On the equivalence of integral norms on the space of measurable polynomials with respect to a convex measure
We prove that, for a convex product-measure μ on a locally convex space, for any set A of positive measure, on the space of measurable polynomials of degree d, all Lp(μ)-norms coincide with the norms obtained by restricting μ to A.
Збережено в:
| Дата: | 2008 |
|---|---|
| Автор: | |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут математики НАН України
2008
|
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/4531 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | On the equivalence of integral norms on the space of measurable polynomials with respect to a convex measure / V. Berezhnoy // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 1. — С. 7–10. — Бібліогр.: 6 назв.— англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-4531 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-45312009-11-26T12:00:36Z On the equivalence of integral norms on the space of measurable polynomials with respect to a convex measure Berezhnoy, V. We prove that, for a convex product-measure μ on a locally convex space, for any set A of positive measure, on the space of measurable polynomials of degree d, all Lp(μ)-norms coincide with the norms obtained by restricting μ to A. 2008 Article On the equivalence of integral norms on the space of measurable polynomials with respect to a convex measure / V. Berezhnoy // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 1. — С. 7–10. — Бібліогр.: 6 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4531 519.21 en Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
We prove that, for a convex product-measure μ on a locally convex space, for any set A of positive measure, on the space of measurable polynomials of degree d, all Lp(μ)-norms coincide with the norms obtained by restricting μ to A. |
| format |
Article |
| author |
Berezhnoy, V. |
| spellingShingle |
Berezhnoy, V. On the equivalence of integral norms on the space of measurable polynomials with respect to a convex measure |
| author_facet |
Berezhnoy, V. |
| author_sort |
Berezhnoy, V. |
| title |
On the equivalence of integral norms on the space of measurable polynomials with respect to a convex measure |
| title_short |
On the equivalence of integral norms on the space of measurable polynomials with respect to a convex measure |
| title_full |
On the equivalence of integral norms on the space of measurable polynomials with respect to a convex measure |
| title_fullStr |
On the equivalence of integral norms on the space of measurable polynomials with respect to a convex measure |
| title_full_unstemmed |
On the equivalence of integral norms on the space of measurable polynomials with respect to a convex measure |
| title_sort |
on the equivalence of integral norms on the space of measurable polynomials with respect to a convex measure |
| publisher |
Інститут математики НАН України |
| publishDate |
2008 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/4531 |
| citation_txt |
On the equivalence of integral norms on the space of measurable polynomials with respect to a convex measure / V. Berezhnoy // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 1. — С. 7–10. — Бібліогр.: 6 назв.— англ. |
| work_keys_str_mv |
AT berezhnoyv ontheequivalenceofintegralnormsonthespaceofmeasurablepolynomialswithrespecttoaconvexmeasure |
| first_indexed |
2025-07-02T07:45:13Z |
| last_indexed |
2025-07-02T07:45:13Z |
| _version_ |
1836520375922458624 |
| fulltext |
Theory of Stochastic Processes
Vol. 14 (30), no. 1, 2008, pp. 7–10
UDC 519.21
VASILIY BEREZHNOY
ON THE EQUIVALENCE OF INTEGRAL NORMS
ON THE SPACE OF MEASURABLE POLYNOMIALS
WITH RESPECT TO A CONVEX MEASURE
We prove that, for a convex product-measure μ on a locally convex space, for any
set A of positive measure, on the space of measurable polynomials of degree d, all
Lp(μ)-norms coincide with the norms obtained by restricting μ to A.
It is well known that if γ is a Gaussian measure on a locally convex space X , then
all Lp-norms on Pd(γ), the space of measurable polynomials of degree at most d, are
mutually equivalent. In the case where X is a Hilbert space, A.A. Dorogovtsev [1] has
shown that the L2(γ)-norm on Pd(γ) is equivalent to the L2(γ|B)-norm, where γ|B is
the restriction of γ to a unit ball B of X .
In the recent paper [2], the author has reinforced that result and has shown that, in
the case of any locally convex space and an arbitrary measurable set A with γ(A) > 0,
all Lp(γ|A)-norms are equivalent to the Lp(γ)-norms. In particular, they are mutually
equivalent.
The main result of this paper shows that it is valid also for convex measures satisfying
certain additional conditions.
Let us recall some definitions (see, e.g., [3],[4]).
Definition 1. A Borel probability measure μ on Rn is called a convex (or log-concave)
measure if there exists an affine subspace E with μ(E) = 1, on which μ is given by a
density with respect to the Lebesgue measure on E such that, for all x, y ∈ E and
λ ∈ [0, 1], the following inequality holds:
(λx+ (1 − λ)y) ≥ (x)λ (y)1−λ.
Definition 2. Let X be a locally convex space equipped with the σ-algebra σ(X) generated
by the dual space X∗. A probability measure μ on σ(X) is called convex (or log-concave)
if, for all l1, . . . , ln ∈ X∗, its image under the mapping x �→ (l1(x), . . . , ln(x)) to Rn is
a convex measure on Rn.
Recently S.G. Bobkov [5] has obtained the following important result for convex mea-
sures.
Set C := 22/ ln 2. Let ν be a convex probability measure on the space Rk and f be a
polynomial of degree at most d on Rk. Then, for all p ∈ [1,∞), the following inequality
holds:
‖f‖Lp(ν) ≤ pCd‖f‖L1(ν). (1)
In particular, on the space Pd(Rk) of all polynomials of degree at most d on Rk, all
Lp(ν)-norms are equivalent with constants which are independent of k and depend only
on d and p.
2000 AMS Mathematics Subject Classification. Primary 28C20, 60B05.
Key words and phrases. Convex measure, measurable polynomial, equivalent norms.
This article was partially supported by the RFBR project 07-01-00536.
7
8 VASILIY BEREZHNOY
Suppose we are given a sequence (Xk, Bk, μk), k ∈ N , of probability spaces. The
measure μ = ⊗∞
k=1μk is called a product-measure, where we deal with the product of the
measures μk defined on the product of the spaces X =
∏∞
k=1Xk which is equipped with
the σ-algebra B :=
⊗∞
k=1 Bk.
Let X be a locally convex space, and let Pd,fin(X) be the class of all finite-dimensional
polynomials on X of the form
f(x) = P (l1(x), . . . , lk(x)),
where P is a polynomial of degree at most d on Rk and l1, . . . , lk are continuous linear
functionals on X . Let ν be a probability measure on σ(X). Let us denote, by Pd(ν), the
closure of the set Pd,fin(X) in the space of all ν-measurable functions with respect to a
metric corresponding to the convergence in measure ν; e.g., one can take the metric
(f, g) :=
∫
X
|f − g|
1 + |f − g| dμ.
Lemma. Let μ be a convex probability measure on a locally convex space X. Then the
following assertions are true.
(i) For every p ∈ [1,∞), one has Pd(μ) ⊂ Lp(μ).
(ii) On the space Pd(μ), all norms from all spaces Lp(μ), where p ∈ [1,+∞), are
equivalent and Pd(μ) is complete with respect to each of these norms.
(iii) If a sequence {fj} ⊂ Pd(μ) converges in measure μ, then it converges in every
Lp(μ), p ∈ [1,+∞).
Proof. It is known that, in the finite-dimensional case, every convex measure has all
moments (see [3]). So Pd,fin(X) ⊂ Lp(μ) for all p < ∞. Suppose that a sequence of
polynomials ϕj ∈ Pd,fin(X) converges in measure to ϕ. Due to the above-mentioned
result of Bobkov, we have the estimates
‖ψ‖Lp(μ) ≤ C(d, p)‖ψ‖L1(μ), ψ ∈ Pd,fin(X), (2)
where the number C(d, p) depends only on d and p. In particular, we have these estimates
for p = 2 and ψ = ϕj . According to Example 2.8.10 in [4], the norms ‖ϕj‖L1(μ) are
uniformly bounded. Indeed, otherwise passing to a subsequence, we may assume that
{ϕj} converges almost everywhere. Then the aforementioned example applies. The
boundedness in Lp(μ) along with the convergence in measure yield the convergence in
Lr(μ) for r < p. Since this is true for all p < ∞, the sequence {ϕj} converges to ϕ
in all Lp(μ). Thus, we obtain not only the inclusion ϕ ∈ Lp(μ) but also estimate (2)
for all ψ ∈ Pd(μ). If we apply the same reasoning to the whole class Pd(μ), we obtain
all assertions of the lemma. In particular, the equivalence of all Lp-norms follows from
(2) and the inequality ‖f‖L1(μ) ≤ ‖f‖Lp(μ). The completeness of Pd(μ) with respect to
Lp-norms follows from what has been said.
Theorem. Suppose we are given a sequence of finite-dimensional spaces Xk = Rnk
equipped with their Borel σ-algebras Bk. For every k, let μk be a convex probability
measure on Xk. Let us consider the space X =
∏∞
k=1Xk equipped with the product-
measure μ =
⊗∞
k=1 μk. Let us fix a set M � X with μ(M) > 0 and a positive integer d.
Then, the following assertions are true.
(i) If a sequence of functions in Pd(μ) converges in the measure μ on M , then it
converges in measure μ on all of X and in all Lp(μ), p <∞, too.
(ii) For every p ∈ [1,+∞), the norm of Lp(μ) on the space Pd(μ) is equivalent to the
norm of Lp(μ|M ). Therefore, whenever 1 ≤ p, q < ∞, the norm of Lp(μ) on Pd(μ) is
equivalent to the norm of Lq(μ|M ).
EQUIVALENCE OF INTEGRAL NORMS 9
Proof. Let us introduce the following two norms on the space Pd(μ):
‖f‖1 :=
(∫
M
|f |p μ(dt)
)1/p
, ‖f‖2 :=
(∫
X
|f |p μ(dt)
)1/p
.
We have seen that the space Pd(μ) is a Banach one with respect to the norm ‖ · ‖2. Let
us show that the space Pd(μ) is a Banach one with respect to the norm ‖ · ‖1 as well.
Then assertion (ii) will follow by Banach’s theorem on equivalent norms. In addition,
we will show that the convergence of a sequence from Pd(μ) in measure μ on the whole
space X follows from its convergence in measure μ on M , which will yield assertion (i)
by the lemma.
So far it is not even obvious that ‖ · ‖1 is not only a semi-norm but a norm (i.e., it
is not obvious that if a function from Pd(μ) vanishes almost everywhere on M , then it
vanishes almost everywhere on X).
Let a sequence {fj} converge in measure μ on M . For proving its convergence in
measure μ on all of X , it is sufficient to check that every subsequence in it contains a
further subsequence convergent almost everywhere on X . Hence, passing to a subse-
quence, we may assume that the sequence {fj} converges almost everywhere on M . For
simplification of notation, we assume that the measures μk are absolutely continuous
(otherwise we could take their affine supports). Furthermore, it is sufficient to consider
polynomials fj from the class Pd,fin(X), because we can replace the initial sequence {fj}
by a sequence of finite-dimensional polynomials with the same limit in measure on M , as
every element in Pd(μ) is the limit of a sequence of finite-dimensional polynomials which
converges in measure (and in all Lp(μ), too).
We aim at proving the convergence of the sequence {fj} almost everywhere on the
whole space X . Then the application of the above lemma will complete our proof. We
apply a modification of the reasoning from [2] and [6] (see §5.10 in [6]).
Set
E :=
{
x ∈ X : ∃ lim
j→∞
fj(x)
}
.
Then E ∈ B and μ(E) > 0 since M ⊂ E. In order to prove the equality μ(E) = 1,
we apply Kolmogorov’s zero-one law. To this end, as is known (see Theorem 10.10.17
in [4]), it is sufficient to satisfy the following condition: if x ∈ E, then y ∈ E for every
y ∈ X with yk = xk for all sufficiently large k. We shall achieve this condition on some
subset E1 of the set E such that E1 is also of positive measure. Since we assume that
all measures μk are absolutely continuous, for every fixed n, due to Fubini’s theorem, for
almost every x ∈M , the section
Mz
n :=
{
z ∈ Xn: (x1, . . . , xn−1, z, xn+1, . . . ) ∈M
}
has a positive Lebesgue measure in Xn. This implies that almost every point in M has
this property for all n ∈ N . Hence, the measurable set
E1 :=
{
x ∈ E: λn(Ex
n) > 0 ∀n ∈ N
}
has a positive μ measure, where λn is the Lebesgue measure on Xn and
Ez
n :=
{
z ∈ Xn: (x1, . . . , xn−1, z, xn+1, . . . ) ∈ E
}
.
If a sequence of polynomials of degree d on Xn converges on a set of positive Lebesgue
measures, then it converges at every point in Xn. Therefore, for every x ∈ E1, the section
Ex
n coincides with the whole space Xn for every n. Thus, if x ∈ E1, then x + u ∈ E1
for every u of the form u = (u1, . . . , un, 0, 0, . . . ). Due to the zero-one law, one has
10 VASILIY BEREZHNOY
μ(E1) = 1, whence it follows that μ(E) = 1. Hence, {fj} converges almost everywhere
on all of X . Along with the lemma, this proves assertion (i).
Now we can easily complete the proof of assertion (ii). Suppose we are given a sequence
{fj} ⊂ Pd(μ) that is fundamental with respect to the Lp(μ|M )-norm. It converges on M
in measure μ. Hence, as shown above, it converges in Lp(μ) to some function g ∈ Pd(μ).
Clearly, the sequence {fj} converges to g in Lp(μ|M ) too. The proof is completed.
Remark. It would be interesting to extend this theorem to more general cases of convex
measures.
The author thanks V.I. Bogachev for his help and support.
Bibliography
1. A.A. Dorogovtsev, Measurable functionals and finitely absolutely continuous measures on Ba-
nach spaces, Ukranian Math. J. 52 (2000), no. 9, 1194–1204.
2. V.E. Berezhnoy, On the equivalence of norms on the space of γ-measurable polynomials, Moscow
Univ. Bulletin. Ser. Math. (2000), no. 4, 54–56.
3. C. Borell, Convex measures on locally convex spaces, Ark. Math. 12 (1974), 239–252.
4. V.I. Bogachev, Measure theory. V. 1,2, Springer, Berlin, 2007.
5. S.G. Bobkov, Remarks on the growth of Lp-norms of polynomials, Lecture Notes in Math. 1745
(2000), 27–35.
4. V.I. Bogachev, Gaussian measures, Amer. Math. Soc., Providence, Rhode Island, 1998.
-���
�� � �� � ����� $�
%���
��� ����+ ������
�� ��� ��������
��� ���
�
��� ��
.����
�� ��� .����
��� ������
�+
E-mail : waber msu@mail.ru
|