Full measure of a set of singular continuous measures

On the space of structurally similar measures, we construct a nontrivial measure m such that the subclass of all purely singular continuous measures is a set of full m-measure.

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Datum:	2009
Hauptverfasser:	Koshmanenko, V. D., Кошманенко, В. Д.
Format:	Artikel
Sprache:	Ukrainisch Englisch
Veröffentlicht:	Institute of Mathematics, NAS of Ukraine 2009
Online Zugang:	https://umj.imath.kiev.ua/index.php/umj/article/view/3003
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Назва журналу:	Ukrains’kyi Matematychnyi Zhurnal
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Ukrains’kyi Matematychnyi Zhurnal

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author	Koshmanenko, V. D. Кошманенко, В. Д.
author_facet	Koshmanenko, V. D. Кошманенко, В. Д.
author_sort	Koshmanenko, V. D.
baseUrl_str	https://umj.imath.kiev.ua/index.php/umj/oai
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datestamp_date	2020-03-18T19:43:07Z
description	On the space of structurally similar measures, we construct a nontrivial measure m such that the subclass of all purely singular continuous measures is a set of full m-measure.
first_indexed	2026-03-24T02:34:24Z
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fulltext	UDK 517.9 V. D. Koßmanenko (In-t matematyky NAN Ukra]ny, Ky]v) POVNA MIRA MNOÛYNY SYNHULQRNO NEPERERVNYX MIR ∗∗∗∗ On the space of similarly structured measures, we construct a nontrivial measure m such that a subclass of all purely singularly continuous measures is a set of the full m-measure. Na prostranstve strukturno-podobn¥x mer postroena netryvyal\naq mera m takaq, çto pod- klass vsex çysto synhulqrno neprer¥vn¥x mer qvlqetsq mnoΩestvom polnoj m-mer¥. 1. Vstup. U 1981 roci rumuns\kyj matematyk Zamfiresku [1] pokazav, wo pere- vaΩna bil\ßist\ monotonnyx funkcij magt\ synhulqrno neperervnu poxidnu. Evristyçno ce oznaça[, wo zvyçajni funkci] (prosti ta absolgtno neperervni), qki vidpovidagt\ toçkovym ta absolgtno neperervnym miram, skladagt\ ekzo- tyçno malu pidmnoΩynu perßo] katehori] Bera. Analohiçnyj rezul\tat otrymano v roboti Sajmona [2] (dyv. takoΩ [3, 4]), de pokazano, wo v pevnomu sensi praktyçno vsi linijni operatory u hil\bertovomu prostori magt\ synhulqrno neperervnyj spektr i lyße deqki � toçkovyj abo absolgtno neperervnyj: “Operators with singular continuous spectrum are generic in the Baire sense”. U zv�qzku iz zastosuvannqmy ponqttq spektral\no] miry v teori] zburen\ z fraktal\nym potencialom [5] ta v teori] konfliktiv [6, 7], vynykla potreba oxarakteryzuvaty toçniße �vahu� mnoΩyny synhulqrno neperervnyx mir, qki zustriçagt\sq v cyx teoriqx. U cij statti na prostori strukturno-podibnyx mir M ss pobudovano netryvi- al\nu jmovirnisnu �hlobal\nu� miru m, dlq qko] pidklas synhulqrno nepererv- nyx mir Msc ss [ mnoΩynog povno] miry: m( )Msc ss = 1. Vodnoças, pidklasy çysto toçkovyx ta absolgtno neperervnyx mir magt\ nul\ovu m-miru. Zaznaçy- mo, wo prostir M ss [ dosyt\ ßyrokym i vklgça[ v sebe vsi samopodibni miry, vvedeni Xatçinsonom [8]. ZauvaΩymo, wo tra[ktori] dynamiçnyx system konf- liktu, qk pravylo, naleΩat\ prostoru M ss (dyv. [9, 10]). Z inßoho boku, qkwo strukturno-podibni miry traktuvaty qk spektral\ni miry pevnyx samosprqΩenyx operatoriv, to vstanovlenyj fakt zmistovno dopovng[ zhadani tverdΩennq pro potuΩnist\ mnoΩyny synhulqrno neperervnyx funkcij ta operatoriv z synhu- lqrno neperervnym spektrom. Varto zaznaçyty, wo, nezvaΩagçy na potuΩnyj rozvytok fraktal\no] heo- metri] [11], naqvnist\ hlybokyx doslidΩen\ po spektral\nyx asymptotykax dy- ferencial\nyx operatoriv z synhulqrnymy potencialamy, zokrema fraktal\ny- my [5], vynyknennq mul\tyfraktal\no] teori] synhulqrnyx rozpodiliv z bahato- rivnevym analizom ta hradaci[g [12], doslidΩennqmy qvnoho vyhlqdu hranyçnyx staniv dynamiçnyx system konfliktu [13 – 15], poslidovno] teori] synhulqrno neperervnyx spektriv ta mir we ne stvoreno. 2. Strukturno-podibni miry. Vvedemo prostir imovirnisnyx strukturno- podibnyx mir na vidrizku [ , ]0 1 ≡ ∆0 (dyv. [9, 10]), qkyj poznaça[mo çerez M ss( )[ , ]0 1 ≡ M ss. Rozhlqnemo sim�g T = { }Tik i n =1, k = 1, 2, … , 2 ≤ n < ∞, styskugçyx po- dibnostej u R1. Prypustymo, wo dlq vsix k vykonugt\sq umovy 0 < c ≤ cik < 1, de cik — koefici[nt stysku peretvorennq Tik ; ∗ Çastkovo pidtrymano proektom DFG 436 UKR 113/78. © V. D. KOÍMANENKO, 2009 ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 1 83 84 V. D. KOÍMANENKO ∆0 = Tik i n ∆0 1= U ; λ T Tik i k∆ ∆0 0I ′( ) = 0, i ≠ i ′ , λ poznaça[ miru Lebeha. Nexaj Ui i i ik k1 1… ′… ′, : = T T T Ti i k i i kk k1 11 1 1… …′ ′ −( ) , 1 ≤ ik , ′ik ≤ n. MnoΩynu S ⊂ ∆0 nazyva[mo strukturno-podibnog, qkwo dlq deqko] fik- sovano] sim�] styskugçyx podibnostej T z navedenymy vywe vlastyvostqmy cq mnoΩyna dopuska[ neskinçennu poslidovnist\ podribnen\ (rozkladiv) na podibni miΩ sobog pidmnoΩyny: S = si i n 1 1 1= U , si1 = si i i n 1 2 2 1= U , … , si ik1 1… − = si i i i n k k 1 2 1 … = U , … , (1) de dlq koΩnoho fiksovanoho ranhu k = 1, 2, … neporoΩni pidmnoΩyny si i ik1 2 … ⊂ ⊂ T Ti i k1 11 0… ∆ pov�qzani miΩ sobog: si i ik1 2 … = U si i i i i ik k k1 1 1… ′… ′ ′… ′, . (2) Pry c\omu prypuska[mo, wo diam ( )si i ik1 2 … ≤ εk → 0, k → ∞ , ta λ s si i i ik k1 1… ′… ′( )cl clI = 0, qkwo il ≠ ′il xoça b dlq odnoho 1 ≤ l ≤ k , de cl poznaça[ zamykannq mno- Ωyny. Takym çynom, dlq strukturno-podibno] mnoΩyny S, wo dopuska[ zobraΩen- nq (1), usi pidmnoΩyny fiksovanoho ranhu si i ik1 2 … podibni miΩ sobog. Podib- nist\ miΩ sobog pidmnoΩyn riznoho ranhu ne vymaha[t\sq, tobto mnoΩyny S, si1 , si i1 2 , … , si ik1… , … vzahali ne [ podibnymy. Tomu ponqttq strukturno-podibno] mnoΩyny [ zahal\ni- ßym, niΩ ponqttq samopodibno] mnoΩyny (dyv. [8]). Imovirnisnu miru µ na vidrizku [ , ]0 1 nazyva[mo strukturno-podibnog, qkwo heometryçnyj nosij ci[] miry Sµ : = supp µ [ strukturno-podibnog mnoΩynog i vykonugt\sq nastupni umovy: dlq neporoΩnix pidmnoΩyn iz rozkladu (1) vidno- ßennq µ µ ( ) ( ) s s i i i i i k k k 1 1 1 1 … … − − = : pi kk > 0, si0 ≡ ∆0, (3) ne zaleΩat\ vid indeksiv i ik1 1… − ta µ( )s si i i i ik k k1 1 1… ′… ′− I = 0, (4) qkwo il ≠ ′il xoça b dlq odnoho l = 1, … , k . PokaΩemo, wo miry µ ∈Mss moΩna buduvaty metodom Q -zobraΩennq dijs- nyx çysel (dyv. [16]). Z ci[g metog rozhlqnemo neskinçennu vpravo prqmokutnu matrycg Q = { }qk k = ∞ 1 = { } , ,qk i k n = = ∞ 1 1, n > 1, de vektory qk , qki utvorggt\ stovpçyky matryci Q , [ stoxastyçnymy: ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 1 POVNA MIRA MNOÛYNY SYNHULQRNO NEPERERVNYX MIR 85 qk = ( ), ,q qk nk1 … ∈ R+ n , q qk nk1 + …+ = 1. Nexaj vykonano umovy qik > 0, inf , { } i k ikq > 0. (5) Q -zobraΩennq toçok x ∈ ≡∆0 0 1[ , ] zada[t\sq poslidovnistg iteracij. Na per- ßomu kroci ∆0 rozklada[t\sq na vporqdkovanu sim�g, wo mistyt\ n zamkne- nyx vidrizkiv ranhu 1: ∆0 ≡ [ , ]0 1 = ∆i i n 1 1 1= U , λ ( )∆i1 = qi11 . Na druhomu kroci koΩen vidrizok ∆i1 rozklada[t\sq v ob�[dnannnq vporqdkova- nyx vidrizkiv ranhu 2: ∆i1 = ∆i i i n 1 2 2 1= U , λ ( )∆i i1 2 = q qi i1 21 2 tak, wo vsi vidnoßennq λ λ ( ) ( ) ∆ ∆ i i i 1 2 1 = qi22 [ nezaleΩnymy vid nomera vidrizka perßoho ranhu. Analohiçno, na k - mu kroci ∆i ik1 1… − = ∆i i i n k k 1 1 … = U , λ ( )∆i ik1… = q qi i kk11 … , (6) λ λ ( ) ( ) ∆ ∆ i i i i k k 1 1 1 … … − = qi kk . V rezul\tati vynyka[ zliçenna poslidovnist\ pokryttiv: ∆0 = ∆i i n 1 1 1= U = ∆i i i i n 1 2 1 2 1= U … ∆i i i i n k k 1 1 1 … … = U = … . Oçevydno, wo na pidstavi (5) ta (6) linijni rozmiry vidrizkiv ∆i ik1… prqmugt\ do nulq pry zrostanni k : λ ∆i ik1…( ) → 0, k → ∞ . Tomu koΩna fiksovana poslidovnist\ vkladenyx vidrizkiv ∆i1 ⊃ ∆i i1 2 ⊃ … … ⊃ ∆i ik1… ⊃ … v peretyni mistyt\ lyße odnu toçku: ∆0 � x = ∆i i k k1 1 … = ∞ I . Cej fakt pryrodno zapysu[mo u vyhlqdi x = ∆i ik1… …, (7) de indeksy ik nazyvagt\ koordynatamy toçky x , a sam zapys (7) � Q -zobra- Ωennqm toçky x . NevaΩko baçyty, wo spivvidnoßennq ∆i i ik1 2 … = T Ti i kk11 0… ∆ (8) vstanovlggt\ vza[mno odnoznaçnu vidpovidnist\ miΩ mnoΩynog vsix Q -zobra- Ωen\ na ∆0 ta sim�qmy vvedenyx vywe styskagçyx podibnostej T. ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 1 86 V. D. KOÍMANENKO Zrozumilo takoΩ, wo klas strukturno-podibnyx mir M ss vklgça[ v sebe vsi jmovirnisni samopodibni miry na vidrizku [ , ]0 1 = ∆0. Pry fiksovanomu Q -zobraΩenni dlq pobudovy miry µ ∈Mss neobxidno dodatkovo zadaty we odnu stoxastyçnu matrycg P = { }pk k = ∞ 1 = { } , ,pik i k n = ∞ 1, 1 ≥ pik ≥ 0, stovpçyky qko] skladagt\sq z koordynat stoxastyçnyx vektoriv pk n∈ +R . Za matrycqmy Q ta P vvodymo poslidovnist\ kuskovo-rivnomirno rozpodi- lenyx mir µk na ∆0, vyznaçenyx takym çynom: µk : = ci i i i i i n k k k 1 1 1 1 … … … = ∑ λ , (9) de koefici[nty ci ik1… : = p p q q i i k i i k k k 1 1 1 1 … … , a λi ik1… : = λ ∆i ik1… poznaça[ zvuΩennq miry Lebeha. Na pidstavi (6) λ( )∆i ik1… = = q qi i kk11 … . Tomu z (9) vyplyva[, wo hrafik funkci] rozpodilu f xk ( ) = = µk x(( , ))− ∞ [ kuskovo-linijnog lamanog lini[g. Za pobudovog oçevydnog [ spravedlyvist\ spivvidnoßen\ µk i i i i n k k ( )∆ 1 1 1 … … = ∑ = 1, µk i ik ( )∆ 1… = µk i i i i n k k k + … = + + ∑ 1 1 1 1 1 ( )∆ , µk i i ik k+ … +1 1 1 ( )∆ = µk i i i kk k p( )∆ 1 1 1… ++ . Z c\oho vyplyva[, wo dlq poslidovnosti mir (9) isnu[ hranycq, µ = limk k→∞ µ , u sensi rivnomirno] zbiΩnosti. Pyßemo µ = µP , wob pidkreslyty zaleΩnist\ zbudovano] miry µ vid matryci P. Varto zaznaçyty, wo na cylindryçnyx mnoΩynax ∆i ik1… znaçennq mir µP , µk vzahali [ riznymy: µ( )∆i ik1… ≠ µk i ik ( )∆ 1… = p pi i kk11 … . (10) Oçevydno takoΩ, wo pry P = Q mira µP [ mirog Lebeha na ∆0. Perekona[mosq, wo mira µP [ strukturno-podibnog. Z ci[g metog pobudu[- mo pidmnoΩyny si ik1… : = S i ikµ I ∆ 1… # , de # oznaça[, wo vidrizok ∆i ik1… zalyßa[t\sq zamknenym, lyße qkwo p k s s 1 0 , + = ∞ ∏ = pn k s s , + = ∞ ∏ 0 = 0. Vykorystannq ∆i ik1… # zamist\ ∆i ik1… obumovleno nerivnistg (10) i [ neobxidnym dlq zabezpeçennq umov (3), (4). Za pobudovog µ( )si ik1… = µ( )∆i ik1… # = µk i ik ( )∆ 1… = p pi ik1 … . (11) ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 1 POVNA MIRA MNOÛYNY SYNHULQRNO NEPERERVNYX MIR 87 Zrozumilo, wo na pidstavi (8) dlq koΩno] pary pidmnoΩyn si ik1… , si ik′… ′1 fikso- vanoho ranhu k ≥ 1 isnu[ peretvorennq podibnosti, qke ]x po[dnu[: si ik′… ′1 = T si i i i i ik k k′… ′ … …1 1 1, . Teper, vraxovugçy (11), nevaΩko baçyty, wo µP ∈ Mss. Dali matrycg Q vvaΩa[mo fiksovanog i rozhlqda[mo uves\ klas M ss strukturno-podibnyx mir µ = µP , asocijovanyx iz stoxastyçnymy matrycqmy P. Dlq takyx mir pyßemo µ ∈ Mpp , Mac , Msc , qkwo µ [ çysto toçkovog ( )µ µ= pp , çysto absolgtno neperervnog ( µ = = µac) abo çysto synhulqrno neperervnog ( )µ µ= sc vidpovidno. Vidomo (dyv. [6, 7, 9, 10]), wo dlq mir µ ∈ Mss [ spravedlyvym analoh teoremy DΩessena � Vintnera pro çystotu spektral\noho typu. V terminax matryc\ P cej analoh moΩna sformulgvaty takym çynom. Teorema 1. KoΩna mira µ = µP ∈ M ss ma[ çystyj spektral\nyj typ: µ ∈Mpp ⇔ Pmax( )µ : = k kp = ∞ ∏ 1 max, > 0, p kmax, : = max { } i ikp , (12) µ ∈Mac ⇔ ρ µ λ( , ) : = k k = ∞ ∏ 1 ρ > 0, ρk : = i n ik ikp q = ∑ 1 , (13) µ ∈Msc ⇔ Pmax( )µ = 0 = ρ µ λ( , ). (14) 3. Pobudova miry m. U c\omu punkti vvodyt\sq i doslidΩu[t\sq jmovirnis- na mira m na prostori strukturno-podibnyx mir µ = µP ∈ M ss ≡ M pry fik- sovanomu Q -zobraΩenni. Dovodyt\sq, wo klas synhulqrno neperervnyx mir Msc [ mnoΩynog povno] m -miry. Mira m budu[t\sq na σ -alhebri, qka poznaça[t\sq çerez J ss . Cq alhebra porodΩena sim�[g cylindryçnyx pidmnoΩyn { }Ii ik i ik 1 1 … …β β , qki vyznaçagt\sq takym çynom. Na perßomu kroci ( k = 1 ) poklada[mo M = Ii i n 1 1 1= U , Ii1 : = Ii i i 1 1 1 β β ∈B U , de βi1 poznaça[ zvyçajnu borelevu mnoΩynu z vidrizka [ , ]/1 1n , a Ii i 1 1 β : = µ µ β µ µ∈ ∈       = = … M ( ) ( ) { ( )}, max , , ∆ ∆ ∆i i i j n j1 1 1 1 11 . U terminax matryc\ P (nahada[mo, wo µ = µP ) Ii i 1 1 β : = µ βP i ip p∈ = ∈{ }M 1 11 1max, , pmax,1 : = max , , { } j n jp = …1 1 . Zaznaçymo, wo 1/n ≤ max , , { ( )} j n j 1 11= … µ ∆ ≤ 1, oskil\ky µ � jmovirnisna mira. OtΩe, borelivs\ka σ -alhebra B sklada[t\sq lyße z pidmnoΩyn, roztaßovanyx na vidrizku [ , ]/1 1n . ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 1 88 V. D. KOÍMANENKO Na druhomu kroci ( k = 2 ) rozklada[mo koΩnu pidmnoΩynu Ii1 : Ii1 = Ii i i n 1 2 2 1= U , Ii i1 2 : = Ii i i i i i 1 2 1 2 1 2 β β β β, ∈B U , de Ii i i i 1 2 1 2 β β : = µ µ µ β µ µ β ∈ ∈ =      = … Ii i i i i i i j n i j i 1 1 1 2 1 2 1 2 2 1 21 ( ) ( ) ( ) { ( )}, max , , ∆ ∆ ∆ ∆ . Inßymy slovamy, Ii i i i 1 2 1 2 β β : = µ β β P i i iI p pi∈ = ∈{ }1 1 2 22 2max, . OtΩe, Ii i1 2 = Ii i i i 1 2 2 2 β β ∈B U , de Ii i i 1 2 2 β : = µ µ µ β µ µ∈ ∈ =      = … Ii i i i i i i j n i j1 1 2 1 2 1 2 2 1 21 ( ) ( ) ( ) { ( )}, max , , ∆ ∆ ∆ ∆ . Na k -mu ( k = 1, 2, … ) kroci vyznaça[mo Ii ik i ik 1 1 … …β β : = µ µ µ β µ µ β β ∈ ∈ =       … … … … … = … …− − − − Ii i i i i i i i i j n i i jk i ik k k k k k k k1 1 1 1 1 1 1 1 1 11 ( ) ( ) ( ) { ( )}, max , , ∆ ∆ ∆ ∆ z µ( )∆i0 ≡ µ( )∆0 = 1. Zokrema, dlq dovil\nyx çysel b b ni ik1 1 1, , [ , ]/… ∈ ma[mo Ii i b b k i ik 1 1 … … : = µ µ µ µ∈ = =         … … … … = … …− − − − I bi i b b i i i i i j n i i jk i ik k k k k k k1 1 1 1 1 1 1 1 11 ( ) ( ) { ( )}max , , ∆ ∆ ∆ . Xoça moΩe statysq, wo mnoΩyna Ii i b b k i ik 1 1 … … [ poroΩn\og, napryklad, qkwo bik > > bik −1 abo bik < b nik −1 / . OtΩe, za provedenog pobudovog M = i n iI 1 1 1= U = i i n i iI 1 2 1 2 1= U = … = i i n i i k k I 1 1 1… = …U = … , de mnoΩyny Ii ik1… dopuskagt\ vyznaçennq v terminax matryc\ P : Ii ik1… = Ii ik i ik i ik 1 1 1 … … … ∈ β β β β B U = µ βP i l l ip p l k l l ∈ = ∈ ≤ ≤{ }M max, , 1 . Sim�q cylindryçnyx pidmnoΩyn I i n ki i i kk i ik k1 1 1 1 2… … ∈ = … = …{ }β β β, , , , , , ,B heneru[ (zhidno zi standartnog procedurog (dyv., napryklad, [17])) deqke kil\- ce R . Vvodymo na kil\ci R miru m takym çynom. Spoçatku poklada[mo m Ii i 1 1 β( ) : = qi i1 11 ′λ β( ), ′λ β( )i1 : = cn iλ β( ) 1 , cn = n n − 1 , de vraxovano, wo λ( / )[ , ]1 1n = n n − 1 . Tomu m( )Ii1 = qi11 i ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 1 POVNA MIRA MNOÛYNY SYNHULQRNO NEPERERVNYX MIR 89 m( )Ii i n 1 1 1= ∑ = m i n iI i i1 1 1 1 1= ∈            U U β β B = 1. Dali poklada[mo m Ii i i i 1 2 1 2 β β( ) : = q qi i i i1 2 1 21 2 ′ ′λ β λ β( ) ( ). Zvyçajno, m( )Ii i1 2 = q qi i1 21 2 ta m( )Ii i i i n 1 2 1 2 1= ∑ = m i i n i iI i i i i1 2 1 2 1 2 1 2 1= ∈            U U β β β β, B = 1. Dlq dovil\noho k vyznaça[mo m Ii ik i ik 1 1 … …( )β β : = q qi i k l k ik l11 1 … ′ = ∏ λ β( ) i lehko perevirq[mo, wo m( )Ii i i i n k k 1 1 1 … … = ∑ = m i i n i i k k i ik i ik I 1 1 1 1 1… = … … … ∈            U U β β β β B = 1, oskil\ky q qi ii i n kk 11 1 …… =∑ = 1 ta ′λ ( / )[ , ]1 1n = 1. Z cylindryçnyx mnoΩyn mi- ru m rozßyrg[mo na kil\ce R , a potim do zovnißno] miry m∗∗∗∗, vyznaçeno] na usix pidmnoΩynax z M . Pry c\omu vykorystovu[t\sq toj fakt, wo m I Ii i i ik i ik k i ik 1 1 1 1 … … ′… ′ …′ ′( )β β β β I : = l k i i i l i il l l l l q = ′ ′∏ ′( ) 1 δ λ β βI , de vraxovano, wo nastupna mnoΩyna ne [ poroΩn\og, I Ii i i ik i ik k i ik 1 1 1 1 … … ′… ′ …′ ′β β β β I ≠ ∅, lyße qkwo dlq vsix l = 1, … , k il = ′il ta β βi il l I ′ ≠ ∅. Ce zabezpeçu[ σ - adytyvnist\ miry m . Sim�q cylindryçnyx pidmnoΩyn I ki i ik i ik k1 1 1 2… … = … ∈{ }β β β, , , , B pislq zastosuvannq standartno] procedury sta[ σ -alhebrog, qku my poznaça[mo çerez J ss . OtΩe, J ss [ minimal\nog σ -alhebrog m∗∗∗∗ - vymirnyx (za Karateo- dori) pidmnoΩyn, porodΩenyx kil\cem R . Pislq zvuΩennq zovnißn\o] miry m∗∗∗∗ na σ -alhebru J ss oderΩu[mo jmovirnisnu miru, qku znovu poznaça[mo çerez m . Takym çynom, m [ σ -adytyvnog mirog na alhebri J ss . Osnovnym rezul\tatom statti [ nastupna teorema. Teorema 2. PidmnoΩyna strukturno-podibnyx synhulqrno neperervnyx mir [ mnoΩynog povno] m -miry: m Msc ss( ) = 1. Dovedennq vyplyva[ z teoremy 6.1 z [17] ta rivnosti m ∗( )Mpp ss = m ∗( )Mac ss = = 0 z nastupno] lemy. Zokrema, z (15) vyplyva[ vymirnist\ mnoΩyn çysto toç- kovyx ta absolgtno neperervnyx mir, Mpp ss , M Jac ss ss∈ . Tomu, vraxovugçy (14), ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 1 90 V. D. KOÍMANENKO robymo vysnovok, wo M Jsc ss ss∈ , qk dopovnennq do M Mpp ss ac ssU u mnoΩyni M ss . OtΩe, m ∗( )Msc ss = m Msc ss( ) = 1. Lema. Zovnißnq mira m∗ mnoΩyn Mpp ss , Mac ss dorivng[ nulg: m ∗( )Mpp ss = m ∗( )Mac ss = 0. (15) Dovedennq. Zafiksu[mo deqku poslidovnist\ εk → 0, k → ∞ , napryklad, poklademo εk = 1 / k . Vvedemo poslidovnist\ pidmnoΩyn Tk : = Ii i i i n k i ik k 1 1 1 1 … … … = β β U , βi1 = [ , ]a1 1 , … , βik = [ , ]ak 1 , de çysla ak vybrano tak, wob ′λ β( )ik = εk . Poznaçymo Tpp,k : = µ µ∈ ∈{ }M Tpp ss k = T Mk I pp ss . Zrozumilo, wo m∗( ),Tpp k ≤ m( )Tk ≤ m I i i i i n k i ik k 1 1 1 1 … … … = ( )∑ β β = = q qi i k l k i i i n k l k 1 1 1 11 … ′ =… = ∏∑ λ β( ) ≤ q qi i k i i n ik k k1 1 1 1 … ′ … = ∑ λ β( ) = εk , oskil\ky q qi i ki i n kk 11 11 …… =∑ = 1. NevaΩko pokazaty, wo dlq koΩno] miry µ ∈Mpp ss isnu[ nomer k0 = k0( )µ takyj, wo µ ∈Tk dlq vsix k ≥ k0 . Ce vyplyva[ z toho, wo dlq dovil\no] fik- sovano] miry µ µ= ∈P Mpp ss na pidstavi spivvidnoßennq (12) vykonu[t\sq umova p kk max,∏ > 0. U svog çerhu z c\oho vyplyva[, wo p kmax, > 1 1− /k = 1 − εk dlq vsix k ≥ k0 poçynagçy z deqkoho k0 = k0( )µ . Tomu µP k∈T , k ≥ k0 . Ot- Ωe, koΩna mira µ ∈Mpp ss naleΩyt\ usim mnoΩynam Tpp,k poçynagçy z deqkoho k , zaleΩnoho vid µ . OtΩe, δk : = m T Tpp pp, ,\k l l k = −      1 1 U → 0, k → ∞ . (16) Poznaçymo ′Tpp,k : = Tpp,ll k =1U . Zrozumilo, wo ′Tpp,k ⊂ ′ +Tpp,k 1, a takoΩ Mpp ss = = ′ = ∞ Tpp,kk 1U . Tomu zhidno z (16) zavdqky σ -adytyvnosti zovnißn\o] miry m ∗( )Mpp ss = 0. Dijsno, za teoremog pro neperervnist\ miry dlq ob�[dnan\ pid- mnoΩyn (dyv. [17], teorema 6.2) ma[mo m ∗( )Mpp ss = lim , k l l k →∞ ∗ = ′      m Tpp 1 U = lim ( ), k k →∞ ∗m Tpp ≤ lim ( ) k k →∞ m T = lim k k →∞ ε = 0. Analohiçnym çynom moΩemo dovesty, wo m( )Mac ss = 0. Z ci[g metog vvedemo inßu poslidovnist\ pidmnoΩyn Tk : = µ µ ε ε β β ∈ ∈ = − + = …{ }… … Mss I b q q l ki i i i l l i l lk i ik l l l1 1 2 2 1, , , , ,[ / / ] i poznaçymo ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 1 POVNA MIRA MNOÛYNY SYNHULQRNO NEPERERVNYX MIR 91 Tac,k : = µ µ∈ ∈{ }M Tac ss k = T Mk I ac ss. Zokrema, u vypadku, koly vsi qi kk = 1 / n, moΩna poklasty βik = [ / / ],1 1n n k+ ε . NevaΩko zrozumity, wo dlq koΩno] miry µ ∈Mac ss isnu[ nomer k0 = k0( )µ takyj, wo µ ∈Tac,k dlq vsix k ≥ k0 . Ce vyplyva[ iz spivvidnoßennq (13). Tomu δk : = m T Tac ac, ,\k l l k = −      1 1 U → 0, k → ∞ . Poznaçymo ′Tac,k : = Tac,ll k =1U . Zrozumilo, wo ′Tac,k ⊂ ′ +Tac,k 1, a takoΩ Mac ss = = ′= ∞ Tac,kk 1U . Takym çynom, moΩna stverdΩuvaty, wo m ∗( )Mac ss = 0 vnaslidok σ -adytyvnosti zovnißn\o] miry. Dijsno, m ∗( )Mac ss = lim , k l l k →∞ ∗ = ′      m Tac 1 U = lim ( ), k k →∞ ∗m Tac ≤ lim ( ) k k →∞ m T = lim k k →∞ ε = 0, de vykorystano spivvidnoßennq m∗( ),Tac k ≤ m( )Tk = q qi i k i i n ik k k1 1 1 1 … ′ … = ∑ λ β( ) = εk . OtΩe, lemu, a razom z neg i teoremu 2 dovedeno. 1. Zamfirescu T. Most monotone functions are singular // Amer. Math. Mon. – 1981. – 88 . – P. 47 – 79. 2. Simon B. Operators with singular continuous spectrum: I. Genaral operators // Ann. Math. – 1995. – 141. – P. 131 – 145. 3. del Rio R., Jitomirskaya S., Makarov N., Simon B. Operators with singular continuous spectrum are generic // Bull. Amer. Math. Soc. – 1994. – 31. – P. 208 – 212. 4. Jitomirskaya S., Simon B. Operators with singular continuous spectrum: III. Almost periodic Schrodinger operators // J. Communs Math. Phys. – 1994. – 165, # 1. – P. 201 – 205. 5. Triebel H. Fractals and spectra related to Fourier analysis and functional spaces. – Basel etc.: Birkhäuser, 1997. 6. Albeverio S., Koshmanenko V., Pratsiovytyi M., Torbin G. Spectral properties of image measures under infinite conflict interactions // Positivity. – 2006. – 10. – P. 39 – 49. 7. Koshmanenko V., Kharchenko N. Spectral properties of image measures after conflict interactions // Theory Stochast. Process. – 2004. – 10, # 3 – 4. – P. 73 – 81. 8. Hutchinson J. E. Fractals and selfsimilarity // Indiana Univ. Math. J. – 1981. – 30. – P. 713 – 747. 9. Koßmanenko V. D. Vidnovlennq spektral\noho typu hranyçnyx rozpodiliv u dynamiçnyx systemax konfliktu // Ukr. mat. Ωurn. � 2007. � 59, # 6. � S. 771 � 784. 10. Karataieva T., Koshmanenko V. Origination of the singular continuous spectrum in the conflict dynamical systems // Meth. Func. Anal. and Top. – 2009. – 14, # 1. – P. 16 – 29. 11. Falconer K. J. Fractal geometry. – Chichester: Wiley, 1990. 12. Torbin H. M. Mul\tyfraktal\nyj analiz synhulqrno neperervnyx imovirnisnyx mir // Ukr. mat. Ωurn. � 2005. � 57, # 5. � S. 706 � 721. 13. Koshmanenko V. On the conflict theorem for a pair of stochastic vectors // Ukr. Math. J. – 2003. – 55, # 4. – P. 555 – 560. 14. Koshmanenko V. The theorem of conflict for probability measures // Math. Meth. Oper. Res. – 2004. – 59, # 2. – P. 303 – 313. 15. Albeverio S., Bodnarchuk M., Koshmanenko V. Dynamics of discrete conflict interactions between non-annihilating opponent // Meth. Func. Anal. and Top. – 2005. – 11, # 4. – P. 309 – 319. 16. Albeverio S., Koshmanenko V., Pratsiovytyi M., Torbin G. Q̃ -representation of real numbers and fractal probability distributions. – Bonn, 2002. – (Preprint / Bonn Univ., No. 12) arcXiv:math., PR/03 08 007 v1, 2003. 17. Berezanskyj G. M., Us H. F., Íeftel\ Z. H. Funkcyonal\n¥j analyz. � Kyev: Vywa ßk., 1990. � 600 s. OderΩano 29.01.08 ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 1
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institution	Ukrains’kyi Matematychnyi Zhurnal
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language	Ukrainian English
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publishDate	2009
publisher	Institute of Mathematics, NAS of Ukraine
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spelling	umjimathkievua-article-30032020-03-18T19:43:07Z Full measure of a set of singular continuous measures Повна міра множини сингулярно неперервних мір Koshmanenko, V. D. Кошманенко, В. Д. On the space of structurally similar measures, we construct a nontrivial measure m such that the subclass of all purely singular continuous measures is a set of full m-measure. Ha пространстве структурно-подобных мер построена нетривиальная мера m такая, что подкласс всех чисто сингулярно непрерывных мер является множеством полной m-меры. Institute of Mathematics, NAS of Ukraine 2009-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3003 Ukrains’kyi Matematychnyi Zhurnal; Vol. 61 No. 1 (2009); 83-91 Український математичний журнал; Том 61 № 1 (2009); 83-91 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3003/2755 https://umj.imath.kiev.ua/index.php/umj/article/view/3003/2756 Copyright (c) 2009 Koshmanenko V. D.
spellingShingle	Koshmanenko, V. D. Кошманенко, В. Д. Full measure of a set of singular continuous measures
title	Full measure of a set of singular continuous measures
title_alt	Повна міра множини сингулярно неперервних мір
title_full	Full measure of a set of singular continuous measures
title_fullStr	Full measure of a set of singular continuous measures
title_full_unstemmed	Full measure of a set of singular continuous measures
title_short	Full measure of a set of singular continuous measures
title_sort	full measure of a set of singular continuous measures
url	https://umj.imath.kiev.ua/index.php/umj/article/view/3003
work_keys_str_mv	AT koshmanenkovd fullmeasureofasetofsingularcontinuousmeasures AT košmanenkovd fullmeasureofasetofsingularcontinuousmeasures AT koshmanenkovd povnamíramnožinisingulârnoneperervnihmír AT košmanenkovd povnamíramnožinisingulârnoneperervnihmír

Full measure of a set of singular continuous measures

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