COMPACITY IN NARROW LIMIT TOWER SPACES

© Hindawi Publishing Corp.

COMPACITY IN NARROW LIMIT TOWER SPACES

LIVIU C. FLORESCU

(Received January 2000 and in revised form 4 May 2000)

Abstract.We introduce a limit tower structure on the space of all bounded Radon mea- sures on a completely regular space and we extend the Prohorov’s theorem of narrow compactness. In the particular case of Polish spaces, we give a sequential version of this extension.

2000 Mathematics Subject Classification. 28A33, 46E27, 54E70, 54A20, 60B10.

1. Introduction. Let T be a completely regular space,Ꮾ the boreliens ofT, and M^b(T )the set of all bounded Radon measures on(T ,Ꮾ)(i.e., the real bounded mea- suresµ:Ꮾ→Rsuch that|µ|(A)=sup{|µ|(K):Kis compact,K⊆A}, for allA∈Ꮾ, where|µ|is the variation ofµ). Denote byᏯ^b(T )the space of all bounded continuous real functions onT and letf =sup{|f (t)|:t∈T}, for everyf∈Ꮿ^b(T ). We recall that a filterFonM^b(T )is narrowly convergent toµif and only ifVε,f(µ)∈F, for all f∈Ꮿ^b(T ),ε >0, whereVε,f(µ)= {ν:|µ(f )−ν(f )|< ε}.

We say that a setH⊆M^b(T )isrelatively narrowly compact if, for every filterbase B⊆2^Hthere exist a filterFonM^b(T )andµ∈M^b(T )such thatFconverges toµ.

Prohorov’s classical theorem states thata bounded setH⊆M^b(T )is relatively nar- rowly compact if the following condition is satisfied:

∀ε >0, ∃Kεcompact⊆T:|µ|

T\Kε

< ε,∀µ∈H. (1.1)

A setHas in (1.1) is calledtight.

We remark that, ifT is a Polish space (i.e.,T is a separable, completely metriz- able space) orT is a locally compact space, the converse is also true (relative narrow compactness implies tightness) (see [2, Section 5, Theorems 1 and 2]).

Limit tower spaces were first defined in 1997 by Kent and Brock [3] as an isomorphic gradated variant of convergence approach spaces of Löwen [8].

In this paper, we introduce onM^b(T )a limit tower structure ¯p= {pa:a∈[0,+∞]}

(see [3]), wherep0is the narrow convergence structure. Then, for every bounded set H⊆M^b(T ), there exists a numbert=t(H)≥0 such that, for every filterbaseB⊆2^H there exists a filterFonM^b(T ),B⊆F,pt-convergent inM^b(T )(seeTheorem 3.8); we say thatHispt-relatively compact. The numbert(H)estimates the degree of tightness ofH. IfHis tight thent(H)=0, so we obtain Prohorov’s theorem.

IfT is a locally compact space we extend also the converse of Prohorov’s theorem (seeTheorem 3.12).

We give some examples in the particular case ofT=NwhenM^b(N)=¹.

InSection 4, we obtain a sequential version ofTheorem 3.8on the subsetM¹(T )⊆ M^b(T )of all probabilities on the Polish spaceT. So, every sequence(µn)n∈N⊆M¹(T ) contains a subsequence p2t-convergent in M¹(T ), where t= t({µn :n ∈N}) (see Theorem 4.9). In particular, we prove that the limit tower structure ¯p on the set of probabilities is induced by a probabilistic metric on this space.

2. Limit tower structures. LetXbe a set,B(X)the set of all filterbases onX, and 2^X the power set ofX; for everyF∈B(X),Fis the filter generated byF. Fo rx∈X, let ˙xdenote the fixed ultrafilter generated by{x}.

Definition2.1(see [3, Definition 1]). Alimit structureonXis a functionq:B(X)→

2^Xsatisfying

x∈q(x), x˙ ∈X, (2.1)

q(F)=q F

, ∀F∈B(X), (2.2) q

F∩G

=q(F)∩q(G), ∀F,G∈B(X). (2.3) A pair(X,q), whereqis a limit structure onXis called alimit space.

Remark2.2. The statement “x∈q(F)” will be writtenF ^q→x and we say thatF q-converges tox.

Remark2.3. In [3], alimit structureis a functionq:F(X)→2^X, whereF(X)denotes the set of all filters onX, satisfying

x∈q(x), x˙ ∈X, F⊆G ⇒q(F)⊆q(G), x∈q(F) ⇒x∈q(F∩x),˙ x∈q(F)∩q(G) ⇒x∈q(F∩G).

(2.4)

If we extend such a functionqtoB(X)lettingq(F)=q(F), then (2.4) is equivalent to (2.1), (2.2), and (2.3).

Remark2.4. Ifτis a topology onXand we defineF ^qτ→xif and only ifᐂτ(x)⊆F, thenqτ is a limit structure onX(hereᐂτ(x)denotes the neighborhood filter ofxin (X,τ)). More exactly we have the following proposition.

Proposition2.5(see [3, Proposition 2]). Letqbe a limit structure onX; the neces- sary and sufficient condition for a topologyτ to exist onX, such thatq=qτ, is thatq fulfills the following condition:

(F)Let{Fj:j∈J}be a family of filterbases onXand{xj:j∈J} ⊂Xbe such that Fj q

→xjfor allj∈J.

IfΦ∈B(J)is such thatF ^q→x, whereF= {{xj:j∈φ}:φ∈Φ}, then

φ∈Φ

j∈φ

F_j ^q→x. (2.5)

Definition2.6(see [3, Definition 4]). Alimit tower p¯on a setX is a family ¯p= {pa:a∈[0,+∞]}of limit structures onXsatisfying the following conditions:

pa(F)⊆pb(F), ∀a≤b,∀F∈B(X), (2.6) p∞(F)=X, ∀F∈B(X), (2.7) pa(F)= ∩b>apb(F), ∀a∈[0,+∞),∀F∈B(X). (2.8) Ifx∈pa(F), we say thatFispa-convergent toxand we denote this byF ^pa→x. If ¯p is a limit tower onX,(X,p)¯ is called alimit tower space.

The axiom (F) defined inProposition 2.5 has a natural extension to a limit tower space(X,p):¯

(F) Leta,b∈[0,+∞],{Fj:j∈J} ⊆B(X), and{xj:j∈J} ⊆Xsuch thatFj pa

→xj, for allj∈J. IfΦ∈B(J)is such thatF ^pb→x, whereF= {{xj:j∈φ}:φ∈Φ}, then

φ∈Φ

j∈φ

F_j ^pa+b→x. (2.9)

Definition2.7. A limit tower ¯ponXwhich satisfies (F) is called atopological limit tower.

Remark2.8. From [3, Theorems 9, 13 and Proposition 12(b)] we know that a topo- logical limit tower is an isomorphic form of a Löwen’s approach structure (see [8]).

3. Narrow limit tower onM^b(T ). In this section, we introduce a topological limit tower ¯p= {pa:a∈[0,+∞]}on the space of bounded Radon measures on a completely regular space such thatp0-convergence is just the narrow convergence; then we extend the Prohorov’s theorem of narrow compactness.

Let T be a completely regular space, let Ꮾ be the σ-algebra of Borel subsets of T, and letM^b(T ) be the set of all bounded Radon measures on(T ,Ꮾ). Denote by C^b(T )the set of all bounded continuous real functions onT. For everyf∈C^b(T )and µ∈M^b(T ), we denoteµ(f )=

Tf dµ.

Now, for everya∈[0,+∞],µ∈M^b(T ), andf∈C^b(T ), we denote Va,f(µ)=

ν∈M^b(T ):µ(f )−ν(f )≤af

. (3.1)

Then, for everya∈[0,+∞), letpa:B(M^b(T ))→2^{Mb(T )}defined by pa(F)=

µ∈M^b(T ):∀b > a,∀ ∈C^b(T ),Vb,f(µ)∈F

, (3.2)

for all filterbasesFonM^b(T ); letp∞be the indiscrete convergence structure onM^b(T ) (p∞(F)=M^b(T ), for allF∈B(M^b(T ))).

We remark thatF ^pa→µif and only if for allb > a, for allf∈C^b(T ),Vb,f(µ)∈F. Proposition3.1. The limit towerp¯= {pa:a∈[0,+∞]}is a topological limit tower onM^b(T ).

Proof. For everya∈[0,+∞),µ∈M^b(T ), andf∈C^b(T ), we haveµ∈Va,f(µ), so that we have (2.1).

Equations (2.2), (2.3), (2.6), and (2.7) are consequences of the definition of ¯p. From (2.6),pa(F)⊆ b>apb(F), for allF∈B(M^b(T )). IfF ^p→^c µ, for allc > a, then for all b > a, there existscsuch thata < c < bhenceVb,f(µ)∈F, for allf∈C^b(T ). Therefore F ^pa→µand sowe have (2.8).

(F) Leta,b≥0,{Fj :j∈J} ⊆B(M^b(T )), and {µj:j ∈J} ⊆M^b(T ) such that (1) Fj pa

→µj, for all j ∈J. Let Φ be a filterbase onJ such that (2) F ^b→µ,whereF= {{µj}j∈φ:φ∈Φ}. Then for allu > a+b, there existd > a,e > bsuch thatu=d+e.

Then for all f ∈C^b(T ), from (2), Ve,f(µ)∈F hence, there existsφ∈Φ such that {µj}j∈φ⊆Ve,f(µ). Then (3)|µj(f )−µ(f )| ≤ef, for allj∈φ.

From (1), for allj∈J,Vd,f(µj)∈F_j. But, from (3),Vd,f(µj)⊆Vu,f(µ), sothatVu,f∈ F_j, for allj∈φ. Therefore,

Vu,f(µ)∈

j∈φ

F_j⊆

φ∈Φ

j∈φ

F_j. (3.3)

It follows thatµ∈pa+b(

φ∈Φ j∈φF_j), sothat ¯p= {pa:a∈[0,+∞]}is a topological limit tower onM^b(T ).

Definition3.2. We say that ¯p= {pa:a∈[0,+∞]}is thenarrow limit tower on M^b(T ).

Remark3.3. Note thatp0is the narrow convergence structure onM^b(T ). Indeed, F ^p0→µ if and only if for allε >0, for allf∈C^b(T ),Vε,f∈F. But the setsVε,f(µ)= {ν:|µ(f )−ν(f )| ≤εf}form a subbase for the neighbourhood system ofµin the narrow topology onM^b(T ); sothatFis narrowly convergent toµ.

Remark 3.4. If F ^pa→µ then F ^pb→µ, for allb ≥a. Thusp0 is the finest limit structure of ¯p.

Remark3.5. We may interpret inf{a:F ^pa→µ}as the degree of narrow convergence of filterbaseFtoµ.

Remark3.6. For every net(µi)i∈I⊆M^b(T )((I,≤)is a directed set) letF= {{xj: j≥i}:i∈I}be the filterbase generated by(µi)i∈I.

If ¯p= {pa :a∈[0,+∞]}is the narrow limit tower on M^b(T ), then we say that µi a

→µifF ^pa→µ. Therefore,µi a

→µif and only if limsup

i

µi(f )−µ(f )≤a·f, ∀f∈C^b(T ). (3.4) Definition 3.7. We say that a subsetH ⊆M^b(T )is a-relatively compact if for every filterbaseB⊆2^H, there exist a filterFonM^b(T )andµ∈M^b(T )such thatB⊆F andF ^pa→µ.

We remark that H is 0-relatively compact if and only ifH is relatively narrowly compact.

A subset H⊆M^b(T )is bounded if sup{|µ|(T ):µ∈H}<+∞, where|µ|is the variation ofµ. The mappingµ|µ|(T )= µis a norm onM^b(T ).

Let᏷(T )be the family of all compact sets onT; for every bounded setH⊆M^b(T )

we denote

t(H)= inf

K∈᏷(T )sup

µ∈H|µ|(T\K). (3.5) We remark thatt(H)∈[0,+∞)andt(H)=0 if and only ifHis tight. We say thatt(H) is thedegree of tightnessofH.

Now we give an extension of Prohorov’s theorem.

Theorem3.8. Every bounded setH⊆M^b(T )ist(H)-relatively compact.

Proof. LetXbe the Stone-ˇCech compactification ofTandi:T→Xbe the canoni- cal injection ofT inX. We remark thatC^b(X)=C(X)(Xis compact); so(M^b(X),·) is the topological dual of the Banach space(C(X), · )and the narrow topology on M^b(X)is the weak^∗-topology,w^∗, of this dual space.

For every µ∈M^b(T )we define ν=I(µ)∈M^b(X), whereI(ν)(F)=µ(F◦i), fo r everyF∈C(X);ν = |ν|(X)= |µ|(T )= µsothatI:M^b(T )→M^b(X),µI(µ), is an isometric embedding.

LetH be a bounded subset of M^b(T ); then I(H)is a bounded subset ofM^b(X).

ThereforeI(H)is w^∗-relatively compact. For every filterbaseB⊆2^H, I(B)= {I(B): B∈B}is a filterbase onI(H). Sothat, there exists a filterGonM^b(X) w^∗-convergent toa measureν0∈M^b(X)such thatI(B)⊆G. From the definition oft(H), there exists a sequence(Kn)n⊆᏷(T )such that (1)|µ|(T\Kn) < t(H)+1/n, for alln∈N, for all µ∈H. We denoteT0=_∞

n=1Knand (2)X0=_∞

n=1i(Kn)=i(T0).

For everyn∈N,i(Kn)∈᏷(X), sothatX0is a Borel set ofX. On the other hand, for everyn∈N,X\i(Kn)is an open subset ofXsothat the mappingλ|λ|(X\i(Kn))is aw^∗-lower semi-continuous mapping onM^b(X)(see [2, Section 5, Proposition 6(a)]).

FromG ^w∗→ν0, for alln∈N, there existsGn∈Gsuch that (3)|ν0|(X\i(Kn))−1/n <

|λ|(X\i(Kn)), for allλ∈Gn.

The filterbaseBis a filterbase onHsothatB= ∅. LetB0be a set inB; thenI(B0)∈ I(B)⊆G.

For everyn∈N, there existsµn∈B0such thatI(µn)∈Gn(I(B0)∩Gn= ∅). There- fore, from (1) and (3), for everyn∈N,

ν0(X\X0)≤ν0X\i Kn

<I µn

X\i Kn

+1 n

=µn i⁻¹

X\i Kn

+1

n=µnT\Kn +1

n< t(H)+2 n.

(3.6)

Hence (4)|ν0|(X\X0)≤t(H).

Now,Xbeing the Stone-ˇCech compactification ofT, for everyf∈C^b(T )there exists F∈C(X)such thatF◦i=f andF = f(see [10, Theorem 1.4.6, page 25]). Now we defineJ:C^b(T )→Rletting

J(f )=ν0

F·χ_X0

=

X0F dν0. (3.7)

Obviously,Jis a continuous linear mapping onC^b(T ). For everyε >0, from (2), there existsK∈᏷(T )such that|ν0|(X0\i(K)) < εandi(K)⊆X0. Then, for everyg∈C^b(T )

with|g| ≤1 andg|K=0, letG∈C(X)such thatG◦i=g. Therefore, we have J(g)=

ν0

G·χX0

≤ ν0

G·χ_X0\i(K) +

ν0

G·χ_i(K)

≤ν0X0\i(K)

< ε. (3.8)

Hence,Jis a linear mapping satisfying the condition(M)from [2, Section 5, Proposi- tion 5] so that there exists exactly one measureµ0∈M^b(T )such thatµ0(f )=J(f ), for everyf∈C^b(T ). Then we have (5)µ0(f )=ν0(F·χ_X0), for allf∈C^b(T ), whereF is the continuous extension off toX.

Now, for everyf1,...,fn∈C^b(T )withfk>0, for allk=1,...,n, letF1,...,Fn∈ C(X)such thatFk◦i=fkandFk = fk, for everyk=1,...,n.

For allb > t(H), letε=(b−t(H))·min{fk:k=1,...,n}>0. The set G=ⁿ

k=1

λ∈M^b(X):λ Fk

−ν0

Fk< ε

(3.9)

is aw^∗-neighborhood ofν0and sois a member ofG(G ^w∗→ν0). Therefore, for every B∈B,G∩I(B)= ∅(I(B)⊆G). Hence there existsµ∈Bsuch thatI(µ)∈G. Then, for everyk=1,...,n, from (4) and (5), we have

µ fk

−µ0

fk=µ Fk◦i

−µ0

fk=I(µ) Fk

−ν0

Fk·χ_X0

≤I(µ) Fk

−ν0

Fk+ν0

Fk·χX\X0

< ε+Fk·ν0X\X0

< ε+fk·t(H)≤

b−t(H)

·fk+fk·t(H)=b·fk. (3.10) Therefore,µ∈ ⁿ_k=1Vb,fk(µ0). So, for everyb > t(H), n∈N,f1,...,fn∈C^b(T )and B∈B,

n k=1

Vb,fk

µ0

∩B= ∅. (3.11)

LetFbe the filter generated by the filterbase



 n k=1

Vb,fk

µ0

∩B:b > t(H), f1,...,fn∈C^b(T ), B∈B



. (3.12)

ThenB⊆FandF ^pt(H)→µ0, sothatHis at(H)-relatively compact set.

Remark3.9. IfHis tight inM^b(T )thent(H)=0, sothatHis a relatively narrowly compact set and we obtain Prohorov’s theorem.

Remark 3.10. Let a≥b≥0; then, everyb-relatively compact set is a-relatively compact set, also. Therefore, for every bounded setH⊆M^b(T )

t(H),+∞

⊆

a≥0 :Hisa-relatively compact

. (3.13)

Remark3.11. We say thatH⊆M^b₊(T )isa-relatively compactinM^b₊(T )if, for every filterbaseB⊆2^H, there exist a filterFonM^b+(T )andµ∈M^b+(T )such thatB⊆Fand for allb > a, for allf∈C^b(T ),Vb,f(µ)∩M^b+(T )∈F; we say in this case thatF ^pa→µ inM^b₊(T ).

The subset of all positive measures,M^b+(X), is closed in the narrow topology of M^b(X)(see [2, Section 5, Remark 2]) so that, ifH⊆M^b₊(T )is a bounded subset, then I(H)isw^∗-relatively compact inM^b₊(X). Then we follow the proof ofTheorem 3.8 and we obtain that every bounded subsetH⊆M^b+(T )ist(H)-relatively compact in M^b₊(T ). Also, we have

t(H),+∞

⊆

a≥0 :Hisa-relatively compact inM^b+(T )

. (3.14)

In the particular case whereT is locally compact, we have the converse ofTheorem 3.8in the subspaceM^b₊(T ).

Theorem3.12. LetT be a locally compact space andHana-relatively compact set inM^b+(T ); thent(H)≤a.

Proof. We suppose thatHis ana-relatively compact subset ofM^b₊(T )andt(H)= infK∈᏷(T )supµ∈Hµ(T\K) > a. Then, for everyε >0 andK∈᏷(T ), there existsµK∈H such that (1)µK(T\K) > a+ε.

For everyK∈᏷(T )we denoteBK= {µL:L∈᏷(T ), K⊆L}. ThenB= {BK:K∈᏷(T )}

is a filterbase onHsothat there exist a filterFonM^b₊(T )andµ∈M^b₊(T )such that B⊆Fand (2)F ^pa→µ, inM^b₊(T )(seeRemark 3.11). Sinceµis a Radon measure, there existsK0∈᏷(T )such that (3)µ(T\K0) < ε/2.

LetU be a relatively compact neighborhood ofKandf :T →[0,1]a continuous function such that (4)f|K0=0 andf|T\U=1.

We remark thatf∈C^b(T )andf =1. Now letb=a+ε/2> aandf∈C^b(T );

from (2),Vb,f(µ)∈F⊇Bsothat (5)Vb,f(µ)∩BU¯= ∅.

Hence, there existsK∈᏷(T ),K⊇U¯such that (6)|µK(f )−µ(f )| ≤b·f =b.

From (1), (3), (4), and (6) we obtain the following contradiction:

a+ε < µK(T\K)≤µK T\U¯

≤µK(f )≤µ(f )+a+ε 2

≤µ T\K0

+a+ε

2< a+ε.

(3.15)

Remark3.13. IfHis a relatively narrowly compact subset ofM^b+(T )(i.e., 0-relatively compact set), thent(H)=0 sothatH is tight. Therefore, we obtain the converse of Prohorov’s theorem; soTheorem 3.12is an extension of [2, Section 5, Theorem 2].

Remark3.14. FromRemark 3.11andTheorem 3.12, we obtain (in the case of lo- cally compact spaces)

t(H),+∞

=

a≥0 :Hisa-relatively compact inM^b₊(T )

. (3.16)

Example3.15. LetT=Nbe the set of natural numbers andB=ᏼ(N). ThenM^b(N)= ¹(the space of all sequences of real numbers(xn)n∈Nsuch that_∞

n=1|xn|<+∞) and C^b(T )=^∞(the space of all bounded sequences of real numbers). Indeed,

∀x= xn

n∈¹, x:B →R, x(A)=

n∈A

xn,

x(y)=

nxnyn, ∀y=(yn)n∈^∞. (3.17)

Let (x^p)p∈N ⊆M^b(N) and x ∈M^b(N), where x^p =(x^pn)n, for every p∈ Nand x=(xn)n. Thenx^{p a}→xif and only if (1) limsup_p|

n∈N(xn^p−xn)·yn| ≤a·sup_n|yn|, for all(yn)n∈^∞(seeRemark 3.6).

For every bounded setH= {x^p:p∈N} ⊆M^b(N)(2)t(H)=infmsupp_∞

n=m|xn^p|.

Let(xp)p∈N⊆[0,1]be a sequence; we define

xn^p=











1−xp, n=0, xp, n=p, 0, otherwise.

(3.18)

Thenx^p=(xn^p)n∈N∈M^b(N)and, from (2), we obtain t

x^p:p∈N

=limsup

n xn=t. (3.19)

We easily remark thatx^{p t}→x, wherex=(xn)nand

xn=





1, n=0,

0, n >0. (3.20)

FromRemark 3.14, inf{a≥0 :x^{p a}→x} =limsupnxn. If xn→0, then (x^p)p is narrowly convergent tox. In the particular case wherexn=1, for everyn∈N,x^pis the Dirac measureδpandδp 1→δ0.

We remark that

inf

a≥0 :δp a→δ0

=1. (3.21)

4. Probabilistic metric onM¹(T ). Let(T ,d) be a Polish space and letM¹(T )⊆ M^b₊(T )be the subset of all probabilities onT. We say that a net(µi)i∈I⊆M¹(T )is pa-convergent toµ∈M¹(T )(a≥0) if

limsup

i

µi(f )−µ(f )≤a·f, ∀f∈C^b(T ). (4.1) We denote this byµi a→µ. So , ¯p= {pa:a∈[0,+∞]}is the narrow limit tower induced onM¹(T )(seeRemark 3.6). IfX is the Stone-ˇCech compactification ofT, the subset M¹(X)is a compact set ofM^b(X)(see [2, Section 5, Proposition 11]). So, with a similar argument tothat ofRemark 3.11, we deduce that every subsetH⊆M¹(T )ist(H)- relatively compact inM¹(T )(i.e., every net(µi)i∈Ihas a subnetpa-convergent).

Theorem 4.1has a similar proof to that of Portmanteau’s theorem (see [1, Theorem 2.1, Appendix III, Theorem 3]) which we omit.

Theorem4.1. Let(µi)i∈I be a net inM¹(T ),µ∈M¹(T )anda≥0; the following statements are equivalent:

µi a

→µ, (4.2)

limsup

i

µi(f )−µ(f )≤a, ∀f∈C^b(T )withf ≤1, (4.3)

limsup

i µi(F)≤µ(F), ∀F=F¯⊆T , (4.4) liminf

i µi(D)≥µ(D), ∀D=D^◦⊆T , (4.5) limsup

i

µi(A)−µ(A)≤a, ∀A∈ᏮwithµA¯−A^◦

=0. (4.6)

InTheorem 4.1,A¯andA^◦ denote the closure and the interior ofAin the topological space(T ,τd), respectively.

Remark4.2. InTheorem 4.1, we can suppose thata∈[0,1].

Remark4.3. R. Löwen gave a similar result in [7, Theorem 6].

Definition4.4. For everyF=F¯⊆Tandε >0 we denoteF^ε= {t∈T:d(t,F) < ε}.

For everya∈[0,1]we defineLa:M¹(T )×M¹(T )→R+letting La(µ,ν)=inf

ε >0 :µ(F) < ν F^ε

+a+ε,ν(F) < µ F^ε

+a+ε,∀F=F¯⊆T . (4.7) Remark4.5. L0is the metric of Lévy-Prohorov onM¹(T ). Therefore,L0induces the narrow topology onM¹(T )and(M¹(T ),L0)is a Polish space [2, Section 5, Examples 8 and 9].

Remark4.6. The familyᏸ= {La:a∈[0,1]}has the following properties:

La(µ,ν)=0, ∀a≥0⇐⇒µ=ν,

La(µ,ν)=La(ν,µ), ∀µ,ν∈M¹(T ), ∀a∈[0,1],

La+b(µ,ν)≤La(µ,λ)+Lb(λ,ν), ∀µ,ν,λ∈M¹(T ), ∀a,b∈[0,1], La(µ,ν)=sup

b>aLb(µ,ν), ∀µ,ν∈M¹(T ), ∀a∈[0,1).

(4.8)

In [4, Theorem 1] we proved that such a familyᏸis an equivalent gradated form of a probabilistic metric(F,Tm), where, for everyµ,ν∈M¹(T )anda >0,

F(µ,ν)(a)=sup

ε>0inf

F=¯F

min

µ F^a−ε

−ν(F),ν F^a−ε

−µ(F) +1+a

∧1 (4.9) andTm(a,b)=max{a+b−1,0}. For the space of distribution functions, equivalent probabilistic metrics are introduced in [5,6,9].

InTheorem 4.7we compare the narrow limit tower with the convergence structures induced by the family of semi-pseudometricsᏸ= {La:a∈[0,1]}. So, this theorem is an important step to obtain a sequential version ofTheorem 3.8.

Theorem4.7. Let(µi)i∈Ibe a net inM¹(T ),µ∈M¹(T )anda∈[0,1].

IfLa µi,µ

→0, thenµi a→µ, (4.10)

Ifµi a→µ, thenL2a µi,µ

→0. (4.11)

Proof. (i) We suppose thatLa(µi,µ)→0; then, for everyn∈N^∗, there existsin∈I such that, for everyi≥in,La(µi,µ) <1/n. Therefore,

µi(F) < µ F^1/n

+a+1

n, ∀F=F,¯ (4.12)

sothat, for everyF=F¯⊆T, limsup

i µi(F)≤sup

i≥in

µi(F)≤µ F^1/n

+a+1

n. (4.13)

Butµ(F^1/n)→µ(F), sothat limsup_iµi(F)≤µ(F)+a, for allF=F¯. From (4.4) this is equivalent toµi a→µ.

(ii) Let nowµi a

→µ and let ε >0. For everyr >0 and t∈T, let Sr(t)= {s∈T : d(s,t) < r}. ThenSr(t)\S_r^◦(t)⊆ {s∈T :d(s,t)=r} =Cr. ButCr1∩Cr2= ∅, fo r allr1=r2andµ(∪r >0Cr)≤1. It follows that there exists a countable setN⊆(0,+∞) such thatµ(Cr)=0, for allr∈(0,+∞)\N. Therefore,Tbeing separable, there exists a countable family{Srn(tn):n∈N}such that (1)T= ∪^∞₁Srn(tn),µ(Srn(tn)\S_r^◦_n(tn))= 0 andrn< ε/6, for alln∈N.

We denote for alln∈N,Sn=Srn(tn). LetK⊆T be a compact set such thatµ(T\ K) < ε/3 and letp∈Nsuch thatK⊆ ∪^pn=1Sn=A0; then (2)µ(T\A0) < ε/3.

We denoteᏭ= {∪^q_i=1Ski:q∈N, k1,...,kn≤p}; obviously,A0∈Ꮽ. For everyA∈Ꮽ, µ(A\¯ A^◦)=0 sothat, from (4.6),

limsup

i

µi(A)−µ(A)≤a. (4.14)

Therefore there existsi0∈Isuch that, for everyi≥i0andA∈Ꮽ, (3)|µi(A)−µ(A)|<

a+ε/3.

Now, for everyF=F¯⊆T, let AF=

Sn:n≤p, Sn∩F= ∅

∈Ꮽ. (4.15)

Then (4)F⊆AF∪(T\A0),AF⊆F^ε/3.

Indeed,F=(F∩A0)∪(F\A0)⊆AF∪(T\A0). For everyt∈AFthere existsSnsuch thatt∈SnandSn∩F= ∅. Then, from (1),d(t,F)≤2·rn< ε/3, sothatt∈F^ε/3. Then, from (2), (3), and (4), we have

µi(F) < µi AF

+µi T\A0

< µ(F)+a+ε

3+1−µi A0

< µ AF

+a+ε

3+1−µ A0

+a+ε 3=µ

AF +µ

T\A0

+2·a+2ε 3

< µ F^ε/3

+2·a+ε≤µ F^ε

+2·a+ε, µ(F)≤µ

AF +µ

T\A0

< µi AF

+a+ε 3+ε

3

< µi F^ε/3

+a+2ε 3 < µi

F^ε

+2·a+ε,

(4.16)

for everyF=F¯⊆T. ThenL2a(µi,µ)≤ε, for everyi≥i0. ThereforeL2a(µi,µ)→0.

Corollary4.8. LetHbe ana-relatively compact subset inM¹(T ); then, for every sequence(µn)n∈N⊆H, there exist a subsequence(µkn)n∈N andµ∈M¹(T )such that µkn 2·a→µ.

Proof. For every sequence(µn)n∈N ⊆H, there exist a subnet (µn_i)i∈I and µ∈ M¹(T ) such thatµn_i a→µ. From (4.11),L2a(µn_i,µ)→0. So, for every p∈N, there existsip∈Isuch thatnip≥pandL2a(µn_ip,µ) <1/p.

Therefore, we can choose a subsequence(µn)n∈Nof(µn)n∈Nsuch thatL2a(µn,µ)→ 0. From (4.10) it follows thatµ_n ^2·a→µ.

Now we are able to give the sequential version ofTheorem 3.8.

Theorem4.9. Let(µn)n∈N⊆M¹(T )andt=t({µn:n∈N})be the degree of tight- ness of(µn)n∈N. Then there exist a subsequence(µkn)n∈N andµ∈M¹(T )such that

µkn 2·t→µ. (4.17)

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Liviu C. Florescu: Faculty of Mathematics, University Al. I. Cuza, Carol I,11, RO-6600, Romania

E-mail address:[email protected]