Finite absolute continuity of Gaussian measures on infinite-dimensional spaces
We study the notion of finite absolute continuity for measures on infinite-dimensional spaces. For Gaussian product measures on \(\mathbb{R}^{\infty}\) and Gaussian measures on a Hilbert space, we establish criteria for finite absolute continuity. We consider cases where the condition of finite abso...
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| Дата: | 2008 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Російська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2008
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3251 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509304190140416 |
|---|---|
| author | Ryabov, G. V. Рябов, Г. В. Рябов, Г. В. |
| author_facet | Ryabov, G. V. Рябов, Г. В. Рябов, Г. В. |
| author_sort | Ryabov, G. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:49:15Z |
| description | We study the notion of finite absolute continuity for measures on infinite-dimensional spaces. For Gaussian product measures on \(\mathbb{R}^{\infty}\) and Gaussian measures on a Hilbert space, we establish criteria for finite absolute continuity. We consider cases where the condition of finite absolute continuity of Gaussian measures is equivalent to the condition of their equivalence. |
| first_indexed | 2026-03-24T02:38:58Z |
| format | Article |
| fulltext |
UDK 519.21
H. V. Rqbov (Nac. texn. un-t Ukrayn¥ „KPY”, Kyev)
FYNYTNAQ ABSOLGTNAQ NEPRERÁVNOST|
HAUSSOVSKYX MER NA BESKONEÇNOMERNÁX
PROSTRANSTVAX
∗∗∗∗
A notion of finite absolute continuity for measures on infinite-dimensional spaces is studied. For
Gaussian product-measures on ∞
R and Gaussian measures on a Hilbert space, criteria for the finite
absolute continuity are obtained. Some cases are considered, where the condition of the finite absolute
continuity of the Gaussian measures means the equivalence of these measures.
Vyvça[t\sq ponqttq finitno] absolgtno] neperervnosti dlq mir na neskinçennovymirnyx pros-
torax. Dlq haussivs\kyx prodakt-mir na
∞
R ta haussivs\kyx mir na hil\bertovomu prostori ot-
rymano kryteri] finitno] absolgtno] neperervnosti. Rozhlqnuto vypadky, koly umova finitno]
absolgtno] neperervnosti haussivs\kyx mir rivnosyl\na umovi ]x ekvivalentnosti.
1. Vvedenye. Absolgtnaq neprer¥vnost\ mer na beskoneçnomern¥x prostran-
stvax yzuçalas\ mnohymy avtoramy. Dlq nekotor¥x vydov mer udalos\ poluçyt\
kryteryy absolgtnoj neprer¥vnosty. V sluçae prodakt-mer na R
∞
takym kry-
teryem qvlqetsq teorema Kakutany [1, 2]. Dlq haussovskyx mer na banaxovom
prostranstve kryteryj absolgtnoj neprer¥vnosty daetsq teoremoj Haeka –
Fel\dmana [1, 2]. V rabote [3] vvedeno ponqtye fynytnoj absolgtnoj nepre-
r¥vnosty, predstavlqgwee soboj nekotorug al\ternatyvu ponqtyg absolgt-
noj neprer¥vnosty. V πtoj rabote dokazano, çto uslovye fynytnoj absolgtnoj
neprer¥vnosty mer¥ ν otnosytel\no mer¥ µ , kotoroe slabee, çem uslovye su-
westvovanyq kvadratyçno yntehryruemoj plotnosty ν otnosytel\no µ , dos-
tatoçno dlq suwestvovanyq razloΩenyq Yto – Vynera mer¥ ν otnosytel\no
mer¥ µ . Pryvedem sootvetstvugwee opredelenye. Pust\ L — lynejnoe topo-
lohyçeskoe prostranstvo, µ y ν — veroqtnostn¥e mer¥ na σ -alhebre borelev-
skyx podmnoΩestv L , ymegwye slab¥e moment¥ vsex porqdkov, Pn — semejst-
vo vsex mnohoçlenov na L stepeny ne v¥ße n .
Opredelenye&<[3]. Mera ν naz¥vaetsq fynytno absolgtno neprer¥vnoj
otnosytel\no µ , esly
∀ n ≥ 0 ∃ Cn > 0 ∀ Q ( u) ∈ Pn :
Q u du( ) ( )ν∫ ≤ C Q u dun
2 1 2
( ) ( )
/
µ∫( )
( oboznaçaem ν <<0 µ ) .
Zameçanyq.<<1. Opredelenye ymeet sm¥sl tol\ko v sluçae dim L = + ∞ ,
tak kak esly dim L < + ∞ y mera µ takova, çto yz ravenstva Q = 0 ( mod µ )
sleduet ravenstvo Q ≡ 0 dlq lgboho mnohoçlena Q , to lgbaq mera ν , yme-
gwaq vse moment¥, budet fynytno absolgtno neprer¥vnoj otnosytel\no µ .
2. Esly ν � µ y plotnost\
d
d
ν
µ
yntehryruema s kvadratom po mere µ , to
ν �0 µ y posledovatel\nost\ { };C nn ≥ 0 moΩno v¥brat\ ohranyçennoj (po
povodu zameçanyj<<1 y 2 sm. [3]).
3. Fynytnaq absolgtnaq neprer¥vnost\ soxranqetsq pry sdvyhax: esly a ∈
∗
V¥polnena pry podderΩke Mynysterstva obrazovanyq y nauky Ukrayn¥ (proekt GP\F13\0095).
© H. V. RQBOV, 2008
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 10 1367
1368 H. V. RQBOV
∈ L , f u u aa( ) = + — sdvyh na vektor a, µ µa af= −� 1, ν νa af= −� 1 — obraz¥
mer µ y ν pry sdvyhe a, to ν �0 µ tohda y tol\ko tohda, kohda dlq lgboho
vektora a ∈ L νa �0 µa . Dejstvytel\no, pust\ ν �0 µ y Q ( u) ∈ Pn — mno-
hoçlen stepeny ne v¥ße n . Tohda Q ( u – a) ∈ Pn y
Q u dua( ) ( )ν∫ = Q u a du( ) ( )−∫ ν ≤
≤ C Q u a dun
2 1 2
( ) ( )
/
−( )∫ µ = C Q u dun a
2 1 2
( ) ( )
/
µ∫( ) .
V nastoqwej stat\e yssleduetsq svqz\ meΩdu fynytnoj absolgtnoj nepre-
r¥vnost\g y absolgtnoj neprer¥vnost\g haussovskyx mer na nekotor¥x besko-
neçnomern¥x prostranstvax y pryvodqtsq kryteryy fynytnoj absolgtnoj ne-
prer¥vnosty haussovskyx mer na πtyx prostranstvax.
2.&&Fynytnaq absolgtnaq neprer¥vnost\ prodakt-mer. Oboznaçym çerez
R
∞
prostranstvo vsex posledovatel\nostej dejstvytel\n¥x çysel x = ( x1 ,
x2 , … ) s ohranyçennoj metrykoj
ρ ( x, y ) = 2
11
-k k k
k kk
x y
x y
−
+ −=
∞
∑ .
Pust\ { }µk , { }νk — posledovatel\nosty veroqtnostn¥x mer na R , µ =
=
�
k
k
=
∞
1
µ , ν =
�
k
k
=
∞
1
ν — prodakt-mer¥ na R
∞, ymegwye vse moment¥ (t. e. dlq
vsex k ≥ 1 mer¥ µk y νk ymegt vse moment¥). Oboznaçym
M m k( ; )µ =
R
∫ u dum
kµ ( ) ,
D m k( ; )µ = M m M mk k( ; ) ( ; )2 2µ µ− , m ≥ 1.
Teorema&&1. Pust\ ν �0 µ . Tohda
∀ m ≥ 1 :
( )( ; ) ( ; )
( ; )
( ; )
M m M m
D m
k k
kk
D m k
µ ν
µ
µ
−
=
≠
∞
∑
2
1
0
< + ∞ .
Dokazatel\stvo. Rassmotrym Q ( x ) = a a xk k
m
k
n
0 1
+ =∑ , Q ( x) ∈ Pm . Tohda
Q x dx( ) ( )ν∫( )2 = a a M m
k
n
k k0
2
1
2 2+
=
∑ ( ; )ν +
+ 2 20
1 1
a a M m a a M m M m
k
n
k k
k j
k j
n
k j k j
= =
<
∑ ∑+( ; ) ( ; ) ( ; )
,
ν ν ν , (1)
Q x dx2( ) ( )µ∫ = a a M m
k
n
k k0
2
1
2 2+
=
∑ ( ; )µ +
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 10
FYNYTNAQ ABSOLGTNAQ NEPRERÁVNOST| HAUSSOVSKYX MER … 1369
+ 2 20
1 1
a a M m a a M m M m
k
n
k k
k j
k j
n
k j k j
= =
<
∑ ∑+( ; ) ( ; ) ( ; )
,
µ µ µ . (2)
Yz uslovyq ν �0 µ sleduet, çto
∃ Cm > 0 ∀ Q ( x) ≥ Pm : Q x dx( ) ( )ν∫( )2 ≤ C Q x dxm
2( ) ( )µ∫ . (3)
Podstavlqq (1) y (2) v (3), dlq vsex n ≥ 1 y vsex dejstvytel\n¥x a0 , … , an po-
luçaem
a C a C M m M mm
k
n
k m k k0
2
1
2 21 2( ) ( ; ) ( ; )− + −( )
=
∑ µ ν +
+ 2 0
1
2a a C M m M m
k
n
k m k k
=
∑ −( )( ; ) ( ; )µ ν +
+ 2
1k j
k j
n
k j m k j k ja a C M m M m M m M m
,
( ; ) ( ; ) ( ; ) ( ; )
=
<
∑ −( )µ µ ν ν ≥ 0.
Poπtomu dlq kaΩdoho n ≥ 1 matryca
An = { }( )
,aij
n
i j
n
=0 ,
hde
a n
00
( ) = Cm −1,
a j
n
0
( ) = aj
n
0
( ) = C M m M mm j j( ; ) ( ; )µ ν− , j = 1, n ,
aii
n( ) = C M m M mm i i( ; ) ( ; )2 2µ ν− , i = 1, n ,
aij
n( ) = aji
n( ) = C M m M m M m M mm i j i j( ; ) ( ; ) ( ; ) ( ; )µ µ ν ν− , i, j = 1, n , i < j,
budet neotrycatel\no opredelennoj. Sledovatel\no, dlq kaΩdoho n ≥ 1
det An ≥ 0. KaΩd¥j stolbec v matryce An est\ raznost\ dvux stolbcov. Ras-
klad¥vaq det An v summu opredelytelej po vsem stolbcam, ymeem
det An =
k
n
k m
nD m C
=
+∏
1
1( ; )µ –
– C D m M m M m D mm
n
k
n
k
k
n
k k
j
j k
n
j
= = =
≠
∏ ∑ ∏+ −
1 1
2
1
( ; ) ( ; ) ( ; ) ( ; )( )µ µ ν µ ≥ 0.
Esly rassmatryvat\ mnohoçlen Q ( x ) = a a xk k
m
k
D m
n
k
0
1
0
+
=
≠
∑
( ; )µ
, to
∀ n ≥ 1 : Cm ≥ 1
2
1
0
+ −
=
≠
∑ ( )( ; ) ( ; )
( ; )
( ; )
M m M m
D m
k k
kk
D m
n
k
µ ν
µ
µ
,
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 10
1370 H. V. RQBOV
çto y dokaz¥vaet teoremu.
Zameçanye<4. Pust\ ν �0 µ y dlq nekotoroho k D k( ; )1 µ = 0, t. e. µk =
= δ µM k( ; )1 . V πtom sluçae ν k = µk . Dejstvytel\no, predpoloΩym, çto
ν µk kM{ }( ; )1 < 1, y rassmotrym odnomern¥j mnohoçlen Q ( x ) = Q ( xk ) =
= ( )( ; )x Mk k− 1 2µ . Tohda
Q x dx2( ) ( )µ∫ = Q x dxk k k
2( ) ( )µ
R
∫ = Q M k
2 1( )( ; )µ = 0.
Yz uslovyq ν �0 µ sleduet, çto Q x dx( ) ( )ν∫ = Q x dxk k k( ) ( )ν
R∫ = 0, çto ne-
vozmoΩno, poskol\ku Q ( xk ) > 0 na mnoΩestve R \ { }( ; )M k1 µ poloΩytel\noj
mer¥ νk .
Yz zameçanyq<<1 k opredelenyg fynytnoj absolgtnoj neprer¥vnosty sle-
duet, çto na R moΩno postroyt\ takye vzaymno synhulqrn¥e mer¥ µ y ν, çto
ν �0 µ y µ �0 ν . Pryvedem prymer πkvyvalentn¥x mer µ y ν na R
∞
takyx, çto ν /�0 µ y µ /�0 ν.
Prymer. Pust\ { };p kk ≥ 1 y { };′ ≥p kk 1 — dve posledovatel\nosty dejst-
vytel\n¥x çysel, takyx, çto dlq vsex k ≥ 1 0 < pk < 1, 0 < ′pk < 1,
pk
k =
∞
∑
1
< + ∞ ,
k
kp
=
∞
∑ ′
1
< + ∞ .
Pust\ qk = 1 – pk , ′qk = 1 – ′pk , k ≥ 1,
µ2 1k− = p qk kδ δ0 1+ , µ2k = ′ + ′p qk kδ δ0 1,
ν2 1k− = ′ + ′p qk kδ δ0 1, ν2k = p qk kδ δ0 1+
— veroqtnostn¥e mer¥ na R y dlq vsex m ≥ 1 µm ∼ νm ,
d
d
xk
k
ν
µ
2 1
2 1
−
−
( ) =
′ + ′p
p
x
q
q
xk
k
k
k
÷ ÷0 1( ) ( ) ,
d
d
xk
k
ν
µ
2
2
( ) =
p
p
x
q
q
xk
k
k
k′
+
′
÷ ÷0 1( ) ( ) , k ≥ 1.
Rassmotrym µ =
�
m
m
=
∞
1
µ y ν =
�
m
m
=
∞
1
ν — veroqtnostn¥e prodakt-mer¥ na R
∞
.
PokaΩem, çto µ ∼ ν . Po teoreme Kakutany [1, 2], dlq πtoho nado ustanovyt\,
çto sxodytsq beskoneçnoe proyzvedenye
m
m mH
=
∞
∏
1
( , )µ ν , hde H m m( , )µ ν — yn-
tehral Xellynhera, kotor¥j v¥çyslqetsq po formule
H m m( , )µ ν = ∫ d
d
x
d
d
x dxm mµ
λ
ν
λ
λ( ) ( ) ( ),
λ — takaq veroqtnostnaq mera, çto µm << λ y νm << λ . V dannom sluçae
H m m( , )µ ν = ∫ d
d
x dxm
m
m
ν
µ
µ( ) ( ),
H k k( , )µ ν2 1 2 1− − = H k k( , )µ ν2 2 = p p q qk k k k′ + ′ .
Tohda
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 10
FYNYTNAQ ABSOLGTNAQ NEPRERÁVNOST| HAUSSOVSKYX MER … 1371
m
m mH
=
∞
∏
1
( , )µ ν =
k
k k k kH H
=
∞
− −∏
1
2 1 2 1 2 2( , ) ( , )µ ν µ ν =
=
k
k k k kp p q q
=
∞
∏ ′ + ′( )
1
2 =
k
k k k kp p p p
=
∞
∏ ′ + − − ′( )
1
21 1( )( ) =
=
k
k k k k k k k kp p p p p p q q
=
∞
∏ + ′ − − ′ + ′ ′( )
1
1 2 2 =
=
k
k k k kp q p q
=
∞
∏ − ′ − ′( )( )
1
21 .
Poslednee proyzvedenye sxodytsq, tak kak yz sxodymosty rqdov pkk =
∞∑ 1
,
′=
∞∑ pkk 1
sleduet sxodymost\ rqda
p q p qk k k k
k
′ − ′( )
=
∞
∑ 2
1
= p q p q p p q qk k k k k k k k
k
′ + ′ − ′ ′( )
=
∞
∑ 2
1
.
Poπtomu µ ∼ ν .
Esly ν �0 µ , to sohlasno teoreme<<1
k
k k
k k
k k
k k
p p
p q
p p
p q=
∞
∑ − ′ + − ′
′ ′
1
2 2( ) ( )
< + ∞ . (4)
S druhoj storon¥,
∫
d
d
x dxν
µ
µ( ) ( )
2
=
=
k
k
k
k
k
k
k
d
d
x dx
d
d
x dx
=
∞
−
−
−∏ ∫ ∫
1
2 1
2 1
2
2 1
2
2
2
2
ν
µ
µ ν
µ
µ( ) ( ) ( ) ( )
R R
=
=
k
k
k
k
k
k
k
k
k
p
p
q
q
p
p
q
q=
∞
∏ ′ + ′
′
+
′
1
2 2 2 2
=
=
k
k k
k k
k k
k k
p p
p q
p p
p q=
∞
∏ + − ′
+ − ′
′ ′
1
2 2
1 1
( ) ( )
.
Takym obrazom,
d
d
Lν
µ
µ∈ ∞
2( ),R y ν �0 µ tohda y tol\ko tohda, kohda v¥pol-
nqetsq (4).
Analohyçno proverqetsq, çto µ �0 ν tohda y tol\ko tohda, kohda v¥pol-
nqetsq (4). Podbyraq posledovatel\nosty { };p kk ≥ 1 y { };′ ≥p kk 1 tak, çto
pk
k =
∞
∑
1
< + ∞ ,
k
kp
=
∞
∑ ′
1
< + ∞ ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 10
1372 H. V. RQBOV
k
k k
k k
k k
k k
p p
p q
p p
p q=
∞
∑ − ′ + − ′
′ ′
1
2 2( ) ( )
= + ∞
(naprymer, pk = 1 4/ k , ′pk = 1 2/ k ), poluçaem prymer¥ mer µ y ν na R
∞
takyx, çto µ ∼ ν y ν /�0 µ , µ /�0 ν .
Dalee budem rassmatryvat\ haussovskye prodakt-mer¥ na R
∞
y haussovskye
mer¥ na hyl\bertovom prostranstve. V πtyx sluçaqx mer¥ lybo πkvyvalentn¥,
lybo synhulqrn¥, a v sluçae πkvyvalentnosty plotnost\ yntehryruema s kvad-
ratom. Takym obrazom, v sluçae haussovskyx mer µ y ν yz πkvyvalentnosty
ν ∼ µ sleduet fynytnaq absolgtnaq neprer¥vnost\ ν �0 µ .
3.&&Haussovskye prodakt-mer¥ na R
∞∞∞∞
. Pust\ µ =
�
k
k
=
∞
1
µ , ν =
�
k
k
=
∞
1
ν —
haussovskye prodakt-mer¥ na R :
µk ∼ N a tk k( , ), νk ∼ N b sk k( , ).
Teorema&&2. Pust\ µ k ∼ νk dlq vsex k ≥ 1. Tohda ν �0 µ v tom y
tol\ko v tom sluçae, kohda ν ∼ µ .
Dokazatel\stvo. PredpoloΩym, çto ν �0 µ . DokaΩem πkvyvalentnost\
mer µ ′ y ν ′ , poluçenn¥x yz µ y ν sdvyhom na ( – a1 , – a2 , … ) , tak çto ′µk ∼
∼ N tk( , )0 , ′νk ∼ N b a sk k k( , )− , ′µk ∼ ′νk dlq vsex k ≥ 1. Tohda
H k k( , )′ ′µ ν =
4
0
1 0
2
1 4
t s
t s
e t s
t s
k k
k k
b a
t s
k k
k k
k k
k k
( )
, , ,
, .
/
+
>
= =
−
+
Dlq πkvyvalentnosty ν ′ ∼ µ ′ dostatoçno, çtob¥ proyzvedenye H k kk
( , )′ ′=
∞∏ µ ν
1
sxodylos\. Dlq πtoho, v svog oçered\, nado, çtob¥
k
t
k k
k k
k
b a
t s=
≠
∞
∑ −
+1
0
2( )
< + ∞ y
k
t
k k
k k
k
t s
t s=
≠
∞
∑ −
+1
0
2
2
( )
( )
< + ∞ ,
tak kak
k
t
k k
k k
k
t s
t s=
≠
∞
∏ +1
0
2
4
( )
=
k
t
k k
k k
k
t s
t s=
≠
∞
∏ − −
+
1
0
2
21
( )
( )
.
Yz uslovyq ν ′ �0 µ ′ po teoreme<<1 poluçaem
k
t
k k
k
k
b a
t=
≠
∞
∑ −
1
0
2( )
< + ∞ , (5)
k
t
k k k k
k
k
t s b a
t=
≠
∞
∑ − − −
1
0
2 2
2
( ( ) )
< + ∞ ,
(6)
t s b a
t
k k k k
k
− − −( )2
→ 0, k → ∞ ,
( )b a
t
k k
k
− 2
→ 0, k → ∞ ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 10
FYNYTNAQ ABSOLGTNAQ NEPRERÁVNOST| HAUSSOVSKYX MER … 1373
poπtomu
t s
t
k k
k
−
→ 0, k → ∞ , (7)
( )t s
t
k k
k
− 2
2 =
( ( ) )t s b a
t
k k k k
k
− − − 2 2
2 + 2
2t s
t
b a
t
k k
k
k k
k
− −( )
–
( )b a
t
k k
k
−
2 2
. (8)
V sylu (5) – (7) rqd, sostoqwyj yz çlenov pravoj çasty ravenstva (8),
sxodytsq. Poπtomu
k
t
k k
k
k
t s
t=
≠
∞
∑ −
1
0
2
2
( )
< + ∞ .
Otsgda y yz (5) poluçaem sxodymost\ nuΩn¥x rqdov.
Teorema dokazana.
Sledstvye. Pust\ µ = �
k
k
=
∞
1
µ , ν = �
k
k
=
∞
1
ν — haussovskye prodakt-mer¥ na
R
∞ . Tohda µ ∼ ν v tom y tol\ko v tom sluçae, kohda ν �0 µ y µ �0 ν .
Dokazatel\stvo. Pust\ ν �0 µ y µ �0 ν . Sohlasno teoreme<<2, çtob¥
dokazat\ πkvyvalentnost\ ν ∼ µ , dostatoçno proveryt\, çto µk ∼ νk dlq vsex
k ≥ 1. Pust\ µk k kN a t∼ ( , ), νk k kN b s∼ ( , ). V zameçanyy k teoreme<<1
pokazano, çto yz ν �0 µ sleduet, çto esly dlq nekotoroho k ≥ 1 tk = 0, to
µk = νk . Dlq mer µk y νk vozmoΩn¥ sledugwye varyant¥:
a) tk > 0, sk > 0; v πtom sluçae µk ∼ νk
;
b) tk = 0; v πtom sluçae uslovye ν �0 µ vleçet ravenstvo µk = νk ;
v) sk = 0; v πtom sluçae uslovye µ �0 ν vleçet ravenstvo µk = νk .
Teorema&&3. Pust\ µ =
�
k
k
=
∞
1
µ , ν =
�
k
k
=
∞
1
ν — haussovskye prodakt-mer¥ na
R
∞ , µk kN t∼ ( , )0 , νk k kN b s∼ ( , ). Tohda ν �0 µ v tom y tol\ko v tom
sluçae, kohda v¥polnen¥ uslovyq:
1) esly dlq nekotoroho k ≥ 1 tk = 0, to νk = µk (t. e. sk = bk = 0);
2)
k
t
k
k
k
b
t=
≠
∞
∑
1
0
2
< + ∞ ,
k
t
k k
k
k
t s
t=
≠
∞
∑ −
1
0
2
2
( )
< + ∞ .
Dokazatel\stvo. Neobxodymost\. Pust\ ν �0 µ . V zameçanyy k teore-
me<<1 pokazano, çto uslovye<<1 v¥polnqetsq. Yz teorem¥<<1 sleduet, çto
k
t
k
k
k
b
t=
≠
∞
∑
1
0
2
< + ∞ ,
k
t
k k k
k
k
t s b
t=
≠
∞
∑ − −
1
0
2 2
2
( )
< + ∞ .
Tak Ωe, kak v dokazatel\stve teorem¥<<2, yz sxodymosty πtyx rqdov poluçaem
k
t
k k
k
k
t s
t=
≠
∞
∑ −
1
0
2
2
( )
< + ∞ .
Dostatoçnost\. V¥delym v podposledovatel\nost\ { };t nln
≥ 1 vse nuly
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 10
1374 H. V. RQBOV
posledovatel\nosty { };t kk ≥ 1 :
tln = 0 ∀ n ≥ 1.
Podposledovatel\nost\ nenulev¥x znaçenyj { };t kk ≥ 1 oboznaçym
{ };t nkn
≥ 1 . Pust\ ν0 , µ0 — proekcyy mer µ y ν na R0
∞ = Rl1
× Rl2
× … .
V sylu uslovyq<<1 ν0 y µ0 sovpadagt y qvlqgtsq meramy, sosredotoçenn¥my
v nule. Poπtomu dostatoçno proveryt\ fynytnug absolgtnug neprer¥vnost\
ν1 �0 µ1 , hde ν1 , µ1 — proekcyy mer µ y ν na R1
∞ = Rk1
× Rk2
× … , ν =
= ν ν0 1� , µ = µ µ0 1� . Poskol\ku
k
t
k k
k
k
t s
t=
≠
∞
∑ −
1
0
2
2
( )
< + ∞ , mnoΩestvo tkn
ta-
kyx, çto skn
= 0, koneçno. Pust\ πto budet mnoΩestvo { }, ,t tk km′ ′…
1
. Podposle-
dovatel\nost\ tkn
takyx, çto skn
> 0, oboznaçym çerez { },t nkn′′ ≥ 1 . Pust\ ′ν1,
′′ν1 — proekcyy mer¥ ν1 na R
m = R ′k1
× … × R ′km
y R−
∞
m = R ′′k1
× R ′′k2
× …
sootvetstvenno, ′µ1, ′′µ1 — proekcyy mer¥ µ1 na Rm = R ′k1
× … × R ′km
y
R−
∞
m = R ′′k1
× R ′′k2
× … sootvetstvenno, ν1 = ′ ′′ν ν1 1� , µ1 = ′ ′′µ µ1 1� , ′ν1 y ′µ1
— haussovskye mer¥ na koneçnomernom prostranstve R
m, pryçem ′µ1 nev¥-
roΩdena na R
m. Poπtomu dlq kaΩdoho n ≥ 0 suwestvuet Cn > 0 takoe, çto
dlq lgboho mnohoçlena Q ( x ) na R
m
stepeny ne v¥ße n
∫ ′Q x dx( ) ( )ν1 ≤ C Q x dxn ∫ ′( ) ( )µ1 . (9)
Dalee, ′′ν1 =
�
n
kn=
∞
′′
1
ν , ′′µ1 =
�
n
kn=
∞
′′
1
µ y dlq vsex n ≥ 1 ν ′′kn
∼ µ ′′kn
. Yz uslovyq<<2
po teoreme Kakutany poluçaem, çto ′′ν1 ∼ ′′µ1 . Pust\ ρ ( x ) =
d
d
x
′′
′′
ν
µ
1
1
( ), x m∈ −
∞
R .
KaΩd¥j vektor x ∈ ∞
R1 edynstvenn¥m obrazom predstavlqetsq v vyde x =
= x xm m+ − , hde xm
m∈R , x m m− −
∞∈R .
Pust\ P ( x ) = P x xm m( , )− — mnohoçlen na R1
∞
stepeny ne v¥ße n . Yspol\-
zuq (9), poluçaem cepoçku neravenstv
R1
1
∞
∫ ′P x dx( ) ( )ν =
R R
m
m
P x x dx dxm m m m∫ ∫
−
∞
− −′′ ′( , ) ( ) ( )ν ν1 1 =
=
R R
m
m
P x x x dx dxm m m m m∫ ∫
−
∞
− − −′′ ′( , ) ( ) ( ) ( )ρ µ ν1 1 ≤
≤
R R R
m
m m
P x x dx x dx dxm m m m m m∫ ∫ ∫
−
∞
−
∞
− − − −′′
′′
′2
1
1 2
2
1
1 2
1( , ) ( ) ( ) ( ) ( )
/ /
µ ρ µ ν ≤
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 10
FYNYTNAQ ABSOLGTNAQ NEPRERÁVNOST| HAUSSOVSKYX MER … 1375
≤
R R R−
∞
−
∞
∫ ∫ ∫− − − −′′
′′ ′
m
m
m
x dx P x x dx dxm m m m m mρ µ µ ν2
1
1 2
2
1 1
1 2
( ) ( ) ( , ) ( ) ( )
/ /
≤
≤
R−
∞
∫ − −′′
m
x dxm mρ µ2
1
1 2
( ) ( )
/
×
× C P x x dx dxn m m m m
m
m
2
1 2 2
1 1
1 2
/
/
( , ) ( ) ( )
R R
∫ ∫
−
∞
− −′′ ′
µ µ =
= C x dx P x dxn m m
m
2
2
1
1 2
2
1
1 2
1R R−
∞ ∞
∫ ∫− −′′
ρ µ µ( ) ( ) ( ) ( )
/ /
,
çto y dokaz¥vaet fynytnug absolgtnug neprer¥vnost\ ν �0 µ .
Teorema dokazana.
4. Haussovskye mer¥ na hyl\bertovom prostranstve. Pust\ µ y ν —
haussovskye mer¥ na vewestvennom separabel\nom hyl\bertovom prostranstve
H, µ ∼ N a S( , )1 1 , ν ∼ N a S( , )2 2 . Sohlasno teoreme Haeka – Fel\dmana [1, 2],
mer¥ µ y ν lybo πkvyvalentn¥, lybo synhulqrn¥, y
1) µ ∼ ν v tom y tol\ko v tom sluçae, kohda N a S N a S( , ) ( , )1 1 2 1∼ y
N a S N a S( , ) ( , )2 1 2 2∼ ;
2) N S N a S( , ) ( , )0 ∼ tohda y tol\ko tohda, kohda a S H∈ ( ) ;
3) N S N S( , ) ( , )0 01 2∼ tohda y tol\ko tohda, kohda suwestvuet takoj ohra-
nyçenn¥j neotrycatel\no opredelenn¥j obratym¥j operator T, çto S 2 =
= S T S1 1 , T I H− ∈L ( )( )2 . Zdes\ L ( )( )2 H — prostranstvo vsex operatorov
Hyl\berta – Ímydta na H.
V [3] dokazano sledugwee utverΩdenye.
Lemma. Pust\ µ ∼ N S( , )0 , ν ∼ N a S( , ). Tohda ν �0 µ v tom y tol\ko
v tom sluçae, kohda a S H∈ ( ) ( t. e. ν ∼ µ ) .
Teorema&4. Pust\ µ ∼ N S( , )0 1 , ν ∼ N S( , )0 2 . Tohda ν �0 µ v tom y
tol\ko v tom sluçae, kohda suwestvuet takoj ohranyçenn¥j neotrycatel\no
opredelenn¥j operator T, çto S2 = S T S1 1 , T I H− ∈L ( )( )2 .
Dokazatel\stvo. Neobxodymost\. Pust\ { };e nn ≥ 1 — ortonormyrovan-
n¥j bazys v (ker )S1
⊥ , sostoqwyj yz sobstvenn¥x vektorov operatora S1, koto-
r¥e sootvetstvugt poloΩytel\n¥m sobstvenn¥m çyslam { };λn n ≥ 1 . Obozna-
çym αij = ( , )S e ei j2 . Rassmotrym mnohoçlen Q ( x ) = a x e x eij i ji j
n
0 1
+ =∑ α ( , )( , )
,
.
Tak Ωe, kak v dokazatel\stve teorem¥<<1, ubeΩdaemsq v neotrycatel\nosty op-
redelytelq
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 10
1376 H. V. RQBOV
C C C
C C C
C
C
n n n nn
n n n n nn
n n n n n n n n n n n nn
− − − − −
− − − − −
− − − − −
−
−
− − − − − −
1
3
1 11 12 1
1 11 1
2
11
2
11 12 11 1 1
1 11 1 12 1 1 1
2
1
λ α α α λ α
λ α λ α α α α α λ λ α α
α α α α α λ λ α α α
λ
�
�
� � � � � �
�
nn nn n nn nn n n nn n nnC C− − − − −−α λ λ α α α α α α λ α1 11 12 1
2 23�
.
Poπtomu
∀ n ≥ 1 : C ≥ 1
2
2
2
1
2
1
+
−
+
= =
<
∑ ∑( )
,
α λ
λ
α
λ λ
ij i
ii
n
ij
i ji j
i j
n
.
Sledovatel\no,
( )
,
α λ δ
λ λ
ij i ij
i ji j
−
=
∞
∑
2
1
< + ∞ .
Opredelym ( , )Te ei j =
( , )S e ei j
i j
2
λ λ
na (ker )S1
⊥
y T = I na ker S1 ( T — yskom¥j
operator ) .
Dostatoçnost\. Pust\ S2 = S T S1 1 , hde T — ohranyçenn¥j neotryca-
tel\no opredelenn¥j operator, T I H− ∈L ( )( )2 . Proekcyy mer µ y ν na ker S1
sovpadagt — πto mer¥, sosredotoçenn¥e v nule. Poπtomu nado proveryt\ fy-
nytnug absolgtnug neprer¥vnost\ ν ′ �0 µ ′, hde ν ′ y µ ′ — proekcyy µ y ν
na (ker )S1
⊥
.
Pust\ { };e nn ≥ 1 — ortonormyrovann¥j bazys v (ker )S1
⊥
, sostoqwyj yz sob-
stvenn¥x vektorov operatora S1. Çyslo ej , dlq kotor¥x Tej = 0, koneçno,
tak kak T I H− ∈L ( )( )2 . Oboznaçym
H1 = l o. . ;{ }e Tej j = 0 , H2 = (ker )S H1 1
⊥ � .
Pust\ ′µ1, ′µ2 — proekcyy mer¥ ′µ na H1 y H2, ′µ = ′ ′µ µ1 2� . Pust\ ′ν1 — pro-
ekcyq mer¥ ′ν na H1. Suwestvuet takoj nabor haussovskyx mer { };′ ∈νu u H1 ,
zadann¥x na H2, çto
′ν ( )A = ′ ′∫ ν νu u
H
A du( ) ( )
1
1 , (10)
hde A — borelevskoe podmnoΩestvo (ker )S1
⊥ , Au — seçenye mnoΩestva A
vektorom u H∈ 1. Dlq vsex u H∈ 1 ′ ∼ ′ν µu 2. Pust\ ρu( )v =
d
d
u′
′
ν
µ2
( )v . Dalee, ′µ1
y ′ν1 — haussovskye mer¥ na koneçnomernom prostranstve H1, pryçem ′µ1 ne-
v¥roΩdena na H1. Poπtomu dlq kaΩdoho n ≥ 1 najdetsq takaq konstanta
Cn > 0, çto dlq vsex mnohoçlenov Q ( u ) na H1 stepeny ne v¥ße n
Q u du
H
( ) ( )′∫ ν1
1
≤ C Q u dun
H
( ) ( )′∫ µ1
1
. (11)
Teper\ pust\ P ( x ) = P ( u, v ) — mnohoçlen na (ker )S1
⊥ = H H1 2� stepeny ne
v¥ße n . V sylu (10) y (11) poluçaem cepoçku neravenstv
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 10
FYNYTNAQ ABSOLGTNAQ NEPRERÁVNOST| HAUSSOVSKYX MER … 1377
(ker )
( , ) ( , )
S
P u du d
1
⊥
∫ ′v vν =
H H
uP u d du
1 2
1∫ ∫ ′ ′( , ) ( ) ( )v vν ν =
=
H H
uP u d du
1 2
2 1∫ ∫ ′ ′( , ) ( ) ( ) ( )v v vρ µ ν ≤
≤
H H H
uP u d d du
1 2 2
2
2
1 2
2
2
1 2
1∫ ∫ ∫′
′
′( , ) ( ) ( ) ( ) ( )
/ /
v v v vµ ρ µ ν ≤
≤
H H
u
H H
d du P u d du
1 2 1 2
2
2 1
1 2
2
2 1
1 2
∫ ∫ ∫ ∫′ ′
′ ′
ρ µ ν µ ν( ) ( ) ( ) ( , ) ( ) ( )
/ /
v v v v ≤
≤
C d du P u d dun
H H
u
H H
2
1 2 2
2 1
1 2
2
2 1
1 2
1 2 1 2
/
/ /
( ) ( ) ( ) ( , ) ( ) ( )∫ ∫ ∫ ∫′ ′
′ ′
ρ µ ν µ µv v v v =
=
C d du P u du dn
H H
u
S
2
2
2 1
1 2
2
1 2
1 2 1
∫ ∫ ∫′ ′
′
⊥
ρ µ ν µ( ) ( ) ( ) ( , ) ( , )
/
(ker )
/
v v v v ,
çto y dokaz¥vaet fynytnug absolgtnug neprer¥vnost\ ν ′ �0 µ ′ .
Teorema dokazana.
Zameçanye<5. Kryteryy fynytnoj absolgtnoj neprer¥vnosty dlq haus-
sovskyx mer v teoremax<<3 y 4 predstavlqgt soboj oslablenye sootvetstvug-
wyx kryteryev πkvyvalentnosty dlq haussovskyx mer. Naprymer, pust\ { }en n=
∞
1
— ortonormyrovann¥j< bazys v hyl\bertovom prostranstve H,
S x1 = 1
2
1 k
x e ek k
k
( , )
=
∞
∑ , S x2 = 1
2
2 k
x e ek k
k
( , )
=
∞
∑ , µ ∼ N S( , )0 1 , ν ∼ N S( , )0 2 .
Tohda operator T yz teorem¥<<4 — πto proektor na podprostranstvo, poroΩ-
dennoe vektoramy { }en n=
∞
2 . T neobratym, ν µ⊥ , no ν µ<<0 .
1. Ho X.-S. Haussovskye mer¥ v banaxov¥x prostranstvax. – M.: Myr, 1979. – 176 s.
2. Skoroxod A. V. Yntehryrovanye v hyl\bertovom prostranstve. – M.: Nauka, 1975. – 232 s.
3. Dorohovcev A. A. Yzmerym¥e funkcyonal¥ y fynytno absolgtno neprer¥vn¥e mer¥ na
banaxov¥x prostranstvax // Ukr. mat. Ωurn. – 2000. – 52, # 9. – S.<1194 – 1204.
Poluçeno 15.12.06
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 10
|
| id | umjimathkievua-article-3251 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:38:58Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/b4/66ba0ceb6dc408584efbb472799acdb4.pdf |
| spelling | umjimathkievua-article-32512020-03-18T19:49:15Z Finite absolute continuity of Gaussian measures on infinite-dimensional spaces Финитная абсолютная непрерывность гауссовских мер на бесконечномерных пространствах Ryabov, G. V. Рябов, Г. В. Рябов, Г. В. We study the notion of finite absolute continuity for measures on infinite-dimensional spaces. For Gaussian product measures on \(\mathbb{R}^{\infty}\) and Gaussian measures on a Hilbert space, we establish criteria for finite absolute continuity. We consider cases where the condition of finite absolute continuity of Gaussian measures is equivalent to the condition of their equivalence. Вивчається поняття фінітної абсолютної неперервності для мір на нескінченновимірних просторах. Для гауссівських продакт-мір на \(\mathbb{R}^{\infty}\) та гауссівських мір на гільбертовому просторі отримано критерії фінітної абсолютної неперервності. Розглянуто випадки, коли умова фінітної абсолютної неперервності гауссівських мір рівносильна умові їх еквівалентності. Institute of Mathematics, NAS of Ukraine 2008-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3251 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 10 (2008); 1367–1377 Український математичний журнал; Том 60 № 10 (2008); 1367–1377 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3251/3248 https://umj.imath.kiev.ua/index.php/umj/article/view/3251/3249 Copyright (c) 2008 Ryabov G. V. |
| spellingShingle | Ryabov, G. V. Рябов, Г. В. Рябов, Г. В. Finite absolute continuity of Gaussian measures on infinite-dimensional spaces |
| title | Finite absolute continuity of Gaussian measures on infinite-dimensional spaces |
| title_alt | Финитная абсолютная непрерывность гауссовских мер на бесконечномерных пространствах |
| title_full | Finite absolute continuity of Gaussian measures on infinite-dimensional spaces |
| title_fullStr | Finite absolute continuity of Gaussian measures on infinite-dimensional spaces |
| title_full_unstemmed | Finite absolute continuity of Gaussian measures on infinite-dimensional spaces |
| title_short | Finite absolute continuity of Gaussian measures on infinite-dimensional spaces |
| title_sort | finite absolute continuity of gaussian measures on infinite-dimensional spaces |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3251 |
| work_keys_str_mv | AT ryabovgv finiteabsolutecontinuityofgaussianmeasuresoninfinitedimensionalspaces AT râbovgv finiteabsolutecontinuityofgaussianmeasuresoninfinitedimensionalspaces AT râbovgv finiteabsolutecontinuityofgaussianmeasuresoninfinitedimensionalspaces AT ryabovgv finitnaâabsolûtnaânepreryvnostʹgaussovskihmernabeskonečnomernyhprostranstvah AT râbovgv finitnaâabsolûtnaânepreryvnostʹgaussovskihmernabeskonečnomernyhprostranstvah AT râbovgv finitnaâabsolûtnaânepreryvnostʹgaussovskihmernabeskonečnomernyhprostranstvah |