R E S E A R C H
Open Access
Daneš theorem in complete random normed
modules
Yujie Yang
**Correspondence:
Department of Basic Course, Beijing Union University, Beijing, 100101, P.R. China
Abstract
Based on the recent work of random metric theory, namely the Ekeland variational principle on a complete random metric space, this paper studies the Daneš theorem in a complete random normed module. In this paper, we first present the notion of finer ordering on a random normed module. Then we establish the Daneš theorem in a complete random normed module under the locallyL0-convex topology. When the
base space of the random normed module is trivial, our result automatically degenerates to the classical case.
MSC: 58E30; 47H10; 46H25; 46A20
Keywords: random metric space; random normed module; locallyL0-convex
topology; Ekeland variational principle
1 Introduction
In , Daneš [] presented the Daneš theorem. With the classical Ekeland variational principle, Brøndsted [] gave it a new proof in .
Recently, Prof. Guo Tiexin and I [] established the Ekeland variational principle for an
¯
L-valued function on a complete random metric space, whereL¯is the set of equivalence classes of extended real-valued random variables on a probability space (,F,P). Based on this result, this paper establishes the Daneš theorem in a complete random normed module under the locallyL-convex topology.
A random normed module is a random generalization of an ordinary normed space. Different from ordinary normed spaces, random normed modules possess the rich strat-ification structure, which is introduced in this paper. It is this kind of rich stratstrat-ification structure that makes the theory of random normed modules deeply developed and also renders it the most useful part of random metric theory [–]. When the probability (,F,P) is trivial, namelyF={∅,}, our result reduces to the classical Daneš theorem. So our result is a nontrivial random extension.
The remainder of this article is organized as follows: in Section we give some necessary definitions and in Section we give the main results and proofs.
2 Preliminary
Throughout this paper, (,F,P) denotes a probability space,Kthe real number fieldRor the complex number fieldC,Nthe set of positive integers,L¯(F) the set of equivalence classes of extended real-valued random variables onandL(F,K) the algebra of equiva-lence classes ofK-valuedF-measurable random variables onunder the ordinary scalar
multiplication, addition and multiplication operations on equivalence classes, denoted by L(F) whenK=R.
Specially,L
+(F) ={ξ∈L(F)|ξ≥},L++(F) ={ξ∈L(F)|ξ> on}.
As usual,ξ>ηmeansξ≥ηandξ=η, whereasξ>ηonAmeansξ(ω) >η(ω) a.s. onA for anyA∈FandξandηinL¯(F), whereξandηare arbitrarily chosen representatives ofξandη, respectively.
For anyA∈F,Acdenotes the complement ofA,A˜ ={B∈F|P(AB) = }denotes the
equivalence class ofA, whereis the symmetric difference operation,IAthe characteristic
function ofA, andI˜Ais used to denote the equivalence class ofIA; given twoξ andηin ¯
L(F), andA={ω∈:ξ=η}, whereξandηare arbitrarily chosen representatives of
ξ andη, respectively, then we always write [ξ=η] for the equivalence class ofAandI[ξ=η] forI˜A, one can also understand the implication of such notations asI[ξ≤η],I[ξ<η], andI[ξ=η]. For an arbitrary chosen representativeξofξ∈L(F,K), define the twoF-measurable random variables (ξ)–and|ξ|by (ξ)–(ω) =
ξ(ω) ifξ(ω)= , and (ξ)–(ω) =
other-wise, and by|ξ|(ω) =|ξ(ω)|,∀ω∈. Then the equivalence classξ–of (ξ)–is called the generalized inverse ofξand the equivalence class|ξ|of|ξ|is called the absolute value ofξ. It is clear thatξ·ξ–=I
[ξ=].
Definition .([]) An ordered pair (E, · ) is called a random normed space (briefly, anRNspace) overKwith base (,F,P) ifEis a linear space and · is a mapping from EtoL
+(F) such that the following three axioms are satisfied: () x= if and only ifx=θ(the null vector ofE); () αx=|α|x,∀α∈Kandx∈E;
() x+y ≤ x+y,∀x,y∈E,
where the mapping · is called the random norm onEandxis called the random norm of a vectorx∈E.
In addition, ifEis left module over the algebraL(F,K) such that the following is also satisfied:
() ξx=|ξ|x,∀ξ∈L(F,K)andx∈E,
then such an RN space is called anRN module over K with base (,F,P) and such a random norm · is called anL-norm onE.
Definition .([]) Let (E, · ) be anRNmodule overKwith base (,F,P). For any
ε∈L
++, letB(ε) ={x∈E| x ≤ε}andUθ={B(ε)|ε∈L++}. A setG⊂Eis calledTc-open
if for everyx∈Gthere exists someB(ε)∈Uθsuch thatx+B(ε)⊂G. LetTcbe the family
of Tc-open subsets, thenTc is a Hausdorff topology onE, called the locallyL-convex
topology, denoted byTc.
Let (E, · ) be anRNmodule overKwith base (,F,P),pA=I˜A·pis called the
A-strat-ification ofpfor each givenA∈FandpinE. The so-called stratification structure ofE means thatEincludes every stratification of an element inE. Clearly,pA=θwhenP(A) =
andpA=pwhenP(\A) = , which are both called trivial stratifications ofp. Further,
To introduce the main results of this paper, let us first recall the following.
Definition .([]) LetXbe a Hausdorff space andf :X→ ¯L(F), then: () dom(f) :={x∈X|f(x) < +∞on}is called the effective domain off. () f is proper iff(x) > –∞onfor everyx∈Xanddom(f)=∅.
() f is bounded from below (resp., bounded from above) if there existsξ∈L(F)such thatf(x)≥ξ (accordingly,f(x)≤ξ) for anyx∈X.
In all the vector-valued extensions of the Ekeland variational principle, it is of key im-portance to properly define the lower semicontinuity for a vector-valued function [–]. Recently, we have found that a kind of lower semicontinuity forL¯-valued functions is very suitable for the study of conditional risk measures.
Definition .([]) Let (E, · ) be anRNmodule overRwith base (,F,P). A function f :E→ ¯L(F) is calledT
c-lower semicontinuousifepi(f) is closed in (E,Tc)×(L(F),Tc).
There is a kind of countable concatenation property, which is concerned with theL -moduleEitself and is very important for the theory ofRNmodule. Let us recall it.
Definition .([]) Let E be a left module over the algebraL(F,K). A formal sum
n∈N˜IAnxnis called acountable concatenationof a sequence{xn|n∈N}inE with
re-spect to a countable partition{An|n∈N}oftoF. Moreover, a countable
concatena-tionn∈NI˜Anxnis well defined or
n∈N˜IAnxn∈Eif there isx∈Esuch that˜IAnx=˜IAnxn, ∀n∈N. A subset GofE is said tohave the countable concatenation propertyif every countable concatenationn∈N˜IAnxnwithxn∈Gfor eachn∈Nstill belongs toG, namely
n∈N˜IAnxnis well defined and there existsx∈Gsuch thatx=
n∈N˜IAnxn.
Definition .([]) LetEbe a left module over the algebraL(F) andf a function from EtoL¯(F), then:
() f isL(F)-convex iff(ξx+ ( –ξ)y)≤ξf(x) + ( –ξ)f(y)for allxandyinEand
ξ∈L
+(F)such that≤ξ≤(here we make the convention that·(±∞) = and
∞–∞=∞!).
() f is said to have the local property if˜IAf(x) =I˜Af(I˜Ax)for allx∈EandA∈F.
It is well known from [] thatf:E→ ¯L(F) isL(F)-convex ifff has the local property andepi(f) isL(F)-convex.
3 Main results and proofs
Definition . Let (E, · ) be anRN module overRwith base (,F,P),z∈Eandr∈ L
++(F). Denote theTc-closed ball by
Bz(r) :=
x∈E:x–z ≤r.
Definition .([]) Let (E, · ) be anRN module overRwith base (,F,P),Bz(r) a
Tc-closed ball inE,y∈E\Bz(r). Define theL(F)-convex hull of{y} ∪Bz(r) by
D(z,r,y) :=tb+ ( –t)y:b∈Bz(r),t∈L+(F) and ≤t≤
Definition . Let (E, · ) be anRNmodule overRwith base (,F,P),≤and≤be both orderings onE. Then≤is finer than≤, if
x≤y ⇒ x≤y.
In [], we established the precise form of the Ekeland variational principle on a Tc
-completeRN-module. Here we only need its general form as follows.
Lemma .([]) Let(F, · )be aTc-complete RN module over R with base(,F,P)such
that F has the countable concatenation property,ϕ:F→ ¯L(F)have the local property.If G⊂F is aTc-closed subset with the countable concatenation property andϕ|G is proper,
Tc-lower semicontinuous,and bounded from below on G,then for each point x∈dom(ϕ|G), there exists z∈G such that the following are satisfied:
() ϕ(z)≤ϕ(x) –z–x;
() for eachx∈Gsuch thatx=z,ϕ(x)ϕ(z) –x–z,which means thatzis a maximal element in(G,≤ϕ).
Remark .([]) The ordering≤ϕonFis defined as follows:x≤ϕyif and only if either
x=y, orxandy∈dom(ϕ) are such thatx–y ≤ϕ(x) –ϕ(y).
Theorem . Let(X, · )be aTc-complete RN module over R with base(,F,P)such
that X has the countable concatenation property,F⊂X aTc-closed subset with the
count-able concatenation property,and z∈X\F.Let r,R,ρ∈L++(F)with <r<R<ρ on, then there exists x∈∂c(F)such that
x–z ≤ρ
and
D(z,r,x)∩F={x},
where R:={z–a:a∈F},and∂c(F)denotes theTc-boundary of F.
Proof We can, without loss of generality, supposez= . LetE:=F∩B(ρ).
Define an ordering ˜≤onEas follows:x ˜≤x if and only ifx∈D(,r,x). It is easy to check that ˜≤is a partial ordering.
Define a functionϕ:E→L
+(F) byϕ(x) = (ρ+r)(R–r)–x,∀x∈E.
SinceFandB(ρ) areTc-closed and have the countable concatenation property, it
fol-lows thatEisTc-closed and has the countable concatenation property.
For eachA∈F, one can have
˜
IA·ϕ(x) =˜IA·(ρ+r)(R–r)–x=I˜A·(ρ+r)(R–r)–˜IA·x=˜IA·ϕ(I˜A·x), ∀x∈E,
which implies thatϕhas the local property.
SinceϕisTc-continuous, it is easy to see thatϕisTc-lower semicontinuous.
We now prove that ˜≤is finer than the ordering≤ϕ.
Letx,xbe points inEsuch thatx ˜≤x. Then one can havex∈D(,r,x); thus we can suppose
x= ( –t)x+tv, ()
wheret∈L
+(F), ≤t≤, andv∈B(r).
From (), it follows thatx ≤( –t)x+tv, which implies
tx–v
≤ x–x.
FromR–r≤ x–v, one can have
t≤x–x
(R–r)–. ()
Thus by () we have
x–x=tv–x ≤t
v+x
≤t(r+ρ)≤(r+ρ)(R–r)–x–x
,
which impliesx ≤ϕ x, and hence ˜≤is finer than the ordering≤ϕ.
Sincexis a maximal element in (E,≤ϕ) and ˜≤is finer than≤ϕ, it is easy to check that
xis a maximal element in (E, ˜≤). Thus we havex ≤ρand{x}=D(,r,x)∩E, which impliesD(,r,x)∩F={x}.
For eachx∈D(,r,x), we can supposex=tx+ ( –t)v, wherev∈B(r). Thus we have
x=tx+ ( –t)v≤tx+ ( –t)v ≤tρ+ ( –t)r<ρ
on.
It is easy to see that for anyy∈F\E,y ≤ρdoes not hold. Hence we havex∈∂c(F).
Remark . When the base space (,F,P) of theRNmodule is trivial, namelyF={∅,}, our result automatically degenerates to the classical Daneš theorem. So our result is a nontrivial random extension.
Competing interests
The author declares to have no competing interests.
Acknowledgements
This work was supported by the ‘New Start’ Academic Research Projects of Beijing Union University (No. Zk10201304).
Received: 11 May 2014 Accepted: 23 July 2014 Published:21 Aug 2014
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