF Fα β− ++

41

xolo (α, (α+β)/2) da ((α+β)/2, β) intervalebze Fn+1 wrfivia. Zneli ar aris

imis Cveneba, rom amgvarad gansazRvruli TiToeuli Fn funqcia mkacrad

zrdadia. amis garda,

0≤Fn(x)≤Fn+1(x)≤1. amitom funqciaTa (Fn) mimdevroba krebadia raRac araklebadi F funqci-isaken. vaCvenoT, rom F aris mkacrad zrdadi da T.Yy. x∈[0,1] wertilisa-

Tvis ( ) 0.F x′ = vTqvaT x aris [0,1] Sualedis raRac wertili. aviRoT (αn,βn)≡ (k2-n

,(k+1)2-n) saxis Calagebuli intervalebi, romlebic Seicaven x wertils.

advili Sesamowmebelia, rom

[ ]1 1 1 1 1 11( ) ( ) ( ) ( ) .

2n n n n n n n ntF F F Fβ α β α+ + + + + +

±− = −

amis garda, Fn(αn)=F(αn) da Fn(βn)=F(βn). kerZod, F(0)=0 da F(1)=1. amitom

[ ]1 11( ) ( ) ( ) ( ) .

2n n n ntF F F Fβ α β α+ +

±− = −

miviRebT

1

1( ) ( ) , 1.2

nk

n n kk

F F εβ α ε=

+− = = ±∏

aqedan gamomdinareobs, rom

( ) ( )n nF Fβ α− >0 da

( ) ( )n nF Fβ α− ≤1 ;

2

nt+⎛ ⎞⎜ ⎟⎝ ⎠

anu

[ ]lim ( ) ( ) 0.n nnF Fβ α

→∞− =

es ki niSnavs, rom F funqcia uwyvetia da mkacrad zrdadia. maSasadame, sa-sruli warmoebuli arseobs T.y. [0,1] Sualedze (ix. Teorema 4.1.1).

ganvixiloT gamosaxuleba

( ) ( )n n

n n

F Fβ αβ α

−=

− 1

(1 ).n

kk

tε=

+∏

ukanaskneli fardobis zRvari, roca n→∞, an ar arsebobs, an usasrulo-

baa, an tolia 0-is. amrigad, T.y. [0,1] segmentze ( ) 0.F x′ =

arsebobs sxva magaliTic, romelsac aqvs agebuli funqciis analogi-

uri Tvisebebi. SeviswavloT kantoris “safexura” Θ funqcia. vaCvenoT,

rom igi aris [0,1] segmentze araklebadi uwyveti funqcia, romlis

warmoebuli T.y. [0,1] Sualedze 0-is tolia, amasTan Θ(0)=0 da Θ(1)=1.

[0,1] segmentze ganvixiloT kantoris diskontinuumi P (ix. paragrafi 1.1). igi aris srulyofili simravle, romlis zoma 0-is tolia. P simravle ai-

go etapobrivad: jer ganxilul iqna pirveli jgufis intervali (1)1I =(1/3,

2/3); Semdeg aigo meore jgufis ori intervali: (2)1I =(1/32, 2/32),

(2)2I =(7/32,

8/32); amas mohyva mesame jgufis oTxi intervali: (3)1I =(1/33, 2/33),

(3)2I =(7/33,

8/33), (3)3I =(19/33, 20/33),

(3)4I =(25/33, 26/33); da a. S. n–ur nabijze miviRebT 2n-1

cal intervals: ( )1

nI ,( )2

nI ,..., 1( )2n

nI − . P-s mosazRvre intervalTa simravle aRv-

niSnoT G0–iT. davuSvaT Θ(x)=1/2, Tu x∈ (1)1I . Semdeg, Θ(x)=1/4, Tu x∈ (2)

1I ; Θ(x)= 3/4, Tu x∈ (2)

2I . mesame jgfis (3)1I ,

(3)2I ,

(3)3I ,

(3)4I intervalebze Θ funqcia gan-

42

vsazRvroT, Sesabamisad, toli: 1/8-is, 3/8-is, 5/8-is da 7/8-is. sazogadod, n-uri jgufis intervalebze Θ funqcia ganvsazRvroT, Sesabamisad, toli:

1/2n-is, 3/2

n-is, 5/2n

-is,..., (2 1) / 2n n− -is. amrigad, es funqcia gansazRvruli

iqneba P simravlis yvela mosazRvre intervalze. amasTan mosazRvre

intervalTa gaerTianebaze (G0 simravleze) es funqcia zrdadia. ganvsaz-

RvroT igi 0 da 1 wertilebze ase: Θ(0)=0 da Θ(1)=1, xolo P simravlis

sxva wertilebze davuSvaT

Θ(x)=0[0, )

sup { }.t x G

Θ ξ∈

( )∩

advili saCvenebelia, rom [0,1] segmentze amrigad gansazRvruli funq-

cia iqneba zrdadi. es funqcia uwyveticaa [0,1] Sualedze. marTlac, jer

erTi, funqciis mniSvnelobaTa simravle mkvrivia [0,1] segmentze. meorec,

Tu Θ funqciisaTvis, rogorc monotonuri funqciisaTvis x0 wertili aris

wyvetis wertili, maSin erT-erTi Semdegi intervalebidan: (Θ(x0-0),Θ(x0)) da (Θ(x0),Θ(x0-0)), ar Seicavs Θ funqciis arcerT mniSvnelobas, rac SeuZlebe-

lia. amrigad, Θ funqcia aris uwyveti, zrdadi funqcia. G0–is yovel x wertilSi ( )xΘ′ =0. radgan P simravlis zoma 0-is tolia, amitom Θ funq-

cias gaaCnia warmoebuli T.y. da igi 0-is tolia. amrigad, Θ funqciisa-

Tvis adgili aqvs (4.1)-is analogiur utolobas.

am paragrafSi Cven mokled ganvixilavT sasruli variaciis funqciaTa

klasis Teorias. funqciaTa es klasi mWidrod ukavSirdeba monotonur

funqciebs.

gansazRvreba 4.1.5 vTqvaT [a,b] segmentze mocemulia f funqcia da P [a,b] segmentis raime danawilebaa: a=x0< x1<...< xn=b. SevadginoT jami VP =

( ) ( )−∑ 111

.nk+ kk

f x f x−

= ganvixiloT sup

P{VP}. Tu igi sasruli ricxvia, maSin

vambobT, rom f aris sasruli variaciis, xolo sidides supP

{VP}. ewodeba

f funqciis sasruli variacia da aRiniSneba simboloTi ( ).b

aV f

Teorema 4.1.6 [a,b] segmentze monotonuri funqcia aris sasruli var-

iaciis.

damtkiceba. vigulisxmoT, rom f zrdadia [a,b] segmentze. maSin nebismie-ri P danawilebisaTvis gveqneba

VP = { }1 11 11 1

( ) ( ) ( ) ( )n nk k k kk k

f x f x f x f x− −+ += =

− = −∑ ∑

=|f(xn)- f(x0)|=|f(b)- f(a)|. es niSnavs, rom zrdadi (sazogadod, monotonuri) funqciis SemTxvevaSi

( ) ( ) ( ) .b

aV f f b f a= −

ra Tqma unda, sasruli variaciis funqcia SeiZleba iyos wyvetili. ma-

galiTisaTvis SeiZleba ganvixiloT sgn funqcia [-1,1] Sualedze. magram yo-

veli uwyveti funqcia ar aris sasruli variaciis. marTlac, davuSvaT

cos , 0 1,( ) 2

0, 0.

x xf x x

x

π⎧ < ≤⎪= ⎨⎪ =⎩

Tu

Tu

dasaxelebuli n –Tvis ganvixiloT [0,1] segmentis danawileba

Pn : 0< 12n

< 12 1n −

<...< 13

< 12

<1.

43

advili Sesamowmebelia, rom

1 1/ 2 ... 1/ ,nPV n= + + +

saidanac davaskvniT, rom 1

0( ) .V f = ∞

gansazRvreba 4.1.7 vityviT, rom [a,b] segmentze f funqcia akmayofi-lebs lipSicis pirobas, Tu arsebobs iseTi K ricxvi, rom am Sualedis

nebismieri x′ da x′′ wertilebisaTvis

|f(x′′)- f(x′)|≤K| x′′- x′|. martivia imis Cveneba, rom nebismieri funqcia, romelic [a,b] segmentze

akmayofilebs lipSicis pirobas, aris sasruli variaciis.

Teorema 4.1.8 [a,b] Sualedze nebismieri sasruli variaciis funqcia

SemosazRvrulia masze.

damtkiceba. ganvixiloT [a,b] segmentis nebismieri x wertili. cxadia,

|f(x)- f(a)|≤ ( )b

aV f . amitom

|f(x)|≤| f(a)|+|f(x)- f(a)|≤| f(a)|+ ( )b

aV f .

Teorema 4.1.9 sasruli variaciis ori funqciis jami, sxvaoba da nam-

ravli sasruli variaciis funqciaa.

Teorema 4.1.10 [a,b] Sualedze gansazRvruli f funqcisTvis adgili

aqvs tolobas

( )b

aV f = ( )

c

aV f + ( )

b

cV f ,

sadac a<c<b. ukanaskneli ori debulebis marTebulobis Semowmebas mkiTxvels van-

dobT.

Teorema 4.1.11 nebismieri sasruli variaciis funqcia [a,b] Sualedze

warmoidgineba ori zrdadi funqciis sxvaobis saxiT.

damtkiceba. davuSvaT

, ,( )0, .

x

aV a x bg x

x a

⎧ < ≤⎪= ⎨⎪ =⎩

Tu

Tu

4.1.10 Teoremis Tanaxmad g funqcia zrdadia [a,b] Sualedze. SemoviRoT aR-

niSvna h(x)=g(x)-f(x). h funqcia zrdadia. marTlac, vTqvaT a≤x<y≤b. maSin 4.1.9 Teoremis ZaliT

h(y)=g(y)-f(y)= g(x)+ ( )y

xV f - f(y),

anu

h(y)-h(x)= ( )y

xV f - [f(y)-f(x)]≥0.

sabolood miviRebT

f(x)=g(x)-h(x).

Teorema 4.1.12 vTqvaT f sasruli variaciis funqciaa [a,b] segmentze

da aris uwyveti x0∈[a,b] wertilze, maSin g funqcia, sadac g(x)= ( )x

aV f ,

x∈[a,b], uwyvetia x0∈[a,b] wertilze.

44

damtkiceba. vTqvaT x0<b da vaCvenoT, rom g funqcia marjvnidan uwyve-

tia x0 wertilSi. amisaTvis aviRoT nebismieri ε>0 ricxvi da ganvixiloT

[x0,b] Sualedis iseTi danawileba x0< x1<...< xn=b, rom

0

1

10

( ) ( ) ( ) .n b

k k xkf x f x V f ε

−

+=

− > −∑

radgan ukanaskneli jami izrdeba axali wertilebis damatebiT, amitom

SeiZleba vigulisxmoT, rom |f(x1)-f(x0)|<ε. am SemTxvevaSi gveqneba

0

1

11

( ) ( ) ( ) 2nb

k kx kV f f x f xε ε

−

+=

< + − <∑

1

1

11

( ) ( ) 2 ( ).n b

k k xk

f x f x V fε−

+=

+ − ≤ +∑

aqedan davaskvniT, rom 1

0

( ) 2x

xV f ε< . maSasadame, 0≤g(x1)-g(x0)<2ε. es niSnavs, rom

g(x0+0)-g(x0)≤2ε. ε–is nebismierobis gamo g(x0+0)=g(x0). analogiurad vaCvenebT, rom g(x0-0)=g(x0), Tu x0>a.

4.2 stiltiesis integrali

axla ganvixilavT rimanis integralis mniSvnelovan ganzogadebas.

vTqvaT [a,b] segmentze mocemulia ori sasruli f da α funqcia. ganvixi-loT [a,b] segmentis danawileba Π : a=x0< x1<...< xn=b. danawilebis yovel

[xk , xk+1] segmentze avirCioT ξk wertili da SevadginoT jami

σ= [ ]1

10

( ) ( ) ( ) .n

k k kk

f x xξ α α−

+=

−∑

gansazRvreba 4.2.1 vityviT, rom arsebobs stiltiesis integrali -

∫b

afdα , Tu moiZebneba iseTi A ricxvi , rom yoveli ε>0 ricxvisaTvis ar-

sebobs δ=ε(ε)>0, rom [a,b] segmentis nebismieri P danawilebisaTvis, rom-

lis parametri λ(P)<δ da nebismieri ξk∈[xk , xk+1] sistemisaTvis (k=1,2,...,n), gvaqvs |σ- A|<ε.

aRvniSnavT stiltiesis integralis ramdenime cxad Tvisebas.

1. ( ) .= +∫ ∫ ∫1 2 1 2

b b b

a a af f d f d f d+ α α α

2. ∫ ∫ ∫( ) ;b b b

a a afd fd fd1 2 1 2+ = +α α α α

3. Tu k da l raRac mudmivebia, maSin

∫ ∫ ;b b

a akfdl = kl fdα α

(ukanasknel sam punqtSi marjvena mxaris arsebobis SemTxvevaSi ar-

sebobs marcxena mxarec)

4. Tu a<c<b da arsebobs integrali ∫b

afdα , maSin aseve arsebobs orive

integrali, romlebic Semdegi tolobis marjvena mxareSi monawile-

obs da

∫b

afdα = ∫

c

afdα + ∫

b

cfdα .

45

me-4 Tvisebis damtkicebisaTvis sakmarisia ganvixiloT [a,b] segmentis iseTi danawilebebi, romlebic Seicaven c wertils. advilia imis Cveneba,

rom b

afdα∫ integralis arsebobidan gamomdinareobs TiToeuli

c

afdα∫ da

b

cfdα∫ integralis arseboba.

aRsaniSnavia, rom sapirispiro debulebas ar aqvs adgili. vTqvaT

f(x)=0, 1 0,1, 0 1.

xx

− ≤ ≤⎧⎨− < ≤⎩

roca

roca da α(x)=

0, 1 0,1, 0 1.

xx

− ≤ <⎧⎨ ≤ ≤⎩

roca

roca

advili saCvenebelia, rom 0 1

1 00.fd fdα α

−= =∫ ∫ amasTan

1

1fdα

−∫ ar arsebobs.

marTlac, ganvixiloT [-1,1] segmentis iseTi danawileba, romelic ar Sei-

cavs 0-s. SevadginoT jami σ = 110

( )[ ( )nk kk

f xξ α−+=∑ − ( )].kxα advili dasanaxia,

rom Tu 0∈(xi , xi+1), maSin σ jamSi mxolod i–uri Sesakrebi iqneba 0-gan gan-sxvavebuli. amrigad, σ = 1( )[ ( )i if xξ α + ( )]ixα− = f(ξi); amasTan, imis mixedviT

ξi≤0, Tu ξi>0, gveqneba σ=0 an σ=1. amrigad, ar arsebobs σ–s zRvari.

5. erT-erTi ∫b

afdα da ∫

b

adfα integralis arsebobidan gamomdinareobs

meore integralis arseboba da

∫b

afdα + ∫

b

adfα =f(b)α(b)- f(a)α(a).

am formulas ewodeba nawilobiTi integrebis formula stiltiesis

integralisaTvis. davamtkicoT es Tviseba. garkveulobisaTvis vigulis-

xmoT, rom arsebobs b

afdα∫ . davanawiloT [a, b] segmenti da SevadginoT ja-

mi

[ ]110

( ) ( ) ( )nk k kk

f x xσ ξ α α−+=

= −∑

1 110 0

( ) ( ) ( ) ( );n nk k k kk k

f x f xξ α ξ α− −+= =

= −∑ ∑

Zneli ar aris imis Semowmeba, rom

[ ]110

( ) ( ) ( )nk k kk

x f fσ α ξ ξ−−=

= − −∑

1 0 0( ) ( ) ( ) ( )n nf x f xξ α ξ α−+ − = ( ) ( ) ( ) ( )f b b f a aα α− - 0{ ( )[ ( ) ( )]a f f aα ξ −

+ [ ]111

( ) ( ) ( )nk k kk

x f fα ξ ξ−−=

−∑

+ 1( )[ ( ) ( )]}.nb f b fα ξ −−

figurul frCxilebSi moTavsebuli gamosaxuleba warmoadgens integra-

lis b

adfα∫ Sesabamis jams. Tu max(xk+1-xk)→0, maSin max(ξk+1-ξk) →0. ase rom,

figurul frCxilebSi moTavsebuli jami miiswrafis b

adfα∫ integralisken.

aqedan gamomdinareobs dasamtkicebeli.

Teorema 4.2.2 Tu f funqcia uwyvetia [a,b] segmentze, xolo α aris sa-

sruli variaciis funqcia amave segmentze, maSin arsebobs ∫b

afdα .

damtkiceba. sakmarisia (ix. Teorema 4.1.11) vigulisxmoT, rom α funqcia aris zrdadi. ganvixiloT [a, b] segmentis raime Π danawilebis darbus ze-

da da qveda jamebi:

46

s= [ ]110

( ) ( )nk k kk

m x xα α−+=

−∑ , S= [ ]110

( ) ( )nk k kk

M x xα α−+=

−∑ , sadac mk da Mk aris funqciis, Sesabamisad, infimumi da supremumi danawi-

lebis ∆k=[xk, xk+1] qvesegmentze. arcerTi zeda S jami ar aris naklebi arc-

erT qveda s jamze. ukanasknelis Cveneba zustad iseve SeiZleba, rogorc

rimanis integralisaTvis mtkicdeba darbus zeda da qveda jamebis analo-

giuri Tviseba.

aRvniSnoT I–iT s jamebis simravlis zeda sazRvari, anu I=sup{s}. cxa-dia, yoveli danawilebisTvis s≤ I ≤S. radgan s≤σ≤S, amitom |σ - I|≤S-s. radgan f funqcia [a, b] segmentze aris Tanabrad uwyveti, amitom yoveli ε>0 ricxvi-saTvis arsebobs δ=δ(ε)>0, rom roca |x′-x″|<δ, maSin |f(x′) - f(x″)|<ε. aqedan davas-kvniT, rom roca λ(Π)<δ, maSin Mk - mk<ε, k=0, 1, ... , n-1. amrigad,

S-s<ε[α(b) - α(a)]. maSasadame, Tu λ(Π)<δ, gveqneba

|σ - I|<ε[α(b) - α(a)].

es ki imas niSnavs, rom 0

lim Iλ

σ→

= , e.i. arsebobs b

afdα∫ .

Teorema 4.2.3 Tu f funqcia uwyvetia [a,b] segmentze, xolo α funq-ciis warmoebuli funqcia integrebadia [a,b] segmentze, maSin

∫b

afdα = ′∫

b

af α .

damtkiceba. Teoremis pirobidan gamomdinareobs, rom α′aris Semosaz-Rvruli funqcia [a,b] segmentze. amitom α funqcia akmayofilebs lipSicis

pirobas da, maSasadame, aris sasruli variaciis. 4.2.2 Teoremis ZaliT es

niSnavs, rom b

afdα∫ arsebobs; arsebobs aseve .

b

af α′∫ darCa vaCvenoT integ-

ralebis toloba. am mizniT ganvixiloT [a,b] segmentis raime Π danawile-

leba da TiToeuli sxvaobisaTvis gamoviyenoT lagranJis toloba 1( )kxα +

( )kxα− = 1( )( ),k k kx xα ξ +′ − 1( , ).k k kx xξ +∈ Tu b

afdα∫ integralisTvis Sesabamisi σ

jamis Sedgenisas ξk –s rolSi ganvixilavT kξ –s, miviRebT

σ= 111

( ) ( )( ).nk k k kk

f x xξ α ξ−+=

′ −∑

ukanaskneli ki [a,b] segmentze integrebadi f α′funqciis rimanis jamia.

Teoremis dasamtkiceblad sakmarisia ganvixiloT ukanaskneli jamis

zRvari, roca λ(Π)→0.

Teorema 4.2.4 Tu f uwyveti funqciaa, xolo α aris [a,b] segmentze

sasruli variaciis funqcia, maSin

| ∫b

afdα |≤M(f) ⋅

b

aV α,

sadac M(f)=[ , ]

max | ( ) | .x a b

f x∈

damtkiceba. Teoremis pirobebSi arsebobs b

afdα∫ , amasTan [a,b] segmentis

yoveli danawilebisaTvis da ξk wertilebis nebismieri arCevisaTvis

|σ|=| [ ]110

( ) ( ) ( )nk k kk

f x xξ α α−+=

−∑ |

1

10( ) ( ) ( )n

k kkM f x xα α−

+=≤ −∑

( ) .b

aM f V α≤

47

axla ganvixiloT stiltiesis integralSi zRvarze gadasvlis Sesaxeb

Teorema 4.2.5 (heli) vTqvaT α sasruli variaciis funqciaa [a,b] seg-mentze, xolo (fn) aris uwyvet funqciaTa f-ken Tanabrad krebadi mimdevro-

ba. maSin

∫limb

nanf dα

→∞= ∫ .

b

afdα (4.2)

damtkiceba. 4.2.4 Teoremis ZaliT

|b

naf dα∫ -

b

afdα∫ |≤

[ , ]max | ( ) ( ) | .

b

nx a b af x f x V α

∈− ⋅

Teoremis dasamtkiceblad sakmarisia SevniSnoT, rom [ , ]

max | ( ) ( ) |nx a bf x f x

∈− →0,

n→∞.

Teorema 4.2.6 vTqvaT f aris uwyveti [a,b] segmentze da am segmentis yovel x wertilSi (αn) funqciaTa mimdevroba krebadia sasruli α funqci- saken. Tu yoveli n–Tvis

b

naV α Μ( ) ≤ < ∞,

maSin

∫limb

nanfd

→∞α = ∫ .

b

afdα

Cven am Teoremis mtkicebas ar ganvixilavT.

4.3 ramdenime SeniSvna lebeg-stiltiesis

integralTan dakavSirebiT

aqamde Cven vixilavdiT e.w. riman-stiltiesis integrals. axla Gganv-

sazRvroT lebeg-stiltiesis integrali.

vTqvaT α zrdadi funqciaa namdvil ricxvTa simravleze. rogorc

cnobilia, am funqciis calmxrivi zRvari arsebobs simravlis yovel

wertilSi. davuSvaT

mα([a,b])=α(b+0)−α(a-0), mα([a,b))=α(b-0)−α(a-0),

mα((a,b])=α(b+0)−α(a+0), mα((a,b))=α(b-0)−α(a+0).

cxadia, m([a,b])=m([a,b))=m((a,b])=m((a,b))=b-a. gansxvavebiT m funqciisagan,

sazogadod, mα damokidebulia intervalis gvarobazec.

lebegis zomis gansazRvrisas Cven gamoviyeneT lebegis gare zomis

cneba (ix. gansazRvreba 1.2.1). amjeradac SemoviRoT analogiuri

gansazRvreba 4.4.1 namdvil ricxvTa simravlis nebismieri A qvesim-ravlisaTvis gare zoma mα

∗ ganvsazRvroT ase:

mα∗ ( A)=inf ⎧ ⎫

⎨ ⎬⎩ ⎭∑

1 1

( ) : ∪ ii i

m I A Iα

∞∞

= =

⊂ ,

sadac I aRniSnavs nebismier intervals.

48

Tu CavatarebT Sesabamis msjelobas, maSin davrwmundebiT, rom mα∗–s

aqvs yvela is Tviseba, rac hqonda m∗ funqcias (ix. paragrafi 1.2 da 1.3).

amis gamo SeiZleba ganimartos α funqciis mimarT zomadi simravlis cne-

ba.

gansazRvreba 4.4.2 A⊂ simravle aris zomadi lebegis azriT α zrdadi funqciis mimarT, Tu yoveli ε>0 ricxvisaTvis arsebobs iseTi F Caketili da G Ria simravleebi, rom F⊂A⊂G da mα

∗ (G \ F)<ε. am SemTxvevaSi

vambobT, rom mα∗ (A) aris A simravlis lebeg-stiltiesis zoma da mas aR-

vniSnavT simboloTi mα(A). mas Semdeg , rac SemoRebulia lebeg-stiltiesis zoma, SeiZleba ukve

cnobili azriT (ix. gansazRvreba 2.1.1) ganimartos zomadi funqciis cneba lebeg-stiltiesis zomis mimarT. amgvarad gansazRvrul zomad funqciebs

eqnebaT is Tvisebebi, romlebic gamoTqmulia me-2 TavSi. bolos Semovi-

RebT (3.1.3 da 3.1.4 gansazRvrebebis analogiurad) integrebadi funqciis

cnebas lebeg-stiltiesis azriT. am integrals eqneba lebegis integralis

is Tvisebebi, romlebic gadmocemulia me-3 TavSi.

49

5. absoluturad uwyveti funqciebi

lebegis ganusazRvreli integrali

5.1 absolururad uwyveti funqciebi

gansazRvreba 5.1.1 vTqvaT f funqcia aris [a,b] segmentze gansazRvru-li sasruli funqcia. davuSvaT yoveli ε>0 ricxvisaTvis arsebobs δ=δ(ε) >0, rom wyvil-wyvilad TanaukveT intervalTa yoveli sasruli [ak, bk] (k= 1,2,…,n) sistemisaTvis, romlisTvisac ∑ =1

( - )nk kk

b a δ< , sruldeba utoloba

∑ =1( ) - ( )n

k kk| f b f a ε|< .

am SemTxvevaSi vambobT, rom f funqcia aris absoluturad uwyveti da Cav-

werT f∈AC[a,b]. Teorema 5.1.2 absoluturad uwyveti funqcia aris: a) uwyveti; b) sas-

ruli variaciis.

damtkiceba. Teoremis a) nawili cxadia; damtkicebisaTvis sakmarisia

gansazRvreba 5.1.1 CavweroT n=1 SemTxvevaSi. b) nawilis saCveneblad Sev-

niSnoT, rom absoluturad uwyveti funqciis gansazRvrebis Tanaxmad, ε= 1-Tvis moiZebneba δ(1), rom [a,b] Sualedis nebismieri danawilebisaTvis, Tu

1( )n

k kkb a

=−∑ <δ(1), maSin

1| ( ) ( ) | 1.n

k kkf b f a

=− <∑ . ganvixiloT [a,b] segmentis ne-

bismieri danawileba Π : a=x0< x1<...< 0nx =b, romlisTvisac xk+1- xk<δ(1) (k=0,1,...,

n0-1). cxadia, 1

( ) 1.k

k

x

xV f

+

≤ amitom 4.1.10 Teoremis Tanaxmad 0( ) .b

aV f n≤

advili Sesamowmebelia, rom absoluturad uwyvet funqciaTa wrfivi

kombinacia da namravli absoluturad uwyvetia.

vTqvaT f aris integrebadi lebegis azriT [a,b] segmentze. maSin is in-tegrebadi iqneba nebismier qvesegmentze. SemoviRoT aRniSvna

( ) ( ).x

aF x C f t= + ∫ (5.1)

F funqcias ewodeba f funqciis ganusazRvreli integrali.

Teorema 5.1.3 [a,b] segmentze integrebadi f funqciis ganusazRvreli

integrali aris absoluturad uwyveti funqcia.

damtkiceba. integralis gansazRvrebis ZaliT (ix. paragrafi (3.1)) nebis-

mieri ε>0 ricxvisaTvis moiZebneba iseTi δ>0, rom roca m(e)<δ, gveqneba | | .

ef ε<∫ f funqciis integrebadobis ganmartebidan gamomdinareobs iseTi

martivi ϕ funqciis arseboba, rom [ , ]

| | / 2;a b

f ϕ ε− <∫ meore mxriv, dasaxele-

buli ε>0–Tvis moiZebneba iseTi δ=δ(ε)>0, rom [a,b] segmentis nebismieri zo-madi e qvesimravlisaTvis, romlisTvisac m(e)<ε, gveqneba | | / 2.

eϕ ε<∫ amri-

gad,

| |e

f ≤∫ | | | |e e

f ϕ ϕ− + ≤∫ ∫ [ , ]| | | |

a b ef ϕ ϕ− + ≤∫ ∫ .

2 2ε ε ε+ =

50

kerZod, Tu e simravlis rolSi ganvixilavT TanaukveT intervalTa sis-

temas (αk, βk), k=1,2,...,n, romelTa sigrZeTa jami naklebia δ–ze, anu srulde-

ba utoloba ( )1( , )nk k km α β δ= <∪ , miviRebT

1 ( , )| ( ) | .

nk k k

f tα β

ε=

<∫∪

aqedan gveqneba

1| ( ) ( ) |n

k kkF Fβ α

=−∑ =

1( )k

k

n

kf t

β

α=∑ ∫ ≤

1

( )k

k

n

kf t

β

α==∑ ∫

1 ( , )| ( ) | .

nk k k

f tα β

ε=

<∫∪

daumtkiceblad CamovayalibebT Semdeg mniSvnelovan debulebebs.

Teorema 5.1.4 (5.1) ganusazRvreli integralis warmoebuli T.y. tolia

integralqveSa funqciis.

Teorema 5.1.5 absoluturad uwyvet funqcia aris misi warmoebulis

ganusazRvreli integrali.

mkiTxels vTxovT Seadaros ukanaskneli debuleba Teorema 4.1.2-s.

Teorema 5.1.6 vTqvaT f integrebadia [a,b] segmentze, xolo F ganisaz-Rvreba (5.1) tolobiT. maSin

∫b b

αaV F | f |= .

gansazRvreba 5.1.7 funqcias, romelic gansxvavebulia mudmivisagan,

ewodeba singularuli, Tu misi warmoebuli T.y. 0-is tolia.

Teorema 5.1.8 uwyveti, [a,b] segmentze sasruli variaciis funqcia er-

TaderTi saxiT warmoidgineba formiT

f(x)=ϕ(x)+r(x), sadac ϕ aris absoluturad uwyveti da ϕ(a)=f(a), xolo r aris singularu-

li funqcia, an igi aris 0-is toli.

Teorema 5.1.9 Tu f funqcia uwyvetia [a,b] segmentze, xolo α aris ab-soluturad uwyveti masze, maSin

′=∫ ∫b b

a afd fα α .

Teorema 5.1.10 Tu f funqciis warmoebuli [a,b] segmentze arsebobs, sasrulia da integrebadia, maSin nebismieri x∈[a,b]-Tvis

f(x)= f(a)+ .′∫x

af

51

6. lebegis jeradi integrali

am Tavis mizania lebegis integralis idea ganvazogadoT ori da meti

cvladis funqciisaTvis.

iseve rogorc erTi cvladis SemTxvevaSi, mravali cvladis funqciis

lebegis integrali warmoadgens rimanis integralis ganzogadebas. ro-

gorc cnobilia, rimanis jeradi integralisTvis samarTliania fubinis

tipis (da ara fubinis) Teorema. am integralisTvis es debuleba ramden-

adme xelovnuria, radgan aq Tavs iCens zeda da qveda integralebi, ro-

melTa dayvana rimanis integralze mxolod kerZo SemTxvevebSi (magaliT-

ad, uwyveti funqciis SemTxvevaSi) Tu aris SesaZlebeli. sxva mdgomare-

obaa lebegis jeradi integralis SemTxvevaSi. fubinis Teorema saSuale-

bas iZleva lebegis jeradi integralis gamoTvlis problema zogad SemT-

xvevaSic daviyvanoT erTjerad integralze.

Cveni midgoma mravalganzomilebian SemTxvevaSic gulisxmobs Ria da

Caketili simravleebis Teoriis ganviTarebas. am bazaze gare zomisa da

zomadi simravlis cnebebis dafuZnebas, mravali cvladis martivi funq-

ciis ganxilvas da misi saSualebiT lebegis jeradi integralis gansaz-

Rvras, am integralis Tvisebebis dadgenas, fubinisa da tonelis Teore-

mebis ganxilvas.

6.1 zogierTi SeniSvna lebegis jerad

integralTan dakavSirebiT

maTematikuri analizis kursidan kargadaa cnobili n sivrce da am si-

vrceSi Ria da Caketili simravleebis cnebebi. miuxedavad amisa Cven mok-

led SevexebiT maT.

gansazRvreba 6.1.1 vTqvaT n naturaluri ricxvia. n–ganzomilebiani

sivrcis (n sivrcis) x wertili ewodeba x1, x2, ..., xn ricxvTa dalagebul

n-euls, anu x=(x1, x2, ..., xn). manZili am sivrcis or x=(x1, x2, ..., xn) da y=(y1, y2, ..., yn) wertils Soris moicema tolobiT

ρ(x,y)= ∑ 21( ) .n

i iix y

=−

rogorc cnobilia, am sivrceSi ganisazRvreba wertilis marTkuTxova-

ni midamo.

gansazRvreba 6.1.2 vTqvaT x=(x1, x2, ..., xn)∈ n, ε=(ε1, ε2,...,εn), εi>0, i=1,2,...,n. simravles

PP(x,ε)={y=(y1, y2,..., yn) | xi -εi < yi < xi +εi, i=1,2,...,n} ewodeba x wertilis n–ganzomilebiani marTkuTxovani midamo.

Tu n=1, maSin P(x,ε) warmoadgens 2ε sigrZis intervals centriT x=x1 wer-

tilSi. roca n=2, maSin P(x,ε) aris marTkuTxedi centriT x=(x1, x2) wertilSi da 2δ1 da 2δ2 sigrZis gverdebiT.

n–ganzomilebiani Ria I(n) intervali ewodeba simravles

52

I(n)={x=(x1, x2,..., xn) | ai < xi <bi, i=1,2,...,n},

sadac ai da bi raRac namdvili ricxvebia. Tu yoveli i–Tvis (i=1,2,...,n) ai≤ xi≤bi, maSin I(n)

intervals Caketili intervali ewodeba. am da yvela sxva

SemTxvevaSi I(n) simravles intervali ewodeba.

gansazRvrebiT I(n) intervalis farTobia |I(n)

|=1( ).n

i iib a

=−∏

gansazRvreba 6.1.3 E⊂ n simravles ewodeba SemosazRvruli, Tu ar-

sebobs 0=(0,0,...,0)∈ n wertilis iseTi P(0,ε) midamo, rom E⊂ P(0,ε).

iseve rogorc erTganzomilebian SemTxvevaSi, n sivrceSic mimdevro-

bis cneba ganimarteba, rogorc asaxva m→x(m)=( ( )1 ,mx ( )

2 ,mx ..., ( )mnx ) naturalur

ricxvTa simravlisa n sivrceSi. rogorc cnobilia,

n sivrcis x(m)

mim-

devrobis amave sivrcis x elementisken krebadoba niSnavs ( )( )( , )mx xρ ric-

xviTi mimdevrobis 0-ken krebadobas, roca m→∞. n sivrceSic, erTganzo-

milebiani SemTxvevis analogiurad, ganisazRvreba dagrovebis (zRvruli),

izolorebuli wertilisa da Ria simravlis cnebebi.

gansazRvreba 6.1.4 vTqvaT E⊂ n. E simravlis x wertils ewodeba ama-

ve simravlis Siga wertili, Tu arsebobs x wertilis iseTi P(x,ε) midamo, rom P(x,ε)⊂E. simravles, romlis yvela wertili Siga wertilia ewodeba

Ria simravle.

cnobilia, rom n sivrcis x wertilis P(x,ε) midamo Ria simravlea.

Caketili simravle SeiZleba ganisazRvros rogorc iseTi simravle,

romelsac ekuTvnis yvela zRvruli wertili. Ria da Caketil simravle-

ebs aqvT iseTive Tvisebebi, romlebic gadmocemulia TeoremebSi 1.1.1 -1.1.3.

magram erTganzomilebian SemTxvevaSi Ria da Caketili simravleebis yve-

la Tviseba ucvlelad rodi gadaitaneba n sivrcis Ria da Caketili sim-

ravleebisTvis. ukanaskneli SeniSvna exeba, magaliTad, Teorema 1.1.4-s. ma-rTalia mravalganzomilebian SemTxvevaSic yoveli Ria simravle Ria in-

tervalTa araumetes Tvladi gaerTianebis saxiT warmodgeba, magram ne-

bismieri Ria simravle ar warmoidgineba TanaukveTi Ria intervalebis ga-

erTianebiT. es xSir SemTxvevaSi qmnis arcTu advilad gadasalax siZne-

les mravalganzomilebian analizSi.

erTganzomilebiani SemTxvevis analogiurad ganisazRvreba n sivrcis

E qvesimravlis gare zomis cneba.

gansazRvreba 6.1.5 n sivrcis nebismieri E simravlis gare zoma gani-

marteba ase

m∗( E)=inf ⎧ ⎫⎨ ⎬⎩ ⎭∑

( ) ( )

1 1

| | : ∪n n

i ii i

I E I∞∞

= =

⊂ .

gansazRvreba 6.1.6 E⊂ n simravles ewodeba nulsimravle, Tu m∗(E)=0.

1.3.1 gansazRvrebis identurad moicema E⊂ n simravlis zomadobis cne-

ba. aseve ganimarteba n sivrceze gansazRvruli zomadi funqciis cneba

(ix. gansazRvreba 2.1.1):

53

gansazRvreba 6.1.7 f : n → n funqcias ewodeba zomadi, Tu yoveli

V⊂ n Ria simravlisaTvis f -1(V) aris zomadi. mravalganzomilebian SemTxvevaSi adgili aqvs Teorema 2.1.2-is identur

debulebas. aseve samarTliania 2.1.3–2.1.10-isa da 2.2.1-2.2.6-is analogiuri

Teoremebi.

lebegis integrali n

f∫R arauaryofiTi f funqciidan ganimarteba 3.1.3

gansazRvrebis analogiurad. aseve ZalaSia gansazRvreba 3.1.4, im cvlile-

biT, rom unda Seicvalos n-iT. jerad SemTxvevaSi lebegis integrals

aqvs is Tvisebebi, romlebic gadmocemulia 3.1.5-3.1.7 TeoremebSi da 3.2.1-3.2.8 debulebebSi. n-ganzozomilebian I(n)

intervalze f funqciis rimanis

azriT integrebadobidan gamomdinareobs igive funqciis lebegis azriT

integrebadoba.

6.2 fubinisa da tonelis Teoremebi

rogorc zemoT aRiniSna, fubinis Teorema saSualebas gvaZlevs lebe-

gis jeradi integralis gamoTvla daviyvanoT lebegis martivi integra-

lebis gamoTvlaze. simartivisaTvis jer ganvixilavT orjeradi integra-

lis SemTxvevas.

Teorema 6.2.1 (fubini) vTqvaT f funqcia integrebadia (2)0I =[a1,b1]×[a2,b2]

intervalze. maSin [a1,b1] segmentis TiTqmis yvela x wertilisTvis arse-

bobs ∫2

2

( , )b

af x y dy da T.y. y-Tvis [a2,b2]-dan arsebobs integrali ∫

1

1

( , )b

af x y dx

da marTebulia toloba

∫(2)0

( , )I

f x y dxdy =⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

∫ ∫1 2

1 2

( , )b b

a a

f x y dy dx =⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

∫ ∫2 1

2 1

( , )b b

a a

f x y dx dy.

bunebrivad ismeba kiTxva: Tu f funqcia zomadia intervalze da arse-

bobs ganmeorebiTi integralebi

1 2

1 2

( , )b b

a a

f x y dy dx⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

∫ ∫ da 2 1

2 1

( , )b b

a a

f x y dx dy⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

∫ ∫

da isini tolia, maSin aris Tu ara f funqcia integrebadi (2)0I intervalze?

pasuxi uaryofiTia. amasTan dakavSirebiT ganvixiloT

magaliTi 6.2.2 (zigmundi) vTqvaT r0=[0,1] ×[0,1]. ganvixiloT wyvil-wyvi-

lad aragadamfarav kvadratTa mimdevroba (Qk), amasTan vigulisxmoT, rom

Qk⊂r0 (k=1,2,...) da yoveli Qk kvadratis diagonali moTavsebulia y=x wrfe-ze. TiToeuli Qk kvadrati gavyoT oTx kongruentul kvadratad da Qk

kvadratis centris mimarT simetriul kvadratTa erTi wyvilisTvis miviC-

nioT f(x,y)=1/| Qk |, kvadratTa meore wyvilisTvis ki davuSvaT f(x,y)= -1/|Qk |. aRebuli Q1, Q2,..., Qk,... kvadratebis gareT vigulisxmoT f(x,y)=0. cxadia, rom f funqcia zomadia r0-ze. amis garda, advili SesamCnevia, rom

1 1 1 1

0 0 0 0

( , ) ( , ) 0.f x y dy dx f x y dx dy⎡ ⎤ ⎡ ⎤

= =⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

∫ ∫ ∫ ∫

54

f funqcia integrebadi araa r0-ze. marTlac yoveli naturaluri m ric-

xvisaTvis gvaqvs

01

| ( , ) | | ( , ) | .k

m

kr Q

f x y dxdy f x y dxdy m=

≥ =∑∫ ∫

aqedan gamomdinareobs, rom

0

| ( , ) | .r

f x y dxdy = ∞∫

maSasadame, f funqcia integrebadi araa r0-ze. Teorema 6.2.2 (toneli) Tu f zomadi, arauaryofiTi funqciaa

(2)0I =

[a1,b1]×[a2,b2] intervalze, maSin ori ganmeorebiTi integralidan

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

∫ ∫1 2

1 2

( , )b b

a a

f x y dy dx da ⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

∫ ∫2 1

2 1

( , )b b

a a

f x y dx dy

erT-erTis arseboba sakmarisia f funqciis integrebadobisaTvis (2)0I -ze.

damtkiceba. davuSvaT, rom x–is TiTqmis yvela mniSvnelobisaTvis [a1,b1] segmentidan arsebobs integrali

2

2

( ) ( , ) .b

a

g x f x y dy= ∫

amis garda, vigulisxmoT, rom arsebobs da sasrulia 2

2

( ) .b

ag x dx∫ yoveli na-

turaluri m-Tvis ganvsazRvroT ori cvladis fm funqcia

fm(x,y)=( , ), ( , ) ,, ( , ) .

f x y f x y mm f x y m

≤⎧⎨ >⎩

roca

roca

radgan yoveli fm funqcia SemosazRvrulia da zomadia, amitom is integ-

rebadicaa (2)0I intervalze. maSasadame, fubinis 6.2.1 Teoremis Tanaxmad

( 2)0

( , )mI

f x y dxdy =∫1 2

1 2

( , )b b

ma a

f x y dy dx⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

∫ ∫ ≤1

1

( ) .b

a

g x dx < ∞∫

fm funqciaTa mimdevroba zrdadia, amitomATu gadavalT zRvarze, roca

m→∞, gveqneba

1

( 2)10

( , ) ( ) .b

aI

f x y dxdy g x dx≤ < ∞∫ ∫

ganvixiloT fubinisa da tonelis Teorema zogad SemTxvevaSi.

Teorema 6.2.3 (fubini) davuSvaT m+n cvladis f funqcia (m,n∈N) in-

tegrebadia ( )1

mI × ( )2

nI intervalze. maSin TiTqmis yvela x wertilisTvis ( )1

mI

intervalidan arsebobs ∫ ( )( , )

n2I

f x y dy da T.Yy. y–Tvis ( )2

nI –dan arsebobs

∫ ( )( , )

m1I

f x y dx integrali da adgili aqvs tolobas

∫( ) ( )1 2

( , )m nI I

f x y dxdy×

=⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

∫ ∫( ) ( )1 2

( , )m nI I

f x y dy dx = .⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

∫ ∫( ) ( )2 1

( , )n mI I

f x y dx dy

Teorema 6.2.4 (toneli) vTqvaT f zomadi, arauaryofiTi funqciaa seg-

mentze ( )1

mI × ( )2

nI . maSin ori ganmeorebiTi integralidan

55

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

∫ ∫( ) ( )1 2

( , )m nI I

f x y dy dx da ⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

∫ ∫( ) ( )2 1

( , )n mI I

f x y dx dy

erT-erTis arseboba uzrunvelyofs f funqciis integrebadobas ( )1

mI × ( )2

nI in-

tervalze.

Date post:	14-Apr-2016
Category:	Documents
Upload:	nikoloz-machavariani
View:	41 times
Download:	3 times

Zoma Da Lebegis Integrali LeqciaTa Kursi

Documents