arXiv:1411.6256v4 [math.FA] 24 Jan 2016
Randomized versions of Mazur lemma and
Krein–Šmulian theorem with application to
conditional convex risk measures for portfolio
vectors
José M. Zapata
∗
January 26, 2016
Abstract
0
The theory of locally L -convex modules was introduced as the analytic basis for conditional L0 -convex risk measures. In this paper we
first give some preliminaries of this theory and discuss about two kinds of
countable concatenation properties. Second we extend to this framework
some results from classical convex analysis, namely we provide randomized versions of Mazur lemma and Krein-Šmulian theorem. Third, as
application, we stablish a representation theorem for conditional convex
risk measures for portfolio vectors.
Keywords: locally L0 -convex module; countable concatenation property;
Mazur lemma; Krein-Šmulian theorem; conditional convex risk measure for
portfolio vectors
Introduction
The study of risk measures was initiated by Artzner et al. [1], by defining and
studying the concept of coherent risk measure. Föllmer and Schied [11] and,
independently, Frittelli and Gianin [13] introduced later the more general concept of convex risk measure. Both kinds of risk measures are defined in a static
setting, in which only two instants of time matter, today 0 and tomorrow T ,
and the analytic framework used is the classical convex analysis, which perfectly
applies in this simple model cf.[4, 12].
However, when it is addressed a multiperiod setting, in which intermediate
times 0 < t < T are considered, it appears to become quite delicate to apply
convex analysis, as Filipovic et al. [9] explained.
In order to overcome these difficulties Filipovic et al. [9] proposed to consider a modular framework, where scalars are random variables instead of real
numbers. Namely, they considered modules over L0 (Ω, F, P) the ordered ring of
(equivalence classes of) random variables. For this purpose, they established the
∗ Universidad de Murcia, Dpto. Matemáticas, 30100 Espinardo, Murcia, Spain, e-mail:
[email protected]
1
2
concept of locally L0 -convex module and proved randomized versions of some
important theorems from convex analysis.
In this regard, randomly normed L0 -modules have been used as tool for
the study of ultrapowers of Lebesgue-Bochner function spaces by R. Haydon
et al. [23]. Further, it must also be highlighted the extensive research done
by T. Guo, who has widely researched theorems from functional analysis under
the structure of L0 -modules; firstly by considering the topology of stochastic
convergence with respect to the L0 -seminorms cf. [14, 15, 16, 18], and later the
locally L0 -convex topology introduced by Filipovic et al. [9] and the connections
between both, cf. [17, 19, 21, 22].
Working with scalars into L0 instead of R implies some difficulties. For example, L0 neither is a field, nor is endowed with a total order and —among
others difficulties— this is why arguments given to prove theorems from functional analysis often fail under the structure of L0 -module. For this reason all
these works often consider additional ’countable concatenation properties’ on
either the algebraic structure or the topological structure cf.[9, 17, 19, 28].
One of the aims of this article is to prove, in the context of L0 -modules,
theorems of functional analysis that are relevant for financial applications. One
of the main theorems of this paper is a randomized version of the classical
Krein-Šmulian theorem for L0 -normed modules. It becomes clear from [4] and
[12], that some of the basic theorems of representation of risk measures or risk
assessments rely on this result. Furthermore, Krein-Šmulian theorem plays a key
role in the fundamental theorem of the asset pricing (see [5]). Hence we believe
that this result may lead to financial applications in dynamic configurations
of time, or situations in where there exists a flow of extra-information in the
market. Likewise, we provide a randomized version of Mazur lemma for L0 normed modules with the locally L0 -convex topology introduced by Filipovic et
al. [9]. An earlier randomized version of Mazur lemma, but in the topology of
stochastic convergence, was proved in [18, Corollary 3.4].
As financial application of the developed theory, we provide a representation
theorem for conditional convex risk measures for portfolio vectors. This type
of risk measure models the specific situation in which additional information is
available in the financial market at some time 0 < t < T and when the risk
manager wants to deal with portfolio vectors X = (X1 , ..., Xd ), measuring the
join risk of X caused by the variation of the components and their possible
dependence. In this regard, static convex risk measures for portfolio vectors has
been studied in [25].
This manuscript starts by giving some preliminaries and collecting some relevant notions and results. We also make some remarks on two kinds of countable
concatenation property. Section 2 is devoted to prove the randomized version
of Mazur lemma and some corollaries. In section 3 we prove the randomized
Krein-Šmulian theorem. Finally, in the last section it is stated and proved
the announced representation theorem for conditional convex risk measures for
portfolio vectors.
3
1
Preliminaries and some remarks on countable
concatenation properties
First and for the convenience of the reader, let us list some notation. Let be
given a probability space (Ω, F, P), let us consider L0 (Ω, F, P), or simply L0 ,
the set of equivalence classes of real valued F-measurable random variables.
It is known that the triple L0 , +, · endowed with the partial order of the
almost sure dominance is a lattice ordered ring. For given X, Y ∈ L0 , we will
write “X ≥ Y “ if P (X ≥ Y ) = 1, and likewise, we will write “X > Y ”, if
P (X > Y ) = 1. And, given A ∈ F, we will write X > Y (respectively, X ≥ Y )
on A, if P (X > Y | A) = 1 (respectively
, if P (X ≥ Y | A)
= 1).
We also define L0+ := Y ∈ L0 ; Y ≥ 0 , L0++ := Y ∈ L0 ; Y > 0 and
F + := {A ∈ F; P(A) > 0}. And we will denote by L¯0 , the set of equivalence classes of F-measurable random variables taking values in R = R ∪ {±∞},
and the partial order of the almost sure dominance is extended to L¯0 in a natural
way.
Furthermore, given a subset H ⊂ L0 , then H owns both an infimum and a
supremum in L¯0 for the order of the almost sure dominance that will be denoted
by ess. inf H and ess. sup H, respectively (see [12, A.5]).
The order of the almost sure dominance
also enables us to define a topology.
0
We define Bε := Y ∈ L0 ; |Y | ≤ ε the ball of radius
ε ∈ L++ centered at
0
0
0
0 ∈ L . Then for all Y ∈ L , UY := Y + Bε ; ε ∈ L++ is a neighborhood base
of Y and a Hausdorff topology on L0 can be defined.
In what follows we will expose more concepts that will be important throughout this paper.
Definition 1.1. We denote by
(
Π (Ω, F) :=
{An }n ⊂ F;
)
[
An = Ω, Ai ∩ Aj = ∅ for all i 6= j
n∈N
the set of countable partitions on Ω to F. Note that we allow Ai = ∅ for some
i ∈ N.
Definition 1.2. Let E be a module over L0 —or L0 -module—, we list the
following notions:
1. For X ∈ E, a sequence {Xn }n∈N in E, and a partition {An }n∈N ∈
Π(Ω, F), we say that X is a countable concatenation of {Xn }n and {An }n
if 1An Xn = 1An X for all n ∈ N.
2. K ⊂ E is said to have the countable concatenation property (with
uniqueness), if for each sequence {Xn }n in K and each partition {An }n ∈
Π(Ω, F), it holds that there exists (an unique) X ∈ E such that X is a
countable concatenation of {Xn }n and {An }n .
3. K ⊂ E is said to have the relative countable concatenation property
(with uniqueness), provided that, if X is a countable concatenation of
{Xn }n and {An }n , then X ∈ K (and any other countable concatenation
equals X).
4
Remark 1.1. It is important to keep in mind both kinds of countable concatenation properties. Clearly the relative countable concatenation property is weaker
than the countable concatenation property. Besides, given a L0 -module E, it
does not have necessarily the countable concatenation property, but nevertheless
it is clear that E has always the relative countable concatenation property (see
Example 1.1).
In addition, if E is a L0 -module that has the countable concatenation property then every subset K ⊂ E that has the relative countable concatenation
property, has the countable concatenation property as well.
Example 1.1. ([9]) Consider the probability space Ω = [0, 1], F = B[0, 1] the
Borel σ-algebra and P the Lebesgue measure on [0, 1] and define
o
n
1
E := spanL0 1[1− n−1
,1− 21n ] ; n ∈ N .
2
0
E is a L -module that does not have the countable concatenation property, but
it has the relative countable concatenation property.
For given a sequence {Xn } ⊂ E and {An } ∈ Π(Ω, F), suppose that
P there is
exactly a countable concatenation of both, then it will be denoted by n∈N 1An Xn .
Fixed {Xn } ⊂ E and {An } ∈ Π(Ω, F), one might wonder whether, provided
that there exists a countable concatenation of them, this is always unique. In
fact this is a question asked by M. Wu and T. Guo in [26]. However the example
below provides a negative answer to this question.
Example 1.2. Let (Ω, F, P) be a probability space and {An } ∈ Π(Ω, F) with
P(An ) 6= 0 for all n ∈ N (for example, Ω = (0, 1), F = B(Ω) the σ-algebra of
1
Borel, An = [ 21n , 2n−1
) with n ∈ N and P the Lebesgue measure). We define in
0
L (Ω, F, P) the following equivalence relation
X ∼ Y if 1An X = 1An Y for all but finitely many n ∈ N.
If we denote by X the equivalence class of X, we can define X + Y := X + Y
and Y · X := Y X, obtaining that the quotient L0 / ∼ is a L0 -module.
Then, for X ∈ L0 , we have that 1An X = 1An X = 0. Hence, any element
of L0 / ∼ is a countable concatenation of {0}n and {An }n , hence there is no
uniqueness.
Definition 1.3. Let us denote by cc ({An }n , {Xn }n ) the set of countable concatenations of {Xn }n ⊂ E and {An }n ∈ Π(Ω, F).
Suppose that K is a subset of a L0 -module E, then we define the countable
concatenation hull of K, given by
[
cc
K :=
cc ({An }n , {Xn }n )
where {An }n runs through Π(Ω, F) and {Xn }n runs through the sequences in
K.
Remark 1.2. In many works the countable concatenation hull of a subset K is
defined assuming that E has the countable concatenation property cf. [17, 19].
Notice that the definition provided here does not require such assumption.
cc
Also note that K is the least set with the relative countable concatenation
property containing K. In particular, K has the relative countable concatenation
cc
property if, and only if, K = K.
5
Let us recall some relevant notions:
Definition 1.4. A topological L0 -module E [τ ] is a L0 -module E endowed with
a topology τ such that
1. E [τ ] × E [τ ] −→ E [τ ] , (X, X 0 ) 7→ X + X 0 and
2. L0 [|·|] × E [τ ] −→ E [τ ] , (Y, X) 7→ Y X
are continuous with the corresponding product topologies.
Definition 1.5. A topology τ on a L0 -module E is locally L0 -convex if there is
a neighborhood base U of 0 ∈ E such that each U ∈ U is
1. L0 -convex, i.e. Y X1 + (1 − Y )X2 ∈ U for all X1 , X2 ∈ U and Y ∈ L0
with 0 ≤ Y ≤ 1,
2. L0 -absorbent, i.e. for all X ∈ E there is Y ∈ L0++ such that X ∈ Y U,
3. L0 -balanced, i.e. Y X ∈ U for all X ∈ U and Y ∈ L0 with |Y | ≤ 1.
4. U has the relative countable concatenation property (i.e, U
cc
= U ).
0
In this case, E [τ ] is said to be a locally L -convex module.
Remark 1.3. The notion of locally L0 -convex module was initially introduced
by Filipovic et al. [9], however they only included items 1, 2 and 3 of the above
definition. We will explain in Remark 1.4 the reason why we have included item
4 in the definition of locally L0 -convex module.
The following result gives a sufficient condition on a locally L0 -convex topology to ensure that there is uniqueness on the countable concatenations.
Proposition 1.1. Let E[τ ] be a locally L0 -convex module, and suppose that τ
is Hausdorff, then for given {An } ∈ Π(Ω, F) and {Xn } ⊂ E, there is exactly a
countable concatenation of them.
Proof. Suppose that U is a neighborhoodTbase of 0 ∈ E as in Definition 1.5.
Fist note that, if τ is Hausdorff, then U ∈U U = {0}. For given X, Y ∈
cc
cc({An }n , {Xn }) we have that X − Z ∈ cc ({An }n , 0) ⊂ U = U for all U ∈ U,
hence X = Z.
It is important to note that the condition of being Hausdorff, is not necessary
for the uniqueness. For example we can think of L0 with the indiscrete topology,
which is a locally L0 -convex module with the countable concatenation property
with uniqueness, but it is not Hausdorff.
Hereafter, we always suppose that all the L0 –modules E involved in this
paper have the property that if X, Y ∈ E are both countable concatenations
of {Xn }n and {An }n then X = Y . In this case we use the notation X =
P
k∈N 1Ak Xk .
This assumption is made only for the sake of convenience. Although this
condition is usually required in the literature, as far as we see, it is rarely
necessary in the arguments. Besides, it is not a great restriction since it is
satisfied under natural hypothesis which are fulfilled by the important examples,
for instance being Hausdorff, as occurs with the L0 -normed modules.
6
Definition 1.6. A function k·k : E → L0+ is a L0 -seminorm on E if:
1. kY Xk = |Y | kXk for all Y ∈ L0 and X ∈ E.
2. kX1 + X2 k ≤ kX1 k + kX2 k , for all X1 , X2 ∈ E.
If moreover, kXk = 0 implies X = 0, then k·k is a L0 -norm on E
Definition 1.7. Let P be a family of L0 -seminorms on a L0 -module E. For
finite Q ⊂ P and ε ∈ L0++ , we define
(
)
UQ,ε :=
X ∈ E; sup kXk ≤ ε .
k.k∈Q
Then for all X ∈ E, UX := X + UQ,ε ; ε ∈ L0++ , Q ⊂ P f inite is a neighborhood base of X. This defines a topology on E, referred to as the topology induced
by P, and E endowed with this topology is denoted by E [P].
Remark 1.4. As we pointed out in Remark 1.3, when Filipovic et al. [9]
introduced the concept of locally L0 -convex module, they only included items 1,
2 and 3 of definition 1.5 omitting 4. In classical convex analysis is well-known
that every locally convex topology can be induced by a family of seminorms.
In fact, Filipovic et al. [9] claim that the topology of every locally L0 -convex
module defined in this way can be induced by a family of L0 -seminorms (see [9],
Theorem 2.4).
However, José M. Zapata [28], and independently M. Wu and T. Guo [26],
provided counterexamples showing that Theorem 2.4 of [9] is wrong, and it is
proved that item 4 must be added in order to obtain an analogue of the classical
characterization theorem. Then, under the new definition 1.5 and according to
[28] we have Theorem 1.1.
Theorem 1.1. A topological L0 -module E[τ ] is locally L0 -convex if, and only
if, τ is induced by a family of L0 -seminorms.
Some relevant notions and results of theory of locally L0 -convex modules,
which will used through out this manuscript, are collected in an appendix at
the end thereof.
2
A randomized version of Mazur lemma
A well-known result of classical convex analysis is the Mazur lemma, which
shows that for any weakly convergent sequence in a Banach space there is a
sequence of convex combinations of its members that converges strongly to the
same limit. We will generalize this tool to nets in a L0 -normed module. For this
purpose, we will consider not only L0 -convex combinations, but also countable
concatenations of members of a given net.
First of all, we will recall the notion of gauge function given in [9]:
Definition 2.1. Let E be a L0 -module. The gauge function pK : E → L̄0+ of a
set K ⊂ E is defined by
pK (X) := ess. inf Y ∈ L0+ ; X ∈ Y K .
7
In addition, among other properties, if K is L0 -convex and L0 -absorbent
then, the essential infimum above can be defined by taking Y ∈ L0++ , also pK is
subadditive and Y pK (X) = pK (Y X) for all X ∈ E and Y ∈ L0+ . If, moreover,
K is L0 -balanced, then pK is a L0 -seminorm (see [9, Proposition 2.23]).
Since L0 is not a totally ordered set, we need to take advantage of the notion
of relative countable concatenation. We have the following result:
Lemma 2.1. Let C ⊂ L0 be bounded below (resp. above) and with the relative
countable concatenation property, then for each ε ∈ L0++ there exists Yε ∈
C such that
ess. inf C ≤ Yε < ess. inf C + ε
(resp., ess. sup C ≥ Yε > ess. sup C − ε)
In particular, given a L0 -module E and K ⊂ E, which is L0 -convex, L0 absorbent and with the relative countable concatenation property, we have
that, for ε ∈ L0++ , there exists Yε ∈ L0++ with X ∈ Yε K such that pK (X) ≤
Yε < pK (X) + ε.
Proof. Firstly, let us see that C is downwards directed. Indeed, given Y, Y 0 ∈ C,
define A := (Y < Y 0 ). Then, since C has the relative countable concatenation
property, 1A Y + 1Ac Y 0 = Y ∧ Y 0 ∈ C.
Therefore, for ε ∈ L0++ , there exists a decreasing sequence {Yk }k in C converging to ess. inf C almost surely.
Let us consider the sequence of sets
A0 := ∅ and Ak := (Yk < ess. inf C + ε) − Ak−1 for k > 0.
P
Then {Ak }k≥0 ∈ Π(Ω, F + ), and we can define Yε := k≥0 1Ak Yk . Given that
C has the relative countable concatenation property, it follows that Yε ∈ C.
For the second part, it suffices
to see that if K
has the relative countable
concatenation property then Y ∈ L0++ ; X ∈ Y K has the relative countable
concatenation property as well. Indeed, given {Ak }k ∈ Π(Ω,
P F) and {Yk }k ⊂
L0++ such that X ∈ Yk K for each k ∈ N, let us take Y := n∈N Yn 1An ∈ L0++ .
Then we have that X/Y ∈ cc ({Ak }k , {X/Yk }k ) and X/Yk ∈ K. Since K has
the relative countable concatenation property, we conclude that X/Y ∈ K, and
the proof is complete.
Remark 2.1. Observe that, in the latter result, we have considered on K the
relative countable concatenation property, rather than the stronger condition of
having the countable concatenation property. This is because Y , and therefore
X/Y , is constructed by using the countable concatenation property of L0 . Once
X/Y exists, we just need to ensure that it belongs to K. But that is the case
due to the relative countable concatenation property of K. The same strategy
was used to prove Proposition 2.3 of [28].
Proposition 2.1. Let E [τ ] be a topological L0 -module and let C ⊂ E be L0 convex and L0 -absorbent. Then the following are equivalent:
1. pC : E → L0 is continuous.
2. 0 ∈ int C.
8
In this case, if in addition C has the relative countable concatenation property
C = {X ∈ E; pC (X) ≤ 1} .
Proof. The proof is exactly the same as the real case.
For the equality.
“⊆”: It is obtained from continuity of pC .
“⊇”: Let X ∈ E be satisfying pC (X) ≤ 1. By Lemma 2.1, we have that for
every ε ∈ L0++ there exists Yε ∈ L0++ such that 1 ≤ Yε < 1 + ε, with X ∈ Yε C.
Then, {X/Yε }ε∈L0 is a net (viewing L0++ downward directed) in C con++
verging to X and therefore X ∈ C. So, the proof is complete.
The example below (drawn from [28]) shows that, for the equality proved in
the latter proposition, C is required to have the relative countable concatenation
property.
Example 2.1. Consider the probability space Ω = (0, 1), E = B(Ω) the σ1
) with n ∈ N, P the Lebesgue measure and
algebra of Borel, An = [ 21n , 2n−1
0
E := L (E) endowed with | · |.
We define the set
U := Y ∈ L0 ; ∃ I ⊂ N finite, |Y 1Ai | ≤ 1 ∀ i ∈ N − I .
Inspection shows that U is L0 -convex, L0 -absorbent and U = U .
But, pU (X) = 0 for all X ∈ L0 , and so {X; pU (X) ≤ 1} = E. Indeed,
it suffices to show that pU1 (1) = 0, since pU1 is a L0 -seminorm. By way of
contradiction, assume that pU1 (1) > 0. Then, there exists m ∈ N such that
P [(pU1 (1) > 0) ∩ Am ] > 0. Define A := (pU1 (1) > 0)∩Am , ν := 21 (pU1 (1) +1Ac ),
Y := 1Ac + ν1A and X := 1Ac + ν1 1A . Thus, we have 1 = Y X ∈ Y U1 and
P (pU1 (1) > Y ) > 0, a contradiction.
We need a new notion:
Definition 2.2. Let E be a L0 -module, we say that the sum of E preserves the
relative countable concatenation property, if for every L, M subsets of E with
the relative countable concatenation property it holds that the sum of both L+M
has the relative countable concatenation property.
The following example shows that, in general, the sum of two subsets with
the relative countable concatenation property does not necessarily have the
relative countable concatenation property.
1
) for each k ∈ N, F =
Example 2.2. Let us take Ω = (0, 1), Ak = [ 21k , 2k−1
σ({Ak ; k ∈ N}) the sigma-algebra generated by {Ak }k and P the Lebesgue
measure restricted to F. Then (Ω, F, P) is a probability space. Let us put L0 :=
L0 (Ω, F, P).
Let us define the following L0 -module, which is a L0 -submodule of L0 (R2 , F):
E := spanL0 ({(1, 0)} ∪ {(0, 1Ak ) ; k ∈ N}) .
Let us also consider the following subsets:
cc
L := {(0, −1Ak ) ; k ∈ N}
9
cc
M := {(1Ak , 1Ak ) ; k ∈ N} ,
which obviously have the relative countable concatenation property.
We also have (1, 0) ∈ E and
1Ak (1, 0) = 1Ak ((0, −1Ak ) + (1Ak , 1Ak )) ∈ 1Ak (L + M ),
cc
and therefore (1, 0) ∈ L + M .
However, inspection shows that (1, 0) ∈
/ L + M . We conclude that L + M
does not have the relative countable concatenation property.
The following result is easy to prove by inspection, we omit the proof:
Proposition 2.2. Let E be a L0 -module. If E has the countable concatenation
property, then its sum preserves the relative countable concatenation property.
In virtue of the above result we have that, for a L0 -module E, the countable
concatenation property is stronger than the property of E having sum which
preserves the relative countable concatenation property. The following result
shows that, in fact, the former is strictly stronger than the latter.
Proposition 2.3. There exists a L0 -module E which does not have the countable concatenation property, but whose sum preserves the relative countable concatenation property.
Proof. Let us revisit the L0 -module E defined in Example 1.1:
E := spanL0 {1An ; n ∈ N}
with An := [1 −
1
2n−1 , 1
−
1
2n ]
for each n ∈ N.
E is a L0 -module that does not have the countable concatenation property.
Let us show that the sum of E preserves the relative countable concatenation
property.
Indeed, let us suppose that L, M are subsets of E with the relative countable
concatenation property.
Let z ∈ E and {Bk }k ∈ Π(Ω, F) be such that 1Bk z = 1Bk (lk + mk ) with
lk ∈ L and mk ∈ M for each k ∈ N.
First note that the following set
FL := {k ∈ N ; 1Ak l 6= 0 for all l ∈ L}
is necessarily finite. Let us define FM in an analogue way, which is finite either.
Let us put
X
z=
1Ai ∩Bk (lk + mk ) =
i∈F,k∈N
X
X
1Ai ∩Bk (lk + mk ) +
1Ai ∩Bk (lk + mk ) = l + m,
i∈FL ∪FM −F, k∈N
i∈F,k∈N
with
l=
P
i∈F ∪FL ∪FM
P
k∈N
1Ai ∩Bk lk ,
m=
P
i∈F ∪FL ∪FM
P
k∈N
1Ai ∩Bk mk .
Since L and M has the relative countable concatenation property, we conclude
that l ∈ L and m ∈ M .
10
Definition 2.3. Given A ⊂ E, we define the L0 -convex hull
(
)
X
X
0
coL0 (A) :=
Yi Xi ; I finite, Xi ∈ A, Yi ∈ L+ ,
Yi = 1
i∈I
i∈I
Finally, we provide a version for L0 -modules of the classical Mazur lemma
(for reviewing how the weak topologies for L0 -modules are defined and some
results, see the Appendix).
Theorem 2.1. [Randomized version of the Mazur lemma] Let (E, k·k) be a
L0 -normed module whose sum preserves the relative countable concatenation
property1 , and let {Xγ }γ∈Γ be a net in E, which converges weakly to X ∈ E.
Then, for any ε ∈ L0++ , there exists
Zε ∈ cocc
L0 {Xγ ; γ ∈ Γ} such that kX − Zε k ≤ ε.
Proof. Define M1 := cocc
L0 {Xγ ; γ ∈ Γ}. We may
replacing X by X − Xγ0 and Xγ by Xγ − Xγ0 for
γ ∈ Γ.
By way of contradiction, suppose that for every
F + such that kX − Zk > ε on AZ .
Denote B 2ε := {X ∈ E; kXk ≤ 2ε }, and define M
assume that 0 ∈ M1 , by
some γ0 ∈ Γ fixed and all
Z ∈ M1 there exists AZ ∈
:=
S
Z∈M1
Z + B 2ε .
Then M is a L0 -convex L0 -absorbent neighbourhood of 0 ∈ E, which has the
relative countable concatenation property (this is because M1 and B 2ε has the
relative c. c. property and the sum of E preserves the relative c. c. property).
Besides, for every Z ∈ M there exists CZ ∈ F + with kX − Zk ≥ 2ε on CZ . So
that X ∈
/ M.
Thus, by Proposition 2.1 we have that there exists C ∈ F + such that
pM (X) > 1
on C,
(1)
where pM is the guage function of M .
Further, given Y, Y 0 ∈ L0 with 1C Y X = 1C Y 0 X, it holds that Y = Y 0 on C.
Indeed, define A = (Y − Y 0 ≥ 0)
1C |Y −Y 0 |pM (X) ≤ pM (1C |Y −Y 0 |X) = pM ((1A −1Ac )1C (Y −Y 0 )X) = pM (0) = 0.
In view of 1, we conclude that Y = Y 0 on C.
Then, we can define the following L0 -linear application
µ0 :
spanL0 {X} −→ L0
µ0 (Y X) := Y 1C pM (X).
In addition, we have that
µ0 (Z) ≤ pM (Z)
for all Z ∈ spanL0 {X}.
1 In the earliest version of this paper, was accidentally omitted the hypothesis on E of
having sum which preserves the relative countable concatenation property. The author is
grateful to Tiexin Guo for pointing out to him the possible lack of this condition, and for
propitiating some interesting discussion which motivated Example 2.2 and Proposition 2.3.
11
Thus, by Theorem A.1, µ0 extends to a L0 -linear application µ defined on
E such that
µ(Z) ≤ pM (Z) for all Z ∈ E.
Since M is a neighborhood of 0 ∈ E, by Proposition 2.1, the gauge function
pM is continuous on E. Hence µ is a continuous L0 -linear function defined on
E.
Furthermore, we have that
ess. sup µ(Z) ≤ ess. sup µ(Z) ≤
Z∈M1
Z∈M
≤ ess. sup pM (Z) ≤ 1 < pM (X) = µ(X)
on C.
Z∈M
Therefore, X cannot be a weak accumulation point of M1 contrary to the hypothesis of Xγ converging weakly to X.
We have the following corollaries:
Corollary 2.1. Let (E, k·k) be a L0 -normed module whose sum preserves the
relative countable concatenation property, and let K ⊂ E be L0 -convex and with
the relative countable concatenation property, we have that the closure in norm
k·k
σ(E,E ∗ )
coincides with the closure in the weak topology, i.e. K = K
.
Then, from now on, for any subset K which is L0 -convex and with the
relative countable concatenation property, we will denote the topological closure
by K without specifying whether the topology is either weak or strong.
Let us recall some notions:
Definition 2.4. Let E[τ ] be a topological L0 -module. A function f : E → L̄0
is called proper if f (E) ∩ L0 6= ∅ and f > −∞. It is said to be L0 -convex if
f (Y X1 + (1 − Y )X2 ) ≤ Y f (X1 ) + (1 − Y )f (X2 ) for all X1 , X2 ∈ E and Y ∈ L0
with 0 ≤ Y ≤ 1. It said to have the local property if 1A f (X) = 1A f (1A X) for
A ∈ F + and X ∈ E. Finally, f is called lower semicontinuous if the level set
V (Y0 ) = {X ∈ E; f (X) ≤ Y0 } is closed for all Y0 ∈ L0 .
Corollary 2.2. Let (E, k·k) be a L0 -normed module whose sum preserves the
relative countable concatenation property, and let f : E → L̄0 be a proper L0 convex function. If f is continuous, then f is lower semicontinuous with the
weak topology.
Proof. It is a known fact that, if f is L0 -convex, then it has the local property
(see [9, Theorem 3.2]).
Being f L0 -convex and with the local property, we have that V (Y ) is L0 convex and has the relative countable concatenation property.
Since f is continuous, we have that V (Y ) is closed, and due to Corollary
2.1, it is weakly closed as well.
12
3
A randomized version of Krein-Šmulian theorem
In this section we prove one of the main results of this article, specifically provide
a generalization of the classical theorem of Krein-Šmulian in the context of the
L0 -normed modules. We will use arguments of completeness and the randomized
bipolar theorem (see A.4).
Before proving the main result of this section, let us introduce some necessary
notions.
Let (E, k · k) be a L0 -normed module. Given ε ∈ L0++ define
Wε := {(A, B); A, B ⊂ E, kX − Zk ≤ ε for all X ∈ A, Z ∈ B} .
Then W := {Wε ; ε ∈ L0++ } is the base of an uniformity on E.
Let G be the set of all nonempty closed subsets of E.
A net {Aγ }γ∈Γ in G is said to converge to A ∈ G if for each ε ∈ L0++ there
is some γε with (Aγ , A) ∈ Wε for all γ ≥ γε . Note that A is unique.
{Aγ }γ∈Γ is said to be Cauchy if for each ε ∈ L0++ there is some γε with
(Aα , Aβ ) ∈ Wε for all α, β ≥ γε .
Further, E is said to be complete if every Cauchy net {Xγ }γ∈Γ ⊂ E converges
to some X ∈ E.
Proposition 3.1. If (E, k · k) is complete, then every Cauchy net {Aγ }γ∈Γ in
G
T converges. Moreover, if {Aγ }γ∈Γ is decreasing, then it converges to A :=
γ∈Γ Aγ .
Proof. If {Aγ }γ∈Γ is Cauchy we have that for every ε ∈ L0++ there exists γε
such that
kX − Zk ≤ ε for all X ∈ Aα , Z ∈ Aβ with α, β ≥ γε
(2)
In particular, every net {Xγ }γ∈Γ with Xγ ∈ Aγ is Cauchy and therefore converges as E is complete.
Define A := {lim Xγ ; Xγ ∈ Aγ for all γ ∈ Γ}. Let us show that {Aγ }γ∈Γ
converges to A.
Indeed, given X ∈ A there exists a net {Xγ }γ∈Γ with Xγ ∈ Aγ which
converges to X. Then by (2) it holds that
kXγ − Zk ≤ ε for all Z ∈ Aα , with α, γ ≥ γε .
By taking limits in γ
kX − Zk ≤ ε for all Z ∈ Aα , with α ≥ γε .
Since X ∈ A is arbitrary, it holds that (A, Aα ) ∈ Wε for all α ≥ γε , and the
proof is complete.
Finally, let us prove the randomized version of the Krein-Šmulian theorem
Theorem 3.1. [Randomized version of the Krein-Šmulian theorem] Let (E, k·k)
be a complete L0 -normed module with the countable concatenation property and
let K ⊂ E ∗ be L0 -convex with the relative countable concatenation property.
Then the following statements are equivalent:
13
1. K is weak-∗ closed.
2. K ∩ {Z ∈ E ∗ ; kZk∗ ≤ ε} is weak-∗ closed for each ε ∈ L0++ .
Proof. 1 ⇒ 2: It is clear because {Z ∈ E ∗ ; kZk∗ ≤ ε} = Bεo is weak-∗ closed.
2 ⇒ 1: For each ε ∈ L0++ we have that Bε o ∩ K is weak-∗ closed. Then
the net {(Bε o ∩ K)o }ε∈L0++ is Cauchy. Indeed, for δ, δ 0 ≤ ε/2, by using the
properties of the polar (see A.1) and the bipolar theorem (see A.4):
(Bδ o ∩ K)o + Bε ⊃ (Bδ o ∩ K)o + Bε/2 = ((Bδ o ∩ K)o + Bε/2 )oo ⊃
o
o
⊃ ((Bδ o ∩ K)oo ∩ Bε/2
)o = (Bδ o ∩ K ∩ Bε/2
)o ⊃ (Bδ0 o ∩ K)o .
T
Since the net is decreasing, it converges to C := ε∈L0 (Bε o ∩ K)o .
++
Let us see that K = C o . Indeed, C ⊂ (Bε o ∩ K)o so Bε o ∩ K ⊂ C o for all
ε ∈ L0++ and therefore K ⊂ C o .
Let ε, r ∈ L0++ with r > 1. Since the net converges to C, there is δ ∈ L0++
with
(Bδo ∩ K)o ⊂ C + (r − 1)Bε ⊂ (C ∪ Bε )oo + (r − 1)(C ∪ Bε )oo = r(C ∪ Bε )oo
and by taking polars, it follows that
[r(C ∪ Bε )oo ]o =
1
(C ∪ Bε )o ⊂ (Bδo ∩ K)oo = Bδo ∩ K
r
and thus C o ∩ Bεo ⊂ r(Bδo ∩ K) for all r ∈ L0++ , r > 1. Therefore
\
Bεo ∩ C o ⊂
r(Bδo ∩ K) ⊂ Bδo ∩ K = Bδo ∩ K ⊂ K,
r∈L0++ ,r<1
and the proof is complete.
Remark 3.1. After writing a first version of this paper, the author knew from
Asgar Jamneshan of the works [8] and [7]. So, the author is grateful to Asgar
Jamneshan for pointing out these references.
Namely, K.-T. Eisele and S. Taieb [8] proved a version of Krein-Šmulian
theorem under the less general structure of L∞ -modules by using a different
strategy-proof.
S. Drapeau et al. [7], in an abstract level, created a new framework, namely
the algebra of conditional sets, in which strong stability properties are supposed
on all structures, so that, they obtained the Algebra of Conditional Sets, which
has a structure of complete Boolean algebra in an analogue way as the algebra
of traditional does. This framework is related to the L0 -theory, in the sense
that if we assume the countable concatenation property with uniqueness on a
L0 -module, we obtain a conditional set. In this related context, it was proved
a version of Krein-Šmulian theorem. However we would like to emphasize that
we work under weaker stability properties. For instance, the relative countable
concatenation property does assume the existence of a countable concatenation.
Besides, briefly speaking, topologies used here does not need to be a ’stable family’
in the sense that, as proposed in [2]. We could show examples and further
explanations, but conditional set theory is an extensive theoretical development,
and there is no room in this paper for including more details.
14
4
Financial application: representation of conditional convex risk measures for portfolio vectors
In this section we will show a financial application of the theory displayed in
the previous sections.
Firstly, we will recall the Lp -type modules, which are introduced by Filipovic
et al. [9].
Let (Ω, F, P) be a probability space, F a σ-subalgebra and p ∈ [1, +∞], we
define the application k· | F kp : L0 (E) → L̄0+ (F) by
(
1/p
p
E [|X| | F]
kX | Fkp :=
ess. inf Y ∈ L̄0 (F) | Y ≥ |X|
if p < ∞
if p = ∞
n
o
and the set LpF (E) := X ∈ L0 (E) ; kX | Fkp ∈ L0 (F) .
Then, (LpF (E) , k· | Fkp ) is a complete L0 -normed module with the countable
concatenation property with uniqueness.
Let us introduce the vectorial version of the Lp -type modules, which will be
interpreted as the set of all possible financial positions. For given d ∈ N we
define

1/p
 Pd
kX
|Fk
if p < ∞
i
p
i=1
|||X|F|||p :=
for X = (X1 , ..., Xd )
ess. sup
kX |Fk
if p = ∞
1≤i≤d
i
∞
Qd
d
which is a L0 -norm on LpF (E) := i=1 LpF (E).
Moreover, the relation proved in [20, Proposition 3.3] clearly extends to this
setting. Thus
cc
d
d
LpF (E) = L0 (F) Lp (E) = Lp (E)d .
We also have the natural duality result, which can easily obtained from [17,
Theorem 4.5]:
Proposition 4.1. Let 1h≤ p < +∞
i∗ and 1 < q ≤ +∞ with 1/p + 1/q = 1. Then
d
d
q
p
the map T : LF (E) → LF (E)
, Z → TZ defined by TZ (X) = E [X · Z | F],
Pd
with X · Z := i=1 Xi Zi , is an isometric isomorphism of L0 -normed modules.
d
Let us endow LpF (E) with a partial order. Namely, let K ⊂ Rd be a
d
proper closed convex cone with Rd+ ⊂ K, then for given X, Y ∈ LpF (E) we
+
write X Y if P(X − Y ∈ K) = 1, and for A ∈ F we say X Y on A if
P(X − Y ∈ K|A) = 1.
d
Definition 4.1. An application ρ : LpF (E) → L0 (F) is called a conditional
convex risk measure if ρ is :
1. monotone, i.e. if X Z then ρ(X) ≥ ρ(Z)
2. cash invariant, i.e. if Y ∈ L0 (F) and 1 ≤ i ≤ d, then ρ(X + Y ei ) =
ρ(X) − Y , where ei ∈ Rd are the unit vector.
15
3. convex, i.e. ρ(αX + (1 − α)Z) ≤ αρ(X) + (1 − α)ρ(Z) for all α ∈ R with
0 ≤ α ≤ 1.
n
o
d
We also define Aρ := X ∈ LpF (E) ; ρ(X) ≤ 0 the acceptance set.
We will focus our study on the special case of conditional convex risk mead
sures on L∞
F (E) . This model space, allows to model the following situation:
d
0
1. the fact that L∞
F (E) is a L -module means that we assume that there
exists available extra-information in the market, which is represented by
the σ-subalgebra F. So, in this case E represents the market information
at some time horizon T > 0, and F the available information at some
earlier date 0 < t < T .
d
2. the fact that L∞
F (E) has d coordinates means that we want to measure
the risk of portfolio vectors, i.e. not only the risk of the marginals Xi
separately but to measure the joint risk of X caused by the variation of
the components and their possible dependence.
d
∞
3. the fact that L∞
F (E) is a L -type module means that the financial positions always has bounded value. 1
Filipovic et al. [9] proposed to study conditional risk measures on Lp -type
modules, however the conditional risk measures considered by [9] are L0 -convex.
We would like to emphasize that we are considering the weaker assumption of
convexity in the traditional sense.
In this setting we have the following representation result:
d
0
Theorem 4.1. Suppose ρ : L∞
F (E) → L (F) is a conditional convex risk
measure, and suppose
d
ρ∗ (Z) := ess. sup {EP [X · Z|F] − ρ(X) ; X ∈ L∞
F (E) }
for Z ∈ L1F (E)d ,
its Fenchel conjugate. Then the following conditions are equivalent:
1. ρ can be represented by some penalty function α : L1F (E)d → L¯0 (F), i.e,
ρ(X) = ess. sup
Z ∈ L1F (E)d , Z ≤ 0,
E [X · Z | F] − α(Z);
E [Zi | F] = −1
(3)
d
for X ∈ L∞
F (E) .
2. ρ can be represented by the penalty function α := ρ∗ .
d
0
3. ρ has the L0 (F)-Fatou property, i.e., if {Xn }n ⊂ L∞
F (E) is a L -bounded
∞
d
sequence such that Xn converges P a.s. to some X ∈ LF (E) , then
ρ(X) ≤ ess. liminf ρ(Xn ).
n
1 Every X ∈ L∞ (E) is of the form X = Y X
0
∞
∞ with Y ∈ L (F ) and X∞ ∈ LF (E). The
F
financial interpretation is that we have a quantity Y of a risky but bonded financial asset X∞ ,
where the quantity Y is set in terms of the extra-information F .
16
4. ρ|L∞ (E)d has the Fatou property, i.e., if {Xn }n ⊂ L∞ (E)d is a bounded
sequence such that Xn converges P-a.s. to some X ∈ L∞ (E)d , then
ρ(X) ≤ ess. liminf ρ(Xn ).
n
d
1
d
5. ρ is lower semicontinuous for the topology σ(L∞
F (E) , LF (E) ).
d
1
d
6. The acceptance set Aρ := {X; ρ(X) ≤ 0} is σ(L∞
F (E) , LF (E) )-closed.
d
Working on L∞
F (E) has some advantages as we can see in the lemma below:
d
0
Proposition 4.2. Let ρ : L∞
F (E) → L (F) be a conditional convex risk measure, then the following properties hold:
1. ρ is Lipschitz continuous. Specifically
d
|ρ(X) − ρ(Z)| ≤ d|||X − Z | F|||∞ , for X, Z ∈ L∞
F (E)
2. ρ has the local property, i.e 1A ρ(X) = 1A ρ(1A X) for all A ∈ F + .
3. ρ is L0 -convex.
Pd
Proof.
1. Clearly, X Z+ i=1 |||X − Z | F|||∞ ei , and so ρ(Z)−d|||X − Z | F|||∞ ≤
Z + ρ(X) by monotonicity and cash invariance. Reversing the roles of X
and Z yields the assertion.
2. In virtue of 1, we have that
|1A ρ(X)−1A ρ(1A X)| ≤ 1A |ρ(X)−ρ(1A X)| ≤ 1A d|||X − 1A X | F|||∞ = 0.
Then the result follows.
d
3. Given Y ∈ L0 (F), 0 ≤ Y ≤ 1 and X, Z ∈ L∞
F (E) . Let us choose
a countable dense subset H ⊂ R, H = {h1 , h2 , ...}. For an arbitrary
ε ∈ L0++ (F), we define A1 := (Y ≤ h1 < Y + ε) and Ak := (Y ≤ hk <
Sk−1
Y + ε) − i=1 Ai for k > 1, and put
!
X
Yε := 1 ∧
1Ak hk .
k∈N
Then, {Yε }ε∈L0++ (F ) is a net (viewing L0++ (F) downwards directed), which
converges to Y .
On the other hand, we have that, since ρ is convex and has the local
property, ρ(Yε X + (1 − Yε )Z) ≤ Yε ρ(X) + (1 − Yε )ρ(Z). Now, since ρ
is continuous, we obtain, by taking limits, that ρ(Y X + (1 − Y )Z) ≤
Y ρ(X) + (1 − Y )ρ(Z).
d ∗
be, we write µ ≥ 0 (resp. µ ≤ 0) if X 0 implies that
Let µ ∈ L∞
F (E)
ρ(X) ≥ 0 (resp. ρ(X) ≤ 0).
We have the following prior representation result:
17
d
0
Lemma 4.1. Any conditional convex risk measure ρ : L∞
F (E) → L (F) is of
the form
n
o
d ∗
ρ(X) = ess. sup µ(X) − ρ∗ (µ) ; µ ∈ L∞
, µ ≤ 0, µ(ei ) = −1, 1 ≤ i ≤ d
F (E)
(4)
d
for X ∈ L∞
F (E) .
P
d
Furthermore, if µ ≤ 0 and µ
Y
e
= −1 for some Yi ∈ L0 (F) with
i
i
i=1
Pd
∗
0 ≤ Yi ≤ 1 and
i=1 Yi = 1, we have that ρ (µ) = ess. sup µ(X), and this
X∈Aρ
Sd
supremum is finite on i=1 (Yi = 1) and equals +∞ on the complement.
Proof. Due to Corollary
with respect to the weak
∞2.2,dρ∗is lower semicontinuous
d
d
topology σ(L∞
) and since L∞
F (E) , LF (E)
F (E) has the countable concatenation property, we can use the Fenchel-Moreau theorem A.5 obtaining that
ρ = ρ∗∗ . Hence,
ρ(X) = ess. sup {µ(X) − ρ∗ (µ)} .
d ∗
µ∈[L∞
F (E) ]
Finally, we will prove that if ρ∗ (µ) < +∞, then µ ≤ 0 and µ(ei ) = −1 for
1 ≤ i ≤ d.
d
+
with
Indeed, suppose that there exist X0 ∈ L∞
F (E) , X0 0, and A ∈ F
0
µ(X0 ) > 0 on A. Then, for λ ∈ L++ (F)
+∞ > ρ∗ (λX0 ) ≥ µ(λX0 ) − ρ(λX0 ) ≥ λµ(X0 ) − ρ(0)
on A
which is impossible as µ(X0 ) > 0 on A and λ ∈ L0++ (F) is arbitrary.
Furthermore, for α ∈ L0++ (F) and 1 ≤ i ≤ d
µ(αei ) − ρ(αei ) = α[µ(ei ) + 1] − ρ(0) < +∞.
Since α is arbitrary, it follows that µ(ei ) = −1.
For the second part, notice that ρ∗ (X) = ess. sup {µ(X) − ρ(X)} ≥ ess. sup µ(X).
X∈Aρ
Pd
In addition, for all X we have that X 0 := X + ρ(X) i=1 Yi ei ∈ Aρ .
Then
ess. sup µ(X) ≥ µ(X 0 ) = µ(X) − ρ(X).
X∈Aρ
d
By taking essential supremum on X ∈ L∞
F (E) , we see that ess. sup µ(X) ≥
X∈Aρ
ρ(X)∗ .
We turn now to prove Theorem 4.1
Proof. [Theorem 4.1]
2 ⇒ 1, 3 ⇒ 4 and 5 ⇒ 6 are clear.
d
6 ⇒ 2 : Let X0 ∈ L∞
F (E) be and put
m := ess. sup E [X0 · Z | F] − ρ∗ (Z) ; Z ∈ L1F (E)d .
(5)
d ∗
Since L1F (E)d is embedded into L∞
via Z 7→ E [Z· | F], in view of
F (E)
Lemma 4.1, it suffices to show that m ≥ ρ(X0 ) or, equivalently, that M + X0 ∈
18
Pd
Aρ with M := m d1 i=1 ei (note that this equivalence is due to cash invariance,
and we could have chosen any other L0 -convex combination of e1 , ..., ed ).
Assume, by sake of contradiction, that B := (ρ (M + X0 ) > 0) ∈ F + .
Let us take some X1 with ρ(X1 ) > 0. Then X2 = (M + X0 )1B + X1 1B c
satisfies 1A X2 ∈
/ 1A Aρ for all A ∈ F + . In addition, Aρ is nonempty weak-∗
0
closed, L -convex and has the countable concatenation property. By applying
d
Theorem A.2, we obtain a continuous linear functional µ on L∞
F (E) such that
µ(X) ≤ ess. sup µ(X) < µ(X2 ),
for all X ∈ Aρ .
(6)
X∈Aρ
In particular,
β := ess. sup µ(X) < µ(M + X0 ) =: γ
on B.
(7)
X∈Aρ
By Theorem A.3, we have that µ must be of the form µ(X) = E [X · Z|F] for
some Z ∈ L1F (E)d .
In fact, Z ≤ 0. To show this, fix X 0 and note that ρ(λX) ≤ ρ(0) for
λ ∈ L0+ (F), by monotonicity. Hence λX + M ∈ Aρ for all λ ∈ L0+ (F). It follows
from (6) that
+∞ > µ(X2 ) > µ (λX + M ) = λµ(X) + µ (M ) .
Since λ ∈ L0+ (F) is arbitrary, we have that µ(X) ≤ 0. We conclude that
E [X · Z|F] ≤ 0 for all X ≥ 0. Thus, necessarily Z ≤ 0.
On the other hand, (6) yields that there exists X such that µ(X) > 0 (it
suffices to put X := X2 − X 0 for some X 0 ∈ Aρ ). Therefore
0 < µ(X) = E [X · Z|F] ≤ |||X|F|||∞ (−d)E Z|F ,
and then we have that E Z|F < 0. (Here, Z denotes the arithmetic mean of
Z).
i
h
Z
If we define Z̃ := − E Z|F
, and µ̃ defined by µ̃(X) := E Z̃ · X|F for X ∈
[ ]
d
(E)
.
Then
µ̃
(M
)
=
−1,
and so, due to Lemma 4.1, we have that
L∞
F
β
ρ∗ (Z̃) = ess. sup µ̃(X) = − µ(M
)
on B.
X∈Aρ
In addition,
h
i
−E Z̃X0 |F + m =
µ(X0 )
E[Z|F ]
Thus, from (7) it follows that
h
i
−E Z̃X0 |F + m =
)
+ m µ(M
µ(M ) =
µ(M +X0 )
µ(M )
γ
µ(M )
= −ρ∗ (Z̃)
<
β
µ(M )
=
γ
µ(M )
on B.
on B,
in contradiction to (5).
1 ⇒ 3 : Let {Xn }n be a sequence in L∞
F (E) such that X = limn Xn P-a.
s. and such that there exists Y ∈ L0 (F) with |Xni | < Y for all n ∈ N, where
Pd
Xn = i=1 Xni ei ..
19
Fix Z ∈ L1F (E)d , Z ≤ 0 with E [Zi | F] = −1 for 1 ≤ i ≤ d, and take
i
probability measures Qi , Qi ∼ P with dQ
di P = −Zi for 1 ≤ i ≤ d. It holds
E [Z · Xn | F] =
d
X
d
X
EQi Xni | F E [Zi | F] = −
EQi Xni | F .
i=1
i=1
Then, by the dominated convergence theorem for conditional expectations (see
[3, Proposition 10.4.4]), the latter converges P-a.s. to E [Z · X | F]. Hence,
n
o
ρ(X) = ess. sup limE [Xn · Z | F] − α(Z); Z ∈ L1F (E)d , Z ≤ 0, E [Zi | F] = −1 ≤
n
≤ ess. liminf ess. sup E [−Xn · Z | F] − α(Z); Z ∈ L1F (E)d , Z ≤ 0, E [Zi | F] = −1
n
= ess. liminf ρ(Xn ).
n
4 ⇒ 3: Let {Xn }n be a sequence such that Xn → X P-a.s. and Y ∈ L0+ (F)
Pd
with |Xni | ≤ Y with Xn = i=1 Xni ei .
For each k ∈ N, define Ak := (k − 1 ≤ Y < k). Then {Ak } ∈ Π(Ω, F) and
|1An Xni | ≤ 1An Y ≤ k for every k ∈ N.
Since ρ|L∞ (E) has the L0 -Fatou property, we see that
1Ak ρ(1Ak X) ≤ 1Ak ess. liminf ρ(1Ak Xn ).
Further, since ρ has the local property, we have the following inequality
1Ak ρ(X) ≤ 1Ak ess. liminf ρ(Xn ),
which yields that ρ(X) ≤ ess. liminf ρ(Xn ).
3 ⇒ 5 : We will show that V (Y ) is weak-∗ closed for Y ∈ L0 (F).
Fix ε ∈ L0++ and define
d
Wε := V (Y ) ∩ X ∈ L∞
F (E) ; |||X|F|||∞ ≤ ε .
If {Xγ }γ∈Γ is a net in Wε converging to X ∈ Wε with |||·|F|||1 , then by induction
we can choose a increasing sequence {γn }n in Γ such that |||Xγn − X|F|||1 ≤ 1/n.
Thus, the sequence X̃n := Xγn converges in norm to X in the space L1d (E).
Therefore, there exists a subsequence which converges P-a.s. to X, and the
L0 (F)-Fatou property of ρ implies that X ∈ Wε . Hence Wε is closed in L1F (E)d .
Since Wε is L0 -convex, closed in L1F (E)d and has the relative countable
concatenation property, due to Corollary 2.1, it holds that Wε is weak closed in
L1F (E)d .
Now, since the injection
d
d
d
d
d
d
∞
1
1
1
∞
L∞
(E)
,→
L
(E)
,
σ
L
(E)
,
L
(E)
,
σ
L
(E)
,
L
(E)
F
F
F
F
F
F
d
d
d
1
is continuous, we have that Wε is σ L∞
(E)
,
L
(E)
-closed in L∞
F
F
F (E) .
Finally, it follows from Theorem 3.1 that V (Y ) is weak-∗ closed.
References
20
References
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A
Appendix
Let us collect some important results from the theory of locally L0 -convex modules:
Theorem A.1. [9, Hahn–Banach extension theorem] Let E be a locally L0 convex module. Consider an L0 -sublinear function p : E → L0 , an L0 -submodule
C of E and an L0 -linear function µ : C → L0 such that
µ(X) ≤ p(X)
for all X ∈ C
Then µ extends to an L0 -linear function µ̃ : E → L0 such that µ̃(X) ≤ p(X)
for all X ∈ E.
For the next theorem we need to introduce the following notion:
22
Definition A.1. A topological L0 -module E[τ ] has the topological countable
concatenation property if it has the countable concatenation property and for
every countable collection {Un } of neighborhoods of 0 ∈ E and for every partition
{An } ∈ Π(Ω, F) the set
[
cc({An }, {Un }) :=
{cc({An }, {Xn }); Xn ∈ Un }
again is a neighborhood of 0 ∈ E.
Notice that any L0 -normed module has the topological countable concatenation property.
A first version of the theorem below was introduced by Filipovic et al. [9],
but instead we enunciate the reviewed version provided by Guo et al. [19] under
the notions and considerations made in this manuscript.
Theorem A.2. Let E[τ ] be a locally L0 -convex module that has the topological
countable concatenation property and let K ⊂ E with the relative countable
concatenation property, closed, L0 -convex and non-empty. If X ∈ E satisfies
1A X ∩ 1A K = ∅ for all A ∈ F + then there is ε ∈ L0++ and a continuous
L0 -linear function µ : E → L0 such that
µ(Z) + ε ≤ p(X)
for all Z ∈ K.
Analogously to the classical case, we have weak topologies, polar sets and a
randomized bipolar theorem for L0 -modules.
Namely, given a topological L0 -module E [τ ], let E[τ ]∗ denote, or simply
∗
E , the L0 -module of continuous L0 -linear functions µ : E → L0 . We define
the weak topology σ(E, E ∗ ) on E as the topology induced by the family of L0 seminorms {qX ∗ }X ∗ ∈E ∗ defined by qX ∗ (X) := |X ∗ (X)| for X ∈ E. Analogously,
the weak-∗ topology σ(E ∗ , E) on E ∗ is defined.
Theorem A.3. [19, Theorem 3.22] Let E [τ ] be a topological L0 -module then
E[σ(E, E ∗ )]∗ = E ∗ and E ∗ [σ(E ∗ , E)]∗ = E.
Definition A.2. Given A ⊂ E, we define the polar of A the subset Ao of E ∗
given by
Ao := {Z ∈ E ∗ ; |hX, Zi| ≤ 1, for all X ∈ A}.
Proposition A.1. Let E[τ ] be a locally L0 -convex module, D, Di ⊂ E for i ∈ I,
then:
1. Do is L0 -convex, weak-∗ closed and with the relative countable concatenation property.
2. 0 ∈ Do , D ⊂ Doo . If D1 ⊂ D2 , then D2o ⊂ D1o .
o
3. For ε ∈ L0++ , we have that (εD) = 1ε Do .
o T
S
4.
= i∈I Dio
i∈I Di
23
Proof. Let us see that Do has the relative countable concatenation property.
Given Z ∈ cc({An }n , {Zn }n ) with Zn ∈ Do , it follows that for X ∈ D
!
X
X
X
|hX, Zi| =
1An |hX, Zi| =
1An |hX,
1An Zi| =
n∈N
X
1An |hX,
n∈N
X
n∈N
1An Zn i| =
X
1An |hX, Zn i| ≤ 1.
n∈N
The rest is analogue to the known proofs for locally convex spaces.
Theorem A.4. [19, Theorem 3.26] Let E[τ ] be a locally L0 -convex module with
the countable concatenation property and D ⊂ E then:
Doo = coL0 cc (D ∪ {0})
Given a function f : E → L̄0 , we define the Fenchel conjugate of f as
f : E ∗ → L̄0 , f ∗ (µ) := ess. sup (µ (X) − f (X)) .
∗
X∈E
Likewise, we define f ∗∗ : E → L̄0 , f ∗∗ (X) := ess. sup (µ (X) − f ∗ (µ)) .
µ∈E ∗
We have a Fenchel-Moreau type theorem for L0 -modules. A version of this
theorem was proved for first time in [9], but we show below the later version
appeared in [19]:
Theorem A.5. [Fenchel-Moreau theorem] Let E[τ ] be a locally L0 -convex module with the countable concatenation property. If f : E → L̄0 is a proper lower
semicontinuous L0 -convex function, then f = f ∗∗ .
Finally, let us recall that, given a L0 -normed module (E, k · k), we have that
for all µ ∈ E ∗ there exists ξ ∈ L0+ such that |µ(X)| ≤ ξkXk for all X ∈ E.
Thus, we can define
kµk∗ := ess. inf {ξ ∈ L0+ ; |µ(X)| ≤ ξkXk for all X ∈ E} =
= ess. sup {|µ(X)|; kXk ≤ 1 for all X ∈ E}.
Then (E ∗ , k · k∗ ) is a L0 -normed module.