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 Metric Space 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, distribution, 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 lim s f(s) =1byD +.Fora [0, ), the element ɛ a D + is defined as ε a (x) = { 0 if x a 1 if x>a and ε (x) = { 0, 0 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 ɛ 0 is the maximal of +. Convergence in + is assumed to be weakly convergence, i.e f n f if and only if f n (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 for all x in (0, 1 h ). The modified Lévy distance is the function d L defined on Δ + Δ + by d L (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 d L is well-defined function and d L (f,g) 1. Lemma 1.2 [3] For any f in Δ + d L (f,ε 0 ) = inf{h : (f,ε 0 ; h) holds} = inf{h : lim s h +f(s) > 1 h}; and for any t>0, f(t) > 1 t iff d L (f,ε 0 ) <t. If f and g are in Δ + and f g, then d L (g, ε 0 ) d L (f,ε 0 ). 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: T p (a, b) =ab, T M (a, b) =Min(a, b) and T L (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 ɛ 0 as identity. Typical continuous triangle function is where T is a continuous t-norm. τ T (f,g)(t) =sup{t (f(u),g(v)) : u + v = t}. 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) =ε 0. (ii) F (x, y) ε 0 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) byf xy. 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 N p (t) ={q M : F pq (t) > 1 t}. and the strong neighborhood system for M is {N p (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 {N p (t) :t>0} is a neighborhood system Definition 1.6 [3] Let {x n } be a sequence in a PM space (X, F, τ). Then (i) The sequence {x n } is said to be convergent to x X if for all t>0 there exist a positif integer N such that F xnx(t) > 1 t for n N. (ii) The sequence {x n } is called a Cauchy sequence if for all t>0 there exist a positif integer N such that F xnx m (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 {x n } be a sequence in a PM space (X, F, τ). Then (i) The sequence {x n } to be convergent to x X iff lim n F xnx = ε 0. (ii) The sequence {x n } is a Cauchy sequence iff lim n,m F xnx m = ε 0. Lemma 1.8 [3] If (X, F, τ) is a PM space, (x n ) and (y n ) are sequences such that x n x and y n y, then F xny n F xy. 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 D A ( ) =1 and D A (t) =L ϕ A (t) on [0, ). Where ϕ A (t) = inf{f pq (t) p, q in A} It is immediate that D A is in Δ + for any A M. Lemma 1.10 [3] The probabilistic diameter D A has the following properties: i. D A = ε 0 iff A is a singleton set. ii. If A B, then D A D B. iii. For any p, q A, F pq D A. iv. If A = {p, q}, then D A = F pq. v. If A B is nonempty, then D A B τ(d A,D B ). vi. D A = D A, 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 D A is in D +. (ii) Semi-bounded if 0 < lim t D A (t) < 1. (iii) Unbounded if lim t D A (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 F d pq = ε d(p,q).
Uniform limit theorem and completion of PM space 459 It is easy to check that (M,F d,τ Min ) is a PM (Menger) space, and N p (t) ={q M : d(p, q) <t}, for t in (0, 1). So (M,F,τ Min ) is a complete PM space if and only if (M,d) is a complete metric space. Moreover, for A a nonempty subset of M we have D A = ε diam(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 {F n } such that D Fn ɛ 0 have nonempty intersection. Proof. Let {x n } be a Cauchy sequence in M. Consider A n = {x i : i n}. It is obvious that {A n } that is a creasing sequence. Now we claim that D An ε 0. For given s>0. Let s>t>0, since {x n } is Cauchy sequence then ɛ >0 there exists N IN such that for all n, p NF xnx p (t) 1 ɛ. So for any n N. It follows that ϕ An (t) 1 ɛ. ϕ An (t ) 1 ɛ. for any s>t >t>0. Letting t s, we obtain It follows from Lemma 1. 10 that D An (s) 1 ɛ. D An (s) =D An (s) 1. Since s is arbitrarily positive number. This clearly means that D An ε 0. Hence by hypothesis n A n. Take x n A n then F xnx D An. So F xnx ε 0, this means that x n x as n. Hence (M,F,τ) is complete PM space Conversely, suppose that (M,F,τ) is complete PM space and {F n } is a creasing sequence of nonempty closed sets of M such that D Fn ɛ 0. Since F n
460 A. Mbarki, A Ouahab and R. Naciri there exists x n F n. Continuing in this manner we can construct by induction a sequence {x n } such that for each n IN,x n F n. Next we claim that {x n } is a Cauchy sequence. Indeed, lets n>p>0, then F xnx p D Fp. which implies that F xnx p ε 0 as n, p, this means that {x n } is a Cauchy sequence, since (M,F,τ) is complete PM space then there exists x M such that x n x. Now for each fixed n, x k F k F n for all k n. Hence x F n since F n is closed set. Therefore x n F n. 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 {F n } 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 {f n } 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 n 0 IN such that d L (F fn(x)f(x),ε 0 ) <t for all n n 0 and for all x M. Theorem 3.2 Let f n : M Y be a sequence of continuous functions from a topological space M to a PM space Y.If{f n } converges uniformly to f then f is continuous. Proof. Firstly, since τ is uniformly continuous on (Δ +,d L ) then, for ɛ>0 there is η ɛ > 0 such that d L (G, ε 0 ) <η ɛ, d L (Q, ε 0 ) <η ɛ and d L (R, ε 0 ) <η ɛ implies that d L (τ(g, τ(r, Q)),ε 0 ) <ɛ. Let V be any open set in Y. Given x 0 f 1 (V ) and let y 0 = f(x 0 ). Since V is open, there is ɛ>0such that N y0 (ɛ) V. On the other hand, since {f n } converges uniformly to f, given η ɛ > 0 there exists n 0 IN such that d L (F fn(x)f(x),ε 0 ) <η ɛ for all n n 0. Since, for all n IN, f n is continuous, we can find a neighborhood U of x 0, for a fixed N n 0, such that f N (U) N fn (x 0 )(η ɛ ). Hence d L (F fn (x)f N (x 0 ),ε 0 ) <η ɛ for all x U. which implies that d L (τ(f f(x)fn (x),τ(f fn (x)f N (x 0 ),F fn (x 0)f(x 0 ))),ε 0 ) <ɛ, for all x U. It follows from Lemma 1. 2 that d L (F f(x)f(x0 ),ε 0 ) d L (τ(f f(x)fn (x),τ(f fn (x)f N (x 0 ),F fn (x 0 )f(x 0 ))),ε 0 ) ɛ.
Uniform limit theorem and completion of PM space 461 Thus f(x) N y0 (ɛ) for all x U. Therefore f(u) V and hence f is continuous. 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