Int. Journal of Math. Analysis, Vol. 8, 2014, no. 10, 455 - 461 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2014.4120

On Uniform Limit Theorem and Completion of Probabilistic

Abderrahim Mbarki

National school of Applied Sciences P.O. Box 669, Oujda University, Morocco MATSI Laboratory

Abedelmalek Ouahab

Department of Mathematics Oujda university, 60000 Oujda Morocco MATSI Laboratory

Rachid Naciri

MATSI Laboratory Oujda university, 60000 Oujda Morocco

Copyright c 2014 Abderrahim Mbarki et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribu- tion, and reproduction in any medium, provided the original work is properly cited. Abstract A necessary and sufficient condition for a probabilistic metric space to be complete is given and the uniform limit theorem [2] is generalized to probabilistic metric space. Mathematics Subject Classification: 54A40, 54E50, 54D65

Keywords: Uniform Limit Theorem, Completion of PM space

1 Introduction and Preliminaries

Our terminology and notation for probabilistic metric spaces conform of that B. Schweizer and A. Sklar [3, 4]. A nonnegative real function f defined on 456 A. Mbarki, A Ouahab and R. Naciri

R+ ∪ {∞} is called a distance distribution function (briefly, a d.d.f.) if it is nondecreasing, left continuous on (0, ∞). with f(0) = 0 and f(∞) = 1. The set of all d.d.f’s will be denoted by Δ+; and the set of all f in Δ+ for which + + lims→∞ f(s)=1byD .Fora ∈ [0, ∞), the element a ∈ D is defined as  0ifx ≤ a εa(x)= 1ifx>a and  0, 0 ≤ x<∞, ε∞(x)= 1,x= ∞. By setting f ≤ g whenever f(x) ≤ g(x) for all x ∈ [0, ∞), one introduces a + + natural ordering in  , in this ordering the d.d.f 0 is the maximal of  . + Convergence in  is assumed to be weakly convergence, i.e fn → f if and only if fn(x) → f(x) at each continuity point x of f.

Definition 1.1 Let f and g be in Δ+ , let h be in (0, 1], and let (f,g; h) denote the condition 0 ≤ g(x) ≤ f(x + h)+h 1 for all x in (0, h ). + + The modified L´evy distance is the function dL defined on Δ × Δ by

dL(f,g) = inf{h : both (f,g; h) and (g, f; h) hold}.

+ Note that for any f and g in Δ , both (f,g; 1) and (g, f; 1) hold, whence dL is well-defined function and dL(f,g) ≤ 1.

Lemma 1.2 [3] For any f in Δ+

dL(f,ε0) = inf{h :(f,ε0; h) holds}

= inf{h : lims→h+ f(s) > 1 − h}; and for any t>0, f(t) > 1 − t iff dL(f,ε0)

A t-norm is a binary operation on [0, 1] which is associative, commutative, nondecreasing in each place and has 1 as identity. Three typical examples of continuous t-norms are:

Tp(a, b)=ab , TM (a, b)=Min(a, b) and TL(a, b)=max{a + b − 1, 0}. Uniform limit theorem and completion of PM space 457

A triangle function is a mapping τ : + ×+ →+ that is associative, commutative, nondecreasing in each place and has 0 as identity. Typical continuous triangle function is

τT (f,g)(t)=sup{T (f(u),g(v)) : u + v = t}. where T is a continuous t-norm.

Definition 1.3 A probabilistic metric space (briefly,PM space) is a triple (X, F, τ) where X is a nonempty set , F is a function from X × X into +, τ is a continuous triangle function , and the following conditions are satisfied for all x, y, z in X, (i) F (x, x)=ε0. (ii) F (x, y) = ε0 if x = y. (iii) F (x, y)=F (y, x). (iv) F (x, z) ≥ τ(F (x, y),F(y, z)).

Throughout this paper, we shall frequently denoted F (x, y)byFxy.

Definition 1.4 Let (M,F) be a probabilistic semimetric space (i.e. (i), (ii) and (iii) are satisfied). For p in M and t>0, the strong t-neighborhood of p is the set Np(t)={q ∈ M : Fpq(t) > 1 − t}. and the strong neighborhood system for M is

{Np(t); p ∈ M, t > 0}.

Lemma 1.5 [3] Let (M,F,τ) be a PM space. If τ is continuous, then the family Υ consisting of ∅ and all unions of elements of strong neighborhood system for M determines a Hausdorff topology for M.

An immediate consequence of Lemma 1.5 is that the family {Np(t):t>0} is a neighborhood system

Definition 1.6 [3] Let {xn} be a sequence in a PM space (X, F, τ). Then (i) The sequence {xn} is said to be convergent to x ∈ X if for all t>0 there exist a positif integer N such that Fxnx(t) > 1 − t for n ≥ N. (ii) The sequence {xn} is called a Cauchy sequence if for all t>0 there exist a positif integer N such that Fxnxm (t) > 1 − t for n, m ≥ N. (iii) APMspace(X, F, τ) is said to be complete if each Cauchy sequence in X is convergent to some point x in X. 458 A. Mbarki, A Ouahab and R. Naciri

Lemma 1.7 [3] Let {xn} be a sequence in a PM space (X, F, τ). Then

(i) The sequence {xn} to be convergent to x ∈ X iff limn→∞ Fxnx = ε0.

(ii) The sequence {xn} is a Cauchy sequence iff limn,m→∞ Fxnxm = ε0.

Lemma 1.8 [3] If (X, F, τ) is a PM space, (xn) and (yn) are sequences such that xn → x and yn → y, then Fxnyn → Fxy.

Here and in the sequel, when we speak about a probabilistic metric space (M,F,τ), we always assume that τ is continuous and M be endowed with the topology Υ. Recall the Definition of probabilistic diameter of a set in PM space.

Definition 1.9 [3] Let A a nonempty subset of a PM space (X, F, τ). The probabilistic diameter of A is the function defined on [0, ∞] by DA(∞)=1 − and DA(t)=L ϕA(t) on [0, ∞). Where

ϕA(t) = inf{Fpq(t)|p, q in A}

+ It is immediate that DA is in Δ for any A ⊂ M.

Lemma 1.10 [3] The probabilistic diameter DA has the following proper- ties: i. DA = ε0 iff A is a singleton set. ii. If A ⊂ B, then DA ≥ DB. iii. For any p, q ∈ A, Fpq ≥ DA. iv. If A = {p, q}, then DA = Fpq. v. If A ∩ B is nonempty, then DA∪B ≥ τ(DA,DB). vi. DA = DA, where A is the strong closure of A.

The diameter of a nonempty set A in a metric space is either finite or infinite; accordingly, A is either bounded or unbounded. In a PM space, on the other hand , there are three distinct possibilities. These are captured in

Definition 1.11 [3] A nonempty set A in a PM space is + (i) Bounded if DA is in D . (ii) Semi-bounded if 0 < limt→∞ DA(t) < 1. (iii) Unbounded if limt→∞ DA(t)=0.

Example 1.12 Let (M,d) be a metric space. Define F d : M × M → Δ+ the probalistic metric induced by d as

d Fpq = εd(p,q). Uniform limit theorem and completion of PM space 459

d It is easy to check that (M,F ,τMin) is a PM (Menger) space, and

Np(t)={q ∈ M : d(p, q)

DA = εdiam(A), where diam(A)=sup{d(p, q): p, q ∈ A}.

Let us now state our results.

2 Completion of probabilistic metric space

Theorem 2.1 APMspace(M,F,τ) is complete if and only if for each creasing sequence of nonempty closed sets {Fn} such that DFn −→ 0 have nonempty intersection.

Proof. Let {xn} be a Cauchy sequence in M. Consider An = {xi : i ≥ n}.It { } is obvious that An that is a creasing sequence. Now we claim that DAn ε0. For given s>0. Let s>t>0, since {xn} is Cauchy sequence then ∀>0 there exists N ∈ IN such that for all n, p ≥ NFxnxp (t) ≥ 1 − . So

ϕAn (t) ≥ 1 − . for any n ≥ N. It follows that

 ϕAn (t ) ≥ 1 − . for any s>t >t>0. Letting t → s, we obtain

DAn (s) ≥ 1 − .

It follows from Lemma 1. 10 that → DAn (s)=DAn (s) 1.

Since s is arbitrarily positive number. This clearly means that D ε0.   An ∅ ∈ ≥ Hence by hypothesis n An = . Take x n An then Fxnx DAn .So

Fxnx ε0, this means that xn → x as n →∞. Hence (M,F,τ) is complete PM space Conversely, suppose that (M,F,τ) is complete PM space and {Fn} is a creasing sequence of nonempty closed sets of M such that DFn 0. Since Fn = ∅ 460 A. Mbarki, A Ouahab and R. Naciri

there exists xn ∈ Fn. Continuing in this manner we can construct by induction a sequence {xn} such that for each n ∈ IN ,xn ∈ Fn. Next we claim that {xn} is a Cauchy sequence. Indeed, lets n>p>0, then Fxnxp ≥ DFp . which implies that Fxnxp ε0 as n, p →∞, this means that {xn} is a Cauchy sequence, since (M,F,τ) is complete PM space then there exists x ∈ M such that xn → x. ∈ ⊂ ≥ ∈ Now for each fixed n, xk  Fk Fn for all k n. Hence x Fn since Fn is . Therefore x ∈ n Fn. This completes our proof.

As consequence of Theorem 2.1 and Example 1.12 we have Corollary 2.2 A necessary and sufficient condition that a metric space (M,d) be complete if that every nested sequence of nonempty closed sets {Fn} with diameter tending to zero have nonempty intersection.

3 Uniform Limit Theorem

In order to state the uniform limit theorem in PM space, let us to recall the following definition Definition 3.1 Let M be any nonempty set and (Y,F,τ) a PM space. Then a sequence {fn} of functions from M to Y is said to converge uniformly to a function f from M to Y if given t>0 there exists n0 ∈ IN such that

dL(Ffn(x)f(x),ε0)

Theorem 3.2 Let fn : M → Y be a sequence of continuous functions from a M to a PM space Y .If{fn} converges uniformly to f then f is continuous.

+ Proof. Firstly, since τ is uniformly continuous on (Δ ,dL) then, for >0 there is η > 0 such that dL(G, ε0) <η,dL(Q, ε0) <η and dL(R, ε0) <η implies that dL(τ(G, τ(R, Q)),ε0) <. Let V be any open set in Y . Given −1 x0 ∈ f (V ) and let y0 = f(x0). Since V is open, there is >0 such that

Ny0 () ⊂ V . On the other hand, since {fn} converges uniformly to f, given

η > 0 there exists n0 ∈ IN such that dL(Ffn(x)f(x),ε0) <η for all n ≥ n0. Since, for all n ∈ IN , fn is continuous, we can find a neighborhood U of x0, for ≥ ⊂ a fixed N n0, such that fN (U) NfN (x0)(η). Hence dL(FfN (x)fN (x0),ε0) <η for all x ∈ U. which implies that

dL(τ(Ff(x)fN (x),τ(FfN (x)fN (x0),FfN (x−0)f(x0))),ε0) <, for all x ∈ U. It follows from Lemma 1. 2 that ≤ dL(Ff(x)f(x0),ε0) dL(τ(Ff(x)fN (x),τ(FfN (x)fN (x0),FfN (x0)f(x0))),ε0) ≤ . Uniform limit theorem and completion of PM space 461

Thus f(x) ∈ Ny0 () for all x ∈ U. Therefore f(U) ⊂ V and hence f is contin- uous.

References

[1] Menger. K. Statistical metrics. Proc. Nat. Acad. Sci. 28, (1942), 535–7.

[2] J.R. Munkres, Topology- A First Course. Prentice-Hall, Delhi, (1999).

[3] Schweizer B. and A.Sklar, Probabilistic Metric Spaces, North-Holland Series in Probability and Applied Mathimatics, 5, (1983).

[4] Schweizer B. and A.Sklar, Statistical metric spaces. Pacific J. Math. 10 (1960), 313–34.

Received: January 25, 2014