Weak bases of vector measures
We solve the problem of representation of measures with values in a Banach space as the limits of weakly convergent sequences of vector measures whose basis is a given nonnegative measure.
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Institute of Mathematics, NAS of Ukraine
2007
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| author | Romanov, V. A. Романов, В. А. Романов, В. А. |
| author_facet | Romanov, V. A. Романов, В. А. Романов, В. А. |
| author_sort | Romanov, V. A. |
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| datestamp_date | 2020-03-18T19:53:10Z |
| description | We solve the problem of representation of measures with values in a Banach space as the limits of weakly convergent sequences of vector measures whose basis is a given nonnegative measure. |
| first_indexed | 2026-03-24T02:41:51Z |
| format | Article |
| fulltext |
UDK 519.53 + 517.987
V. A. Romanov (Kyrovohrad. ped. un-t)
SLABÁE BAZYSÁ VEKTORNÁX MER
We solve a problem of the representation of measures with values in a Banach space as limits of weakly
convergent sequences of vector measures for which a given nonnegative measure is a basis.
Rozv’qzano pytannq pro zobraΩennq mir iz znaçennqmy v banaxovomu prostori qk hranyc\ slabko
zbiΩnyx poslidovnostej vektornyx mir, wo magt\ svo]m bazysom danu nevid’[mnu miru.
1. Vvedenye. Meru m nazovem slab¥m bazysom mer¥ µ, esly µ est\ predel
slabo sxodqwejsq posledovatel\nosty mer, ymegwyx m svoym bazysom. Ta-
koj obobwenn¥j podxod pryvodyt k suwestvovanyg mer, kotor¥e sluΩat sla-
b¥m bazysom dlq vsex druhyx mer (y daΩe vektorn¥x) v dannom polnom separa-
bel\nom metryzuemom topolohyçeskom lynejnom prostranstve. Pry klassyçes-
kom Ωe podxode uΩe v hyl\bertovom prostranstve nykakaq veroqtnostnaq (a po-
tomu y syhma-koneçnaq) mera m ne moΩet sluΩyt\ bazysom daΩe dlq semejstva
vsex svoyx sdvyhov, poskol\ku ee sdvyh na vektor, ne prynadleΩawyj obrazu ne-
kotoroho operatora Hyl\berta – Ímydta, pryvodyt k mere, vzaymno synhulqr-
noj s m [1, s. 144]. Suwestvovanye „unyversal\noho” slaboho bazysa moΩet
predstavlqt\ ynteres pry yssledovanyy dyfferencyal\n¥x uravnenyj dlq
mer, kohda stanovytsq vozmoΩn¥m rassmotrenye sootvetstvugwyx uravnenyj
dlq approksymyrugwyx plotnostej, t. e. dlq funkcyj toçky, a takΩe pry ys-
sledovanyy svojstv mer v termynax approksymyrugwyx plotnostej.
2. Postanovka zadaçy. Pust\ X — polnoe separabel\noe metryzuemoe to-
polohyçeskoe lynejnoe prostranstvo, Y — banaxovo prostranstvo. Pod Y-
znaçnoj meroj v X ponymaem syhma-addytyvnug funkcyg mnoΩestva koneçnoj
polnoj varyacyy, opredelennug na vsex borelevskyx podmnoΩestvax prostran-
stva X y prynymagwug znaçenyq v Y. Napomnym, çto neotrycatel\naq mera m
naz¥vaetsq bazysom Y-znaçnoj mer¥ µ, esly µ predstavyma kak proyzvedenye
yntehryruemoj po Boxneru Y-znaçnoj funkcyy na meru m.
Oboznaçym çerez Cb ( X ) mnoΩestvo vsex neprer¥vn¥x na X ohranyçenn¥x
funkcyj s çyslov¥my znaçenyqmy. Posledovatel\nost\ Y-znaçn¥x mer µn na-
z¥vaem slabo sxodqwejsq k Y-znaçnoj mere µ, esly dlq kaΩdoj funkcyy f yz
Cb ( X )
lim ( ) ( )
n
X
nf x d x
→∞ ∫ µ = f x d x
X
( ) ( )µ∫ . (1)
Opredelenye 1. Neotrycatel\naq mera m v prostranstve X naz¥vaet-
sq slab¥m bazysom Y -znaçnoj mer¥ µ v X, esly suwestvuet slabo sxodq-
waqsq k µ posledovatel\nost\ Y-znaçn¥x mer, ymegwyx m svoym bazysom.
Cel\ dannoj rabot¥ sostoyt v tom, çtob¥ dat\ opysanye vsex takyx neotry-
catel\n¥x mer m v prostranstve X, çto dlq kaΩdoj Y-znaçnoj mer¥ v XAAmera
m est\ slab¥j bazys. Zatem rezul\tat o slabom bazyse prymenqetsq kAslabo
sxodqwymsq posledovatel\nostqm dyfferencyruem¥x vektorn¥x mer.AAOsnov-
n¥e ponqtyq dyfferencyruem¥x vektorn¥x mer soderΩatsq v [2] (hl. 4).
3. Formulyrovka rezul\tatov.
Teorema 1. Pust\ X — polnoe separabel\noe metryzuemoe topolohyçeskoe
lynejnoe prostranstvo, Y — banaxovo prostranstvo. Dlq toho çtob¥ ne-
otrycatel\naq mera m v prostranstve X b¥la slab¥m bazysom vsex Y -
znaçn¥x mer v X, neobxodymo y dostatoçno, çtob¥ ny na odnom nepustom ot-
kr¥tom mnoΩestve ona ne prynymala nulevoho znaçenyq.
© V. A. ROMANOV, 2007
1436 ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 10
SLABÁE BAZYSÁ VEKTORNÁX MER 1437
Teorema 2. Pust\ ( x k
) — proyzvol\naq posledovatel\nost\ s plotnoj ly-
nejnoj oboloçkoj L v polnom separabel\nom metryzuemom lokal\no v¥puklom
prostranstve X. Tohda kaΩdaq Y -znaçnaq mera µ v X est\ predel slabo
sxodqwejsq posledovatel\nosty beskoneçno dyfferencyruem¥x po vsem nap-
ravlenyqm yz L Y-znaçn¥x mer, ymegwyx svoym bazysom odnu y tu Ωe neotry-
catel\nug meru ν, ne zavysqwug ot µ.
4. Dokazatel\stva. Dokazatel\stvo teorem¥ 1. PredpoloΩym, çto dlq
nekotoroho nepustoho otkr¥toho mnoΩestva U m ( U) = 0. Pust\ f — funkcyq
yz Cb ( X ) , prynymagwaq znaçenye 1 v nekotoroj toçke a mnoΩestva U y
obrawagwaqsq v nul\ vne U, ( µn
) — posledovatel\nost\ Y-znaçn¥x mer, yme-
gwyx m svoym bazysom, µ — dyskretnaq Y-znaçnaq mera, sosredotoçennaq v
toçke a. Tohda levaq çast\ formul¥ (1) ravna nulg, a pravaq A— net. Otsgda
sleduet neobxodymost\.
DokaΩem dostatoçnost\. Pust\ m — fyksyrovannaq neotrycatel\naq
mera v X s poloΩytel\n¥my znaçenyqmy nepust¥x otkr¥t¥x mnoΩestv, a µ —
proyzvol\naq Y-znaçnaq mera koneçnoj polnoj varyacyy. V polnom separa-
bel\nom metryzuemom prostranstve varyacyq v ( µ ) mer¥ µ, kak y lgbaq
druhaq koneçnaq neotrycatel\naq mera, dolΩna b¥t\ meroj Radona [2, s. 19], a
poπtomu v X najdetsq posledovatel\nost\ neperesekagwyxsq kompaktov Kn ,
dlq kotor¥x
v( ) \µ X K
nj
n
j
=
<
1
1∪ . (2)
Kompakt K1 pokroem koneçn¥m çyslom dyzægnktn¥x yzmerym¥x mnoΩestv
A i n1, , dyametra men\ße
1
n
, vnutrennosty kotor¥x peresekagtsq s K1. Pust\
T n1, — obæedynenye πtyx mnoΩestv.
MnoΩestvo K2A\AT n1, pokroem koneçn¥m çyslom neperesekagwyxsq meΩdu
soboj y s T n1, yzmerym¥x mnoΩestv A i n2, , dyametra men\ße
1
n
, vnutrennosty
kotor¥x peresekagtsq s K2. Pust\ T n2, — obæedynenye πtyx mnoΩestv.
ProdolΩym πtot process dalee. Na j-m ßahe mnoΩestvo KjA\A
p
j
p nT
=
−
1
1
∪ , po-
kroem koneçn¥m çyslom neperesekagwyxsq meΩdu soboj y s Tp n, (pry p < j )
yzmerym¥x mnoΩestv Aj i n, , dyametra men\ße
1
n
, vnutrennosty kotor¥x pere-
sekagtsq s Kj. Zdes\ yndeks i moΩet yzmenqt\sq ot 1 do nekotoroho natural\-
noho N ( j, n ), zavysqweho ot j y n. Pust\ Tj, n — obæedynenye πtyx mnoΩestv.
Zametym, çto
p
j
p
p
j
p nK T
= =
⊂
1 1
∪ ∪ , , (3)
pryçem mnoΩestva v pravoj çasty vklgçenyq (3) dyzægnktn¥.
Process takyx postroenyj zaverßym dlq çysla j = n vklgçytel\no. Obo-
znaçym
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 10
1438 V. A. ROMANOV
Fn n0 , =
j
n
j nT
=1
0
∪ , =
j
n
i
N j n
j i nA
= =1 1
0
∪ ∪
( , )
, , , (4)
hde n0 moΩet b¥t\ proyzvol\n¥m natural\n¥m çyslom, ne prev¥ßagwym n.
Poskol\ku kaΩdoe yz mnoΩestv Aj i n, , ymeet nepustug vnutrennost\, mera
m prynymaet na nem poloΩytel\noe znaçenye. Pust\ mj i n, , — veroqtnostnaq
mera, poluçagwaqsq normyrovkoj proyzvedenyq yndykatora πtoho mnoΩestva
na meru m; yj i n, , — znaçenye Y-znaçnoj mer¥ µ na ukazannom mnoΩestve.
Rassmotrym posledovatel\nost\ Y-znaçn¥x mer, zadavaem¥x formulamy
µn j i n
i
N j n
j
n
j i ny m= ⋅
==
∑∑ , ,
( , )
, ,
11
. (5)
Qsno, çto dlq kaΩdoj yz µn mera m est\ bazys. Yz postroenyq µn takΩe
sleduet, çto ee varyacyq ne prev¥ßaet varyacyg µ.
Teper\ zafyksyruem proyzvol\nug funkcyg f yz Cb ( X ) y çyslo ε > 0.
Pust\ M = sup ( ) :f x x X∈{ } ≠ 0.
Yz uslovyq (2) sleduet, çto najdetsq takoe natural\noe n0 , çto
v( ) \µ X K
j
n
j
=
1
0
∪ < ε
M
.
Tohda yz formul (3) – (5) sleduet, pry n > n0
v( ) \ ,µn n nX F
0( ) ≤ v( ) \ ,µ X Fn n0( ) < ε
M
. (6)
Çyslo n0 dalee sçytaem fyksyrovann¥m. Rassmotrym modul\ neprer¥vnos-
ty funkcyy f na mnoΩestve
j
n
jK=1
0∪ :
ω δn0
( ) =
sup ( ) ( ) : ( , ) , ,f z f x z x z K x X
j
n
j− < ∈ ∈
=
ρ δ
1
0
∪ .
Poskol\ku obæedynenye koneçnoho çysla kompaktov snova est\ kompakt, to pry
δ → 0 πtot modul\ neprer¥vnosty ymeet nulevoj predel, a poπtomu najdetsq
takoe n1 > n0, çto pry n > n1
ω εn n0
1
< . (7)
V kaΩdom yz mnoΩestv Aj i n, , najdetsq toçka z j i n, , , prynadleΩawaq kom-
paktu Kj. Poskol\ku na ukazannom mnoΩestve mer¥ µn y µ prynymagt odyna-
kov¥e znaçenyq, to s uçetom (7) pry n > n1 y pry j ≤ n0
f x d xn
Aj i n
( ) ( )( )
, ,
µ µ−∫ = f x f z d xj i n
A
n
j i n
( ) ( ) ( )( ), ,
, ,
−[ ] −∫ µ µ ≤
≤ 2ε µv( ) , ,Aj i n[ ],
a poπtomu s uçetom (4) pry n > n1
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 10
SLABÁE BAZYSÁ VEKTORNÁX MER 1439
f x d xn
Fn n
( ) ( )( )
,
µ µ−∫
0
≤
2
0
ε µv( ) ,Fn n[ ] ≤ 2ε µVar .
Otsgda y yz (6) v¥tekaet, çto pry n > n1
f x d xn
X
( ) ( )( )µ µ−∫ ≤ 2 1ε µ( )+ Var .
Sledovatel\no, postroennaq posledovatel\nost\ Y-znaçn¥x mer µ n slabo
sxodytsq k µ.
Teorema dokazana.
Dokazatel\stvo teorem¥ 2. Bez umen\ßenyq obwnosty moΩno sçytat\,
çto πlement¥ xk lynejno nezavysym¥. Yz rezul\tatov [3, s. 63 – 68] y [ 4 ]
(predloΩenye 4) sleduet suwestvovanye takoho vklgçagweho L lynejnoho
podprostranstva H prostranstva X y takoho skalqrnoho proyzvedenyq na H,
çto H stanovytsq separabel\n¥m hyl\bertov¥m prostranstvom s kompaktn¥m
kanonyçeskym vloΩenyem v X, a L — vsgdu plotn¥m v H. Pust\ m est\ haus-
sova mera v H, obraz kvadratnoho kornq yz korrelqcyonnoho operatora kotoroj
vklgçaet L. Tohda m dyfferencyruema, a tem bolee neprer¥vna po vsem nap-
ravlenyqm yz L [5] (teorema 5.3.1). No tohda L-neprer¥vn¥ y absolgtno ne-
prer¥vn¥e otnosytel\no m mer¥ [6] (teorema 1), v tom çysle mer¥ mj i n, , ,
postroenn¥e v xode dokazatel\stva pred¥duwej teorem¥. Sledovatel\no, pro-
yzvedenyq πtyx neotrycatel\n¥x mer na yntehryruem¥e po Boxneru Y-znaçn¥e
funkcyy L -neprer¥vn¥ v topolohyy sxodymosty po varyacyy [7] (predloΩe-
nyeA2). No tohda varyacyonno L-neprer¥vn¥ y summ¥ koneçnoho çysla takyx
proyzvedenyj, v tom çysle Y-znaçn¥e mer¥ µ n
, zadavaem¥e formulamy (5).
Sledovatel\no [8] (teorema 2), najdutsq takye beskoneçno dyfferencyruem¥e
po vsem napravlenyqm yz L Y-znaçn¥e mer¥ λn
, dlq kotor¥x
Var ( )µ λn n n
− < 1
.
No dlq kaΩdoj funkcyy f yz Cb ( X )
f x d xn
X
( ) ( ) ( )λ µ−∫ ≤
1
n
f x
x X
sup ( )
∈
+ f x d xn
X
( ) ( ) ( )µ µ−∫ .
Otsgda y yz slaboj sxodymosty µn k µ sleduet, çto posledovatel\nost\ λn
takΩe slabo sxodytsq k µ.
Pry πtom yz postroennoj v [8] konstrukcyy sleduet, çto znaçenyq vektorn¥x
mer λn zadagtsq formulamy
λ µn n
H
nE E A h dm h( ) ( ) ( )= +( )∫ ,
hde m En( ) = m nE( ) , pryçem operator A s qdern¥m kvadratom moΩno v¥brat\
zavysqwym ot mer¥ m, sluΩawej obwym bazysom dlq vektorn¥x mer µ n
, a
potomu vektorn¥e mer¥ λn ymegt obwym bazysom meru ν, zadavaemug formu-
loj
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 10
1440 V. A. ROMANOV
ν( ) ( ) ( )E m E A h dm hn
n H
n= +( )
=
∞
∑ ∫1
21
.
Teorema dokazana.
1. Skoroxod A. V. Yntehryrovanye v hyl\bertovom prostranstve. – M.: Nauka, 1975. – 232 s.
2. Daleckyj G. L., Fomyn S. V. Mer¥ y dyfferencyal\n¥e uravnenyq v beskoneçnomern¥x
prostranstvax. – M.: Nauka, 1983. – 384 s.
3. Ho X. Haussovskye mer¥ v banaxov¥x prostranstvax. – M.: Myr, 1979. – 176 s.
4. Bohaçev V. Y. PrenebreΩym¥e mnoΩestva v lokal\no v¥pukl¥x prostranstvax // Mat. za-
metky. – 1984. – 36, # 1. – S. 51 – 64.
5. Averbux V. Y., Smolqnov O. H., Fomyn S. V. Obobwenn¥e funkcyy y dyfferencyal\n¥e
uravnenyq v lynejn¥x prostranstvax. 1 Dyfferencyruem¥e mer¥ // Tr. Mosk. mat. o-va. –
1987. – 24. – S. 133 – 174.
6. Romanov V. A. Ob H-neprer¥vn¥x merax v hyl\bertovom prostranstve // Vestn. Mosk. un-
ta. Mat., mex. – 1977. – 32, # 1. – S. 81 – 85.
7. Romanov V. A. Vektorn¥e mer¥ razlyçn¥x klassov hladkosty y yx predel¥ // Ukr. mat.
Ωurn. – 1995. – 47, # 4. – S. 512 – 516.
8. Romanov V. A. Predel¥ analytyçeskyx vektorn¥x mer // Tam Ωe. – 1992. – 44 , # 8. –
S.A1133 –1135.
Poluçeno 19.09.2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 10
|
| id | umjimathkievua-article-3402 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:41:51Z |
| publishDate | 2007 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/d4/60f2af6fd9429316c3b2dc9ad966fcd4.pdf |
| spelling | umjimathkievua-article-34022020-03-18T19:53:10Z Weak bases of vector measures Слабые базисы векторных мер Romanov, V. A. Романов, В. А. Романов, В. А. We solve the problem of representation of measures with values in a Banach space as the limits of weakly convergent sequences of vector measures whose basis is a given nonnegative measure. Розв'язано питання про зображення мір із значеннями в банаховому просторі як границь слабко збіжних послідовностей векторних мір, що мають своїм базисом дану невід'ємну міру. Institute of Mathematics, NAS of Ukraine 2007-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3402 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 10 (2007); 1436–1440 Український математичний журнал; Том 59 № 10 (2007); 1436–1440 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3402/3547 https://umj.imath.kiev.ua/index.php/umj/article/view/3402/3548 Copyright (c) 2007 Romanov V. A. |
| spellingShingle | Romanov, V. A. Романов, В. А. Романов, В. А. Weak bases of vector measures |
| title | Weak bases of vector measures |
| title_alt | Слабые базисы векторных мер |
| title_full | Weak bases of vector measures |
| title_fullStr | Weak bases of vector measures |
| title_full_unstemmed | Weak bases of vector measures |
| title_short | Weak bases of vector measures |
| title_sort | weak bases of vector measures |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3402 |
| work_keys_str_mv | AT romanovva weakbasesofvectormeasures AT romanovva weakbasesofvectormeasures AT romanovva weakbasesofvectormeasures AT romanovva slabyebazisyvektornyhmer AT romanovva slabyebazisyvektornyhmer AT romanovva slabyebazisyvektornyhmer |