An. Şt. Univ. Ovidius Constanţa
Vol. 19(1), 2011, 145–160
A RADON-NIKODYM TYPE THEOREM
Christiane Godet-Thobie and Anca Croitoru
Abstract
In [6] and [7], the authors introduced and studied an integral for
multifunctions with respect to a multimeasure which contains different multivalued integrals as particular cases. If Pk (X) is the family
of nonempty compact subsets of a locally convex algebra X, both the
e of Pk (X)
multifunction and the multimeasure take values in a subset X
which satisfies certain conditions. In this paper, we continue this work
and establish a Radon-Nikodym theorem, using a method of Maynard
[13] which bases on the notion of exhaustion.
Introduction
The study of multifunctions was intensified and diversified in the last period
thanks to their multiple applications in mathematical economics, theory of
games, optimization and optimal control.
In [6] and [7], we constructed an integration theory for multifunctions with
respect to multimeasures. If Pk (X) is the family of nonempty compact subsets
of a locally convex algebra X, then the multifunctions and the multimeasures
e of Pk (X) which satisfies certain conditions. For diftake values in a subset X
ferent choices of the space X, of the multifunctions and of the multimeasures,
this set-valued integral contains, like particular cases, the classical integrals
of Dunford [10], Brooks [3] and the integrals introduced in Sambucini [14],
Croitoru [4].
Key Words: Radon-Nikodym theorem, multimeasure, integrable multifunction, setvalued integral.
Mathematics Subject Classification: 28B20
Received: January, 2010
Accepted: December, 2010
145
146
Christiane Godet-Thobie and Anca Croitoru
One of the most interesting problems in the theory of integration is the
existence of a Radon-Nikodym derivative. In this paper, we obtain a RadonNikodym type theorem in the context of the integration theory constructed
in [6]. According to this result, we can express a multimeasure Γ like a setvalued integral of a multifunction with respect to a multimeasure ϕ, under
a condition of absolute continuity: Γ ϕ. In this case, the construction of
the Radon-Nikodym derivative follows the method of Maynard [13], using the
notion of exhaustion.
1
Terminology and notations
The terminology and different notations are those of [6] and [7]. Let S be a
nonempty set, A an algebra of subsets of S. Let X be a Hausdorff locally
convex commutative algebra and let Q be a filtering family of seminorms
which defines the topology of X and satisfies the following property for every
x, y ∈ X and every p ∈ Q:
(*)
p(xy) ≤ p(x)p(y).
1.1. Examples
(a) X = {f | f : T → IR} where T is a nonempty set.
Let Q = {pt |t ∈ T } where pt (f ) = |f (t)|, for every f ∈ X
(b) X = {f | f : T → IR is bounded} where T is a topological space.
Let K = {K ⊂ T |K is compact} and Q = {pK |K ∈ K} where, for every
f ∈ X, pK (f ) = sup |f (t)|.
t∈K
(c) X = {f | f : T → IR is continuous } = C(T ) where T is a topological
space. Let K = {K ⊂ T |K is compact} and Q = {pK |K ∈ K} where,
for every f ∈ X, pK (f ) = sup |f (t)|.
t∈K
(d) As a particular case, we may consider X = L1 (R) which is a Banach
algebra with the sum and the convolution as operations.
We denote by Pk (X) = Pk the family of all nonempty compact subsets
of X. For every p ∈ Q and every A, B ∈ Pk , let hp (A, B) be the Hausdorff
semimetric defined by p on Pk analogously to the definition of Hausdorff metric
[12]. We define kAkp = hp (A, O) = sup p(x) where O = {0}. It is known that
x∈A
{hp }p∈Q is a filtering family of semimetrics on Pk which defines a Hausdorff
topology on Pk .
147
A RADON-NIKODYM TYPE THEOREM
For greater convenience of the reader, we now recall some definitions and
properties which be used in the following.
1.2. Definition.
M : A → Pk is said to be an additive multimeasure if:
(i) M (∅) = O,
(ii) M (A ∪ B) = M (A) + M (B), for every A, B ∈ A such that A ∩ B = ∅.
1.3. Examples and applications.
I. If ν1 , ν2 are two finite measures defined on A, so that ν1 ≤ ν2 and ν2 is
a probability measure, then one obtains a particular multimeasure M : A →
P0 ([0, 1]), M (A) = [ν1 (A), ν2 (A)], ∀A ∈ A, which is the simplest example of a
probability multimeasure. We recall that a multimeasure M : A → P0 ([0, 1])
is said to be a probability multimeasure if 1 ∈ M (S). These probability multimeasures are used in control, robotics and decision theory (in Bayesian estimation).
II. We now give an example of such a multimeasure used by Wasserman
[16] in robust Bayesian inference. In this paper, Wasserman generalizes previous works of Shafer [15] and Dempster [8] who defines the upper and lower
probabilities generated by a multifunction. Following [16] p.454-455, let Θ
be a Polish space with Borel σ-algebra B(Θ) and let X be a convex compact
metrizable subset of a locally convex topological vector space with Borel σalgebra B(X). Let µ be a probability measure on (X, B(X)) and let Γ be
a multifunction defined on X with values in Pf (Θ) the family of nonempty
closed subsets of Θ. For each A ⊂ Θ, we denote
A∗ = {x ∈ X; Γ(x) ⊂ A} and A∗ = {x ∈ X; Γ(x) ∩ A 6= ∅}.
Now, if, as in [16], we define on (Θ, B(Θ)) the belief function Bel and the
plausibility function P l by, for any A ∈ B(Θ),
Bel(A) = µ(A∗ ) and P l(A) = µ(A∗ ),
then we can consider the set Π of all probability measures P satisfying, for
any A ∈ B(Θ)), Bel(A) ≤ P (A) ≤ P l(A). It can be shown that Π is non
empty and that, for every A ∈ B(Θ),
Bel(A) = inf P (A) and P l(A) = sup P (A).
P ∈Π
P ∈Π
So, Bel and P l may be thought as the lower and upper bounds of the family of
the selections measures of the multimeasure M such that, for every A ∈ B(Θ),
M (A) = [Bel(A), P l(A)].
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Christiane Godet-Thobie and Anca Croitoru
The next definition is a natural extension of the concept of the variation
of a vector measure [10].
1.4. Definition.
Let M : A → Pk . For every p ∈ Q, the p-variation of M is the nonnegative (possibly infinite) set function vp (M, ·) defined on A as follows:
( n
X
vp (M, A) = sup
kM (Ei )kp (Ei )ni=1 ⊂ A, Ei ∩ Ej = ∅ for i 6= j,
i=1
n
[
)
Ei = A, n ∈ IN
∗
, ∀A ∈ A.
i=1
If M is an additive multimeasure, then vp (M, ·) is finitely additive for every
p ∈ Q. We say that M is with bounded p−variation iff vp (M, ·) is bounded
for every p ∈ Q.
In the sequel, multimeasures and multifunctions take their values in a subset
e of Pk satisfying the conditions:
X
e is complete with respect to {hp }p∈Q ,
•
X
e
•
O ∈ X,
e for all A, B ∈ X,
e
•
A + B, A · B ∈ X,
e
•
A · (B + C) = A · B + A · C, for all A, B, C ∈ X.
1.5. Examples
e = {{f }|f ∈ X} for X like in Example 1.1, (a) and (b).
(a) X
e = {A | A ⊂ [0, +∞[, A is nonempty compact convex} for X = IR.
(b) X
e = {[f, g] | f, g ∈ X, 0 ≤ f ≤ g} for X like in 1.1-(a), where [f, g] =
(c) X
{u ∈ X | f ≤ u ≤ g} = {u ∈ X |f (t) ≤ u(t) ≤ g(t), for every t ∈ T } and
[f, f ] = {f }.
e is the family of nonempty compact subsets of X like in Example 1.1
(c) X
(c).
e is an additive multimeasure
In the sequel, we also suppose that ϕ : A → X
such that its p−variation vp (ϕ, ·), denoted by νp , is bounded and there exists
at least one p ∈ Q such that (S, A, νp ) is complete (cf. [10]-III).
149
A RADON-NIKODYM TYPE THEOREM
1.6. Definition
e is said to be absolutely continuous with
A multimeasure Γ : A → X
respect to the multimeasure ϕ if, for every p ∈ Q and ε > 0, there exists
δ(p, ε) = δ > 0 such that for every E ∈ A,
νp (E) < δ ⇒ vp (Γ, E) < ε
that is denoted: Γ ϕ.
Now we recall some integral notions already used in [6] and [7].
n
X
e is the simple multifunction F =
e
If F : S → X
Bi · XAi , where Bi ∈ X,
i=1
n
[
Ai ∈ A, i ∈ {1, 2, ..., n}, Ai ∩ Aj = ∅ for i 6= j,
Ai = S and XAi is the
i=1
characteristic function of Ai , the integral of F over E ∈ A with respect to ϕ
is:
Z
n
X
e
F dϕ =
Bi · ϕ(Ai ∩ E) ∈ X.
E
i=1
1.7. Definition (Definition 2.2 of [6])
e is called ϕ-totally measurable in semiA multifunction F : S → X
norm if for every p ∈ Q, there is a sequence (Fnp )n of simple multifunctions
νp
e such that hp (Fnp , F ) −→
0.
Fnp : S → X
1.8. Definition (Definition 2.3 of [6])
e is called ϕ−integrable in seminorm if, for
A multifunction F : S → X
every p ∈ Q, there exists a sequence (Fnp )n of simple multifunctions, Fnp : S →
e satisfying the following conditions:
X,
νp
(i) hp (Fnp , F ) −→ 0 (that is: F is ϕ-totally measurable in seminorm),
(ii) hp (Fnp , F ) is νp −integrable, for every n ∈ IN,
Z
(iii) lim
hp (Fnp , F )dνp = 0, for every E ∈ A,
n→∞ E
e such that, for every p ∈ Q,
(iv) For every
exists IE ∈ X
Z E ∈ A, there
lim hp
Fnp dϕ, IE = 0.
n→∞
E Z
We denote IE = F dϕ and call it the integral of F on E with respect
E
to ϕ. The sequence (Fnp )n is said to be a p−defining sequence for F .
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Christiane Godet-Thobie and Anca Croitoru
1.9. Remark (Connections with previous integrals)
I. In the above definition of ϕ-integrability in seminorm, the p-defining
sequence depends on the seminorm. This setting is weaker and differs from
that of [5], where the defining sequence is independent of the seminorm.
II. In [6] and [7], we have shown that this integral contains different classical
integrals or multivalued integrals ( [1], [2], [3], [4], [10] and [14]) and has some
of the classical properties of an integral.
For examples:
(a) If X is a real Banach algebra, then we obtain the integral defined in [4].
e = {A|A ⊂ [0, +∞), A is a nonvoid compact convex set}
(b) If X = R, X
and ϕ = {µ} (where µ is finitely additive), then we obtain the integral
(defined in [14]) of the multifunction F with respect to µ.
The next convergence theorem of Vitali type (Theorem 3.1 of [7]) will be
used in the next section.
1.10. Theorem(Vitali)
e be a multifunction and, for every p ∈ Q, (Fnp )n∈IN∗ be
Let F : S → X
e . We
a sequence of ϕ−integrable in seminorm multifunctions Fnp : S Z→ X
denote, for every E ∈ A, every n ∈ IN∗ and every p ∈ Q, Γpn (E) = kFnp kp dνp
and, for every p ∈ Q, we suppose the following conditions:
E
νp
(i) hp (Fnp , F ) −→ 0,
(ii) Γpn νp , uniformly in n ∈ IN∗ (i.e. for every p ∈ Q and ε > 0, there is
δ(p, ε) = δ > 0 such that Γpn (E) < ε for all E ∈ A with νp (E) < δ and
for every n ∈ IN∗ ).
ThenZthe multifunction
F is ϕ−integrable in seminorm and, for every E ∈
Z
p
A, lim
Fn dϕ = F dϕ.
n→∞ E
2
E
Radon-Nikodym type theorem
In this section, the approach to be used in obtaining a Radon-Nikodym theorem will be analogous to that of Maynard in [13]. We begin by recalling some
definitions.
In the sequel, we denote A+ = {E ∈ A | νp (E) > 0, ∀p ∈ Q}.
151
A RADON-NIKODYM TYPE THEOREM
2.1. Definition
(i) A finite or countable family of pairwise disjoint sets (Ei )i ⊂ A+ will be
called an uniform
exhaustion
of S if for every ε > 0, there exists n0 ∈ N
n
S0
Ei < ε for every p ∈ Q.
such that νp S\
i=1
(ii) A set property P is said to be uniformly exhaustive on E ∈ A if there
exists an uniform exhaustion (Ei )i of E such that every Ei has P .
2.2. Definition
A set property P is called ”uniform null difference” if whenever A, B ∈
A+ , from νp (A4B) = 0 for every p ∈ Q, it follows that either A and B both
have P or neither does.
2.3. Theorem
Let P be an uniform null difference property such that P is uniformly
exhaustive on S. Then there exists I ⊂ N∗Sand (Ei )i∈I an uniform exhaustion
of S, such that every Ei has P and S =
Ei .
i∈I
Proof
Since P is uniformly exhaustive on S, there exists I ⊂ N∗ and (Ai )i∈I an
uniform exhaustion of S, such that every Ai has P. Thus, we have:
(1)
∀ε > 0, ∃n0 (ε) = n0 ∈ N∗ such that νp (S\
n0
[
Ai ) < ε, ∀p ∈ Q.
i=1
Let A0 = S\
S
Ai . By the inclusion A0 ⊂ S\
n
S0
Ai and from (1), it results
i=1
i∈I
that νp∗ (A0 ) < ε, ∀ε > 0 (where νp∗ (A0 ) = inf{νp (C)|A0 ⊂ C, C ∈ A}, cf.
Dunford and Schwartz [10]-III). So νp∗ (A0 ) = 0. If there is q ∈ Q such that
(S, A, νq ) is complete, from νq∗ (A0 ) = 0, it follows that A0 ∈ A and, for every
p ∈ Q, νp (A0 ) = 0.
Let (Ei )i∈I be the family of sets defined by: E1 = A0 ∪ A1 ∈ A and for i ≥ 2,
Ei = Ai ∈ A. We have, for every p ∈ Q, for i = 1, νSp (E1 ) ≥ νp (A1 ) > 0 and
for every i ≥ 2, νp (Ei )=νp (Ai ) > 0. Evidently, S =
Ei .
i∈I
Let ε > 0. For n0 of (1) we have
By the inclusion S\
n
S0
Ei ⊂ S\
i=1
νp S\
n
S0
E i = A0 ∪
i=1
n
S0
n
S0
Ai .
i=1
Ai and from (1), it follows:
i=1
n0
[
i=1
Ei
[
n0
≤ νp S\
Ai < ε, ∀p ∈ Q
i=1
152
Christiane Godet-Thobie and Anca Croitoru
which assures that (Ei )i∈I is an uniformly exhaustion of S. Now, for every
i ≥ 2, Ei = Ai has P . So, we have only to prove that E1 has P. By the
relations:
E1 4A1 = (A0 ∪ A1 )4A1 = A0 \A1 ⊂ A0
it follows
νp (E1 4A1 ) ≤ νp (A0 ) = 0, ∀p ∈ Q
and νp (E1 4A1 ) = 0, for every p ∈ Q. Since P is uniform null difference and
A1 has P , we obtain that E1 has P .
Now, we give two properties of a set-valued integral Γ, properties which will
be of use in the next Radon-Nikodym theorem.
2.4. Theorem
e be a ϕ-integrable in seminorm bounded multifunction and
Let FR : S → X
Γ(E) = E F dϕ, E ∈ A ( Γ is a multimeasure according to Theorem 2.8-(a)
of [6]). Then we have:
(i) Γ ϕ;
(ii) for every p ∈ Q, there exists rp > 0 such that, for every E ∈ A with
νp (E) > 0, kΓ(E)kp ≤ rp νp (E).
Proof
(i) It follows from Theorem 2.8-(c) of [6].
(ii) Since the boundedness of F , for every p ∈ Q, there exists rp > 0 such that:
(2)
kF (s)kp ≤ rp , ∀s ∈ S.
From (2) and Theorem 2.7-(b) of [6], for each E ∈ A with νp (E) > 0, we have:
Z
Z
≤
kF kp dνp ≤ rp νp (E).
F
dϕ
kΓ(E)kp = E
p
E
The next sentence follows from classical properties of the variation of a
vector measure ([9], [10]).
2.5. Proposition
If Γ is a multimeasure which satisfy condition (ii) of Theorem 2.4, then Γ
is a multimeasure with bounded p−variation.
We now give a definition of approximate average ranges which is adapted from
that of [13] for set-valued case.
A RADON-NIKODYM TYPE THEOREM
153
2.6. Definition
e p ∈ Q, ε > 0 and E ∈ A, let:
For a multifunction Γ : A → X,
e | hp (Γ(B), νp (B) · C) ≤ ενp (B), ∀B ∈ A, B ⊂ E},
Dp (Γ, E, ε) = {C ∈ X
e p (Γ, E, ε) = {C ∈ X
e | hp (Γ(B), ϕ(B) · C) ≤ ενp (B), ∀B ∈ A, B ⊂ E}.
D
The next proposition give examples of uniform null difference properties which
will used in this paper. Its demonstration is adapted from that of Theorem
3.6 of [5].
2.7. Proposition
Let Γ be an additive multimeasure with bounded p−variation which is absolutely continuous with respect to ϕ. Then,
(i) Dp (Γ, E, γ) 6= ∅,
e p (Γ, E, γ) 6= ∅ and
(ii) D
e p (Γ, E, γ) 6= ∅
(iii) Dp (Γ, E, γ) ∩ D
are uniform null difference properties.
Proof
It is clear it is sufficient to prove (i) and (ii).
(i) Since Γ ϕ, for every p ∈ Q and every ε > 0, there exists δ(p, ε) = δ > 0
such that, for every E ∈ A such that νp (E) < δ, kΓ(E)kp ≤ vp (Γ, E) < ε.
Now, we have to show that, if A and B ∈ A+ such that νp (A4B) = 0 for
every p ∈ Q , Dp (Γ, A, γ) = Dp (Γ, B, γ) for each p ∈ Q. Now we fix one p ∈ Q
and consider C ∈ Dp (Γ, A, γ).
e and, for every B ∈ A, B ⊂ A, hp (Γ(B), νp (B)C) ≤ γνp (B).
C∈X
Let H ∈ A, H ⊂ B. Since νp (A4B) = 0, νp (H\A) ≤ νp (B\A) = 0 and
νp (H) = νp (H ∩ A) + νp (H\A) ≤ νp (H ∩ A) + νp (B\A) = νp (H ∩ A) ≤ νp (H).
So, νp (H) = νp (H ∩ A) for every H ∈ A, H ⊂ B.
From H ∩ A ⊂ A, it follows hp (Γ(A ∩ H), νp (A ∩ H) · C) ≤ γνp (A ∩ H). Since
νp (H\A) = 0 < δ, kΓ(H\A)kp < ε and this for all ε. So, kΓ(H\A)kp = 0.
Now, we can write:
hp (Γ(H), νp (H) · C) = hp (Γ(H ∩ A) + Γ(H\A), (νp (H ∩ A) + νp (H\A)) · C) ≤
≤ hp (Γ(H ∩ A), νp (H ∩ A) · C) + kΓ(H\A)kp ≤ γνp (H ∩ A) = γνp (H). So
for every H ∈ A, H ⊂ B, we have hp (Γ(H), νp (H) · C) ≤ γνp (H). That is:
C ∈ Dp (Γ, B, γ). The inclusion Dp (Γ, B, γ) ⊂ Dp (Γ, A, γ) is similarly proved
by exchange of A and B. So, the first part ” Dp (Γ, E, γ) 6= ∅ is an uniform
null difference property” is proved.
(ii) For the second property, with same notations as in the first part, if H ∈
A, H ⊂ B, since, for every p ∈ Q, νp (A4B) = 0, it follows from kϕ(H\A)kp ≤
154
Christiane Godet-Thobie and Anca Croitoru
νp (H\A) = 0, as in the first part, that kΓ(H\A)kp = 0 and hp ( Γ(H), ϕ(H) ·
C ) = hp ( Γ( H ∩ A ), ϕ(H ∩ A) · C ) + kΓ( H\A )kp ≤
≤ γνp (H ∩ A) = γνp (H). The proof is finished as in the first part.
2.8. Theorem
e be a ϕ-integrable in seminorm bounded multifunction and
Let FR : S → X
Γ(E) = E F dϕ, E ∈ A. Then,
∀p ∈ Q, ∀ε > 0 and ∀E ∈ A such that νp (E) > 0, there exists B ∈ A, B ⊂ E
e p (Γ, B, ε) 6= ∅.
νp (B) > 0 such that D
Proof
Since F is ϕ−integrable in seminorm, for every p ∈ Q, there exists (Fnp )n a
e So,
p−defining sequence of simple multifunctions Fnp : S → X.
νp
(i) hp (Fnp , F ) −→ 0,
(ii) hp (Fnp , F ) is νp −integrable, for every n ∈ IN,
Z
(iii) lim
hp (Fnp , F )dνp = 0, for every E ∈ A.
n→∞
E
Thanks to (i), there exists a strictly increasing sequence (nk )k∈N∗ ⊂ N such
that νp ({s ∈ S | hp (Fnpk (s), F (s)) > 21k }) ≤ 21k . Let Apk = {s ∈ S | hp (Fnpk (s), F (s)) >
1
}. If we denote Gpk = Fnpk X{Apk , Gpk is a simple function and for every
2k
k ∈ N∗ , hp (Gpk , F ) is νp -measurable. It is easy to see that for every ε > 0
and every k ∈ N∗ , νp ({s ∈ S|hp (Gpk (s), F (s)) > ε}) ≤ νp (Apk ) + νp ({s ∈
νp
S|hp (Fnpk (s), F (s)) > ε}). So hp (Gpk , F ) −→ 0. And, with notations of Theorem
Z 2.4,
Z
Z
1
hp (Gpk , F )dνp =
kF kp dνp +
hp (Gpk , F )dνp ≤ rp νp (Apk ) + k νp (S).
p
p
2
S
Ak
{Ak
Z
Z
p
hp (Gpk , F )dνp = 0, ∀E ∈ A.
hp (Gk , F )dνp = 0 and lim
Then, lim
k→∞
k→∞
S
E
Since νp (E) > 0 and lim νp (Apk ) = 0, lim νp (E ∩ {Apk ) = νp (E). So, there
k→∞
k→∞
exists k0 such that, for every k ≥ k0 , νp (E ∩ {Apk ) > 0.
l
l
X
X
If Gpk =
Ci XAi , νp (E ∩ {Apk ) =
νp (E ∩ {Apk ∩ Ai ) and there exists at
i=1
i=1
least one i = i0 (k) = i0 such that νp (E ∩ {Apk ∩ Ai0 ) > 0.
Denoting B = E ∩ {Apk ∩ ARi0 , for every
R H ∈ A, HR⊂ B, we have
hp (Γ(H), ϕ(H) · Ci0 ) = hp ( H F dϕ, H Gpk dϕ) ≤ H hp (Gpk , F )dνp ≤ 21k νp (H)
e p (Γ, B, 1k ). And, since, for every ε > 0, there exists k ≥ k0
That is: Ci0 ∈ D
2
e p (Γ, B, ε).
such that 21k ≤ ε, we can conclude that Ci0 ∈ D
155
A RADON-NIKODYM TYPE THEOREM
2.9. Remark
e be a bounded multimeasure such that:
Let Γ : A → X
(3)
e p (Γ, E, ε) 6= ∅
for every ε > 0 and every p ∈ Q, Dp (Γ, E, ε) ∩ D
is an uniformly exhaustive property on every E ∈ A+ .
Since (3) and Theorem 2.3, for every p ∈ Q and εS> 0, there is (Eip,ε )i an
uniform exhaustion of each E ∈ A+ , such that E = Eip,ε and:
i
e p (Γ, E p,ε , ε) 6= ∅, ∀i.
Dp (Γ, Eip,ε , ε) ∩ D
i
By induction, following the same way as in Hagood [11], we can obtain a
sequence Eαp,n (n ∈ N, α ∈ Nn ) of uniform exhaustions of S such that:
(4)
(5)
(6)
e p (Γ, Eαp,n , 2−n ) 6= ∅, ∀α ∈ Nn , n ∈ N,
Dp (Γ, Eαp,n , 2−n ) ∩ D
Eαp,n =
S
p,n
Eα,i
, where
i∈N
Eαp,n , ∀α
S=
[
p,n+1
Eα,i
is an uniform exhaustion of
i
∈ Nn , n ∈ N,
Eαp,n and (Eαp,n )α is an uniform exhaustion of S, ∀n ∈ N.
α
From (6), for ε =
1
n,
there exists k(n) = k ∈ N such that:
(7)
νp
S\
k
[
!
Eip,n
<
i=1
If we consider Gpn =
k
P
i=1
Cip,n · XEip,n + O · X
S\
1
.
n
k
S
i=1
Eip,n
, where
e p (Γ, E p,n , 2−n ), Gpn is simple and therefore ϕCip,n ∈ Dp (Γ, Eip,n , 2−n ) ∩ D
i
integrable in seminorm. The sequence (Gpn )n is called associate to Γ.
2.10. Theorem(Radon-Nikodym)
e be a multimeasure satisfying the three following conditions:
Let Γ : A → X
(i) Γ ϕ,
(ii) for every p ∈ Q, there exists rp > 0 such that kΓ(E)kp ≤ rp νp (E), for
every E ∈ A with νp (E) > 0,
156
Christiane Godet-Thobie and Anca Croitoru
e p (Γ, E, ε)6=∅”
(iii) for every p ∈ Q and ε > 0, the property ”Dp (Γ, E, ε) ∩ D
is an uniform exhaustive property on every E ∈ A+ , and the sequence (Gpn )n ,
associate to Γ (see Remark 2.9), is convergent in νp -measure.
e such
Then there exists
a ϕ-integrable in seminorm multifunction F : S → X,
R
that Γ(E) = E F dϕ, for every E ∈ A.
Proof
Thanks to Propositions 2.7 and 2.5, Γ has a bounded p-variation and, for every
e p (Γ, E, ε) 6= ∅”
p ∈ Q and ε > 0, the property ”Dp (Γ, E, ε) ∩ D
is an uniform null difference property on every E ∈ A+ . So, we can use
Theorem2.3.
According to Remark 2.9, the sequence (Gpn )n , associate to Γ, is given by:
Gpn =
k
X
Cip,n · XEip,n + O · X
k
S
S\
i=1
i=1
Eip,n
,
where
e p (Γ, E p,n , 2−n ).
Cip,n ∈ Dp (Γ, Eip,n , 2−n ) ∩ D
i
(8)
e such that:
Since (iii), there exists a multifunction F : S → X
νp
hp (Gpn , F ) −→ 0.
(9)
From (8) and (ii), it results:
1
hp (νp (Eip,n )Cip,n , O) ≤
νp (Eip,n )
1
1
p,n
p,n
p,n
≤
kΓ(Eip,n )kp ≤ 2−n + rp .
p,n hp (νp (Ei )Ci , Γ(Ei )) +
νp (Ei )
νp (Eip,n )
kCip,n kp = hp (Cip,n , O) =
(10)
Since (10), we have for every E ∈ A:
Z
kGpn kp dνp =
Z X
k
E
(11)
E i=1
≤
k
X
i=1
−n
(2
kCip,n kp XEip,n dνp =
k Z
X
i=1
Z
+ rp )
E
E
XEip,n dνp = (2−n + rp )
kCip,n kp XEip,n dνp ≤
k
X
i=1
= (2−n + rp )νp (E).
νp (E ∩ Eip,n ) =
157
A RADON-NIKODYM TYPE THEOREM
If we consider now δ(p, ε) = δ = 2−nε+rp > 0, then for every E ∈ A with
R
νp (E) < δ, using (11), we obtain E kGpn kp dνp < ε. So, we have:
for every p ∈ Q and ε > 0, there exists δ(p, ε) = δ >
R 0
such that for every E ∈ A with νp (E) < δ, it follows E kGpn kp dνp < ε.
From (9) and (12), using Theorem 1.10 (Vitali), it results that F is ϕ-integrable
in seminorm and:
Z
Z
(13)
lim
Gpn dϕ =
F dϕ, ∀E ∈ A.
(12)
n→∞
E
E
Now we prove that Γ(E) = E F dϕ, ∀E ∈ A. Let us fix ε > 0, p ∈ Q and let
δ(p, 3ε ) = δ > 0 given from (i). According to (13), let n ∈ N∗ such that n1 < δ
and:
Z
Z
ε
(14)
hp ( Gpn dϕ,
F dϕ) < .
3
E
E
R
Then we have:
Z
F dϕ) ≤ hp
hp (Γ(E),
Γ(E), Γ(
E
+hp
Γ(
k
[
+ hp
|
+
!
(E ∩
(E ∩
+
Eip,n )),
k
X
!
Cip,n
· ϕ(E ∩
Eip,n )
+
i=1
Gpn dϕ,
E
{z
k
[
Gdϕ < kΓ(E\ (E ∩ Eip,n ))kp +
E
i=1
}
Z
< 3ε , cf.(14)
k
X
hp (Γ(E ∩ Eip,n ), Cip,n · ϕ(E ∩ Eip,n )) +
i=1
≤ v(Γ, E\
k
[
(E ∩ Eip,n )) +2−n
{z
k
X
}
cf. (i) and (7)
2ε
2ε
<
+ 2−n νp (E) ≤
+ 2−n νp (S) < ε,
3
3
R
which shows that Γ(E) = E F dϕ, ∀E ∈ A.
< 3ε ,
ε
≤
3
νp (E ∩ Eip,n ) +
i=1
i=1
|
Eip,n ))
i=1
i=1
Z
k
[
ε
<
3
158
Christiane Godet-Thobie and Anca Croitoru
2.11. Remark
In the previous Theorem of Radon-Nikodym type (in [5]) the first three
conditions fulfilled by Γ are the following:
(a) Γ is uniformly bounded;
(b) Γ νp , uniformly in p ∈ Q;
(c) there exists r > 0 such that kΓ(E)k ≤ rνp (E), for every E ∈ A with
νp (E) > 0 and for every p ∈ Q.
As we may observe, Theorem 2.10 applies to a wider class of multimeasures
Γ, that are not necessarily uniformly bounded.
Acknowledgements. The authors thank the referees for their valuable
suggestions in the improvement of this paper.
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A RADON-NIKODYM TYPE THEOREM
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Christiane GODET-THOBIE
Université de Bretagne Occidentale,
Laboratoire de Mathématiques-UMR 6205,
6, avenue Victor Le Gorgeu - CS 93837
29238 - BREST CEDEX 3, FRANCE
christiane.godet-thobie@univ-brest.fr
Anca CROITORU
Faculty of Mathematics, University “Al.I.Cuza”,
Carol I Street, No. 11, Iaşi 700506, ROMANIA
croitoru@uaic.ro
160
Christiane Godet-Thobie and Anca Croitoru