COMPLETION OF PROBABILISTIC NORMED SPACES

Internat. J. Math. & Math. Sci.

VOL. 18 NO. 4 (1995) 649-652

649

COMPLETION OF PROBABILISTIC NORMED SPACES

BERNARDO LAFUERZA

GUILLIN JOSi

ANTONIORODRiGUE.ZLALLENA

Departamento

deMatemtica, Aplicada

Universidadd,eAhneria 04120 Almeria, Spain

and CARLOSEMPI Dipartimento di Matematica

Universit di Lecce 73100

Lecce,

Italy

(Received

^December6, 1993 and in revised form

January

28,

1994)

ABSTRACT. We prove that everyprobabilistic normed space, either according to the original definition given by

erstnev,

or according tothe recentone introducedby Alsina, Schweizer and Sklar,has acompletion.

KEY

WORDS

AND

PHRASES. Probabilistic Normed

Spaces,

completion, triangle ^function, L6vydistance.

1991 AMS

SUBJECT CLASSIFICATION

CODES. 54E70, 60E99.

1.

INTRODUCTION.

As iswellknown,arealorcomplex normed linear space admits^acompletion, namely, given a normed linear space

(Y, I1" ^),

^{there exists another} linear space

(V’, I1" ^I1")

^{such that}

^V"

^is

isometrictoadensesubspaceof

V.

It wasproved by Mutari

[2],

Sherwood

([7], [8])

and Sempi

[5]

^{that a} probabilistic metric space has a completion.

Here

we answer in the positive the natural question of whether a probabilistic normed space has a completion. In fact, there are twodefinitions ofprobabilistic normedspace

(= Pg-space)"

theoriginalone by

erstnev ([6],

^{but see}

[3]

^{for a}presentation in agreement with our

notation),

^{and a}more recent one by Alsina, Schweizer and Sklar

(see [1]).

The proofwill be givenin both cases. For the notation and the concepts used we refer to the bookbySchweizer andSklar

[3];

^{weshallwrited.f. for}distribution function.

Accordingto

erstnev,

a

PN-space

is atriple

(V,

v,

’),

^whereVis areallinear space; ris a triangle

function ([3],

section

7.1),

^{i.e., a} binary operation ^on A

+,

^the space of distance distribution functions, that is commutative, associative and nondecreasing in each variable and whichhas the d.f. _e0asidentity, i.e.,

(a) VF,

^{G A+}

r(F,G) r(G,F);

(b) VF, G,H

^A+

v(F,r(G,H) T(r(F,G),H);

(c) VHA +F_<G=> v(F, H) _< r(G. H);

(d) ^VF

^{A +}

’(F, e0)

F.

Here

_{0 is}the d.f. definedby

f0, ifx<:_0,

0(z): =/1,

^ifz>0;

650 B. L.

GUILLN,

J. A. R. LALLENA AND C. SEMPI

v is the probabilistic norm, i.e., v is a nap from V into ^/k+ that satisfies the following conditions:

(N.1)

t,(p) e0, if, and only if,p 0,where0is the nullvectorof

V;

(N.2)

VzE

I

+,aE

R,

witha

#

^{0 v(ap)(x)}

^{u(p)(z/[a [);}

(N.3) ^Vp,

q

,z

_t,(p

+

^q)

^>_ r[(p), v(q)].

Inbothdefinitionsthe trianglefunctionisassumedtobecontinuous.

The space A + canbe metrizedby different metrics

([9], [3], [4], [0]),

^{but we shallusehere}

the modified

LSvy

metric d_L

[3].

2.

MAIN

RESULTS.

THEOREM I.

Every PN-space (V,u,r)

^hasacompletion,viz. is isometric toadenselinear subspaceofacomplete

PN-space (V’, v’, r).

PROOF.

Only the steps needed toconplement the treatment in

[7]

^and

IS]

will be given.

Now

V"

is theset of equivalence classes ofCauchysequences of elements ofV.

In

order to prove that

V"

is a linear space, let

p"

and

q"

be elements of V" and let

{p,,}

and

{q,}

^be Cauchy sequences of elements ofVwith

{p,}

6

p"

and

{q,}

6

q’.

Since V is a linear space, onehas,for everyn

N,

p,

+

^{q,EV. Wewish toshow thatit ispossibleto define asumof}

^p"

^and

^q"

^insuch

awaythat

p" + ^q"

^{6V’. Since}

(V, q, ’),

^with(p,q):

v(p

q)^{is a}probabilisticmetric space

([3],

Theorem

15.1.2),

^onehas,ifnandmarelarg enough,

q(P,, +

^q,,,P,,,

+ ^{qm) u((p,} + ^q,,)

^(p,

+ ^q,))

v((p, p,,) + ^{(q, q,))} (because

^of

(N.3))

> ’[u(p,- p,,), v(q,,- q,)l.

Takingintoaccount

Lemma

4.3.4in

[3],

^onehas

dL(q(P,, +

^q,,P,

+ q,,), 0) < dL(r[u(P,,- p,,),v(q,- q,,)], 0) dL([(P,, P,,,),(q,,q,,)],o).

The continuity of both

d

^{and r ensures that, when both m and rt tend to infinity,}

q(p,, +

^q,,,p,,,

+ ^q,,,) -

^{e0. Thus}

^{{p,, + q,}}

^{is aCauchysequence}

^and,

^asaconsequence, it belongs toanelementof

V’,

^{which will}be denotedby

r’.

Thenwedefine

p" + ^q"

^{r’. This}definition does notdependontheelements of

p"

and

q"

selected, for,if

{p,,}, {p,}

6

p"

and

{q,}, {q,} q’,

then

q(p,, + ^q,,,p, + q) v(p,,-p,,q,,-q,) > r[v(p,,- p,),v(q,,-q,)] ’[q(p,,,p,),(q,,,q,)],

sothat

dL((p, + ^q,,p, + ^q,), 0) < dL (r[(P=,P,),(q,,q,)],O)

Since both d_Landrarecontinuousweobtain

(p,, +

^q,,,

^p, + q,)

^-%0,i.e.,

{p,, + ^{q,,} {p,} + ^q,}.

Thus thesumdefined above isagood definition,whichimmediatelysatisfies the properties ofan abelian group.

For every a

R,

and for every Cauchy sequence

{p,,}

of elements of

V,

also

{a p,,}

is a

Cauchysequenceof elements ofV. This is obvious ifa 0. Ifa

#

^{0,onehas,foreveryx}

^>

^0,

q(p., ,a)() (p. p.)(*) (p. p-)(/I I)

r(p., p,,,)(/l I),

and this lends to

^{ap,}

^{is a Cauchy}1,^sequence;for every x^let

>

^us0,^{denoteas n andby u"}mthe element oftend to infinity, i.e.,

V"

towhich

q(ap,,ap,,)

it belongs. Then

- ^o.

^Thuswe define

ap’=

u’. This isagainagooddefinition;infact,let

{p,,}, {p,}

E

p’.

Then

COMPLETION OF PROBABILISTIC NORMED SPACES 651 which tends to for all z

>

^{0when nx,whence}

{op,} {ap}.

Therefore it is inmediateto conclude that V"isalinearspace. All that isleft toshowisthat thedistanced.f.

"

^derivesfrom

aprobabilistic normv"onV’. Forevery

p"

E1,’" set, if

{p,,}

E

p"

with p, Vforeveryn I1 v’(p’): ’(p’,tg") l,m,, (p,,

d)

lim,

t,(p.). (2.1)

Thus

’(

p’,

q’) lira, (p,,q,) tim, v(p, q,) v’(p" q’).

It

isnowaneasytasktoverify that

v"

doesindeedfulfill conditions

(N.1), (N.2)

^and

(N.3).

¹³ We now turn to theproofof the analogousresult for

PN-spaces

accordingto the definition given in

[1].

^This latter differs from theone given above in that condition

(N.2)

^is replaced by the weakerone

(N.2") Vp

CV

t,(-

p)

u(p);

andanew oneisadded:

(N.4)

^VaC

[0,1]Vp

CV v(p)

< r’[v(ap),v((1 cr)p)].

Then a

PN-space

is a quadruple

(V,v,r, r*),

^where

V,

as

above,

is a reallinear space,

r,r"

arecontinuoustrianglefunctionsand

v’:

VA+ isamap such thatconditions

(N.1), (N.2"), (N.3)

and

(N.4)

^{aresatisfied.}

The last part ofthis note isentirely devotedto

PN-spaces

accordingto this latter definition.

LEMMA

2. Let

(V,v,r,r*)

bea

PN-space

and let h and k be tworeal constants such that 0<h<k; ^then

Vp, q

V (kp,kq)

<

(hp,

hq),

where

q(p, q): v(p

q).

PROOF. Thereis

A

q

[0,1]

such that h Ak. Then

5(kp, kq) (kp kq) [k(p q)] <

< r*[t,[Ak(p ^q)], t,[(1 A)k(p q)]] ^<

< r*[[Ak(p ^q)], 0] ^{t[Ak(p q)] [h(p q)] (hp, hq).}

¹³

’I’ItIOtM 3.

Every PN-space (V,u,r,r’)

has a completion, viz. is isometric to a dense linearsubspaceofacomplete

PN-space (V’, r,’,

r,

r’).

PROOF.

Exactly as inthe proofof Theorem 1, one canprove that if both

p"

and

q"

belong to

V’,

then

p’+ q" V’. However,

one can nolongerusethe sameproofof the fact that. if^cEN and

p"

E

V"

then

p" V’,

becauserecoursewasmade toproperty

(N2)

^{which now}may well not hold.

Now

assume

R

and

p" V’,

let

{p,}

E

p"

and consider the sequence

{cp,,}.

As a first step, we shall prove that it is a Cauchy sequence in V. This is obviously true for c 0 and

ct 1. Because^of

(N.2"),

^{it suffices}to consideronly thecase c

>

0. Nowassumethat

{cp,,}

isa

Cauchysequence forcr 0,1, ,k-

l(k I1).

^Then

(kp,kp,,) [k(p,- p,,)] _> u(p ^p,), t,[(k- 1)(p,,- p,)]]

(p,,p,,),q((k-

^1)p,,

(k 1)p,)].

Sinceris continuousand

tim q(p,,p,)= ^lm,

;((k-

1)p,,,(k- 1)p,,)=₀

652 B. L.

GUILLN,

J. A. R. LALLENA AND C. SEMPI

it follows that, also

{ap,,}

^{is a} Cauchy sequence for every aE

Z

_+. If a is positive, but not integer, thereexists

"

^El+ such that k<ⁿ <^k

+

^{1. Lemma2nowgives}

henceit isimmediate to conclude that {op,,}isa Cauchysequence for every o6R_+. Thus there exists element u"6V"suchthat

{op,,}

6u’. Let us defineu"

op’. In

order to check that this is a gd definition, let

{p.}{p:}.

If

a6[O, 1],

^{it follows from} Lemma 2 that

(p.,p)

(op.,op:); since, by assumption

(p.,p.)eo,

also(op,,,ap.)So. If^a k6 +,

above,one has

(’p,,,kp:) v[k(p. p,:)] v(p.

^p:,),

[(k-

1)(p.

p:)]]

(p,,,p:,),((],

^1)p,,,(k-

1)p:)].

The same gumcnt above yields

{kp,} {’p:,}

^forevery k _+. Again, from this it is easy toobtain that,forevery Uonehs

{ap,} {p}.

Therefore

V"

is a linear space. Only conditions

(N.2")

^and

(N.4)

remainnow to beproved.

Proceeding above, let

p"

V"and let

{p,,}

be aCauchysequenceof elements ofVthatbelongs to

p’;

then

{ p.} p’.

Sinceu"is defined by

(2.1),

onehas,onaccount of

(N.2"),

whichholds for

Moreover,

for every

[0,1],

^oneh,because r*iscontinuous,

’(F) i. ^(p.) ,. ^-[(.p.), (( -)p.)]

’[.’(-F),.(( -)F)].

ACKNOWLEDGEMENT. The authors aregrateful to their institutionsfor having made their cperation possible. The third authoris a member ofthe ItMian

G.N.A.F.A.-C.N.R.;

he gratefully acknowledgespartiM supportfromfunds providedbytheItalianM.U.R.S.T..

REFERENC.

^E$

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B. and

SKLAR, A.,

On the definition of a probabilistic normed space, AequationsMath. 46

(1993),

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The completion of random metric spaces, Kazan Gos. Univ.

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SKLAR, A.,

Probabilistic Metric Spaces, Elsevier North-Holland, New York, 1983.

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