aa r X i v : . [ m a t h . F A ] A p r PROBABILISTIC ARZELA-ASCOLI THEOREM
MOHAMMED BACHIR, BRUNO NAZARET
Abstract.
We prove that, in the space of all probabilistic continuous func-tions from a probabilistic metric space G to the set ∆ + of all cumulativedistribution functions vanishing at 0, the space of all 1-Lipschitz functionsis compact if and only if the space G is compact. This gives a probabilisticArzela-Ascoli type Theorem. Keywords:
Probabilistic metric space; Probabilistic 1-Lipschitz map; Proba-bilistic Arzela-Ascoli type Theorem. msc:
Contents Introduction
Classical Notions of Probabilistic Metric Spaces.
Some Properties of Probabilistic continuous and 1-Lipschitzmap
The main result: Probabilistic Arzela-Ascoli theorem
Introduction
The general concept of probabilistic metric spaces was introduced by K. Menger,who dealt with probabilistic geometry [9], [10], [11]. The decisive influence on thedevelopment of the theory of probabilistic metric spaces is due to B. Schweizer andA. Sklar and their coworkers in several papers [13], [14], [15], [16], see also [18], [19][6], [4] and [5]. For more informations about this theory we refeer to the excellentmonograph [12].Recently, the first author introduced in [1] a natural concept of probabilistic Lip-schitz maps defined from a probabilistic metric space G into the set of all cumulativedistribution functions that vanish at 0, classically denoted by ∆ + . In particular,the introduction of the space of all probabilistic 1-Lipschitz maps provides a newmethod for the completion of probabilistic metric spaces extending a result of H. Date : 05/03/2019.1991
Mathematics Subject Classification.
Sherwood in [19]. It also leads to a probabilistic version of the Banach-Stone the-orem (see for instance [1] and [2]).In this paper, we investigate new properties of the space of all 1-Lipschitz prob-abilistic maps defined on probabilistic compact metric spaces. More precisely, wegive in our main Theorem 1 a probabilistic Arzela-Ascoli theorem, proving thatthe space of all 1-Lipschitz probabilistic maps is a compact subset of the spaceof all probabilistic continuous functions equipped with the uniform modified L´evydistance that we introduce in the next section.This paper is organized as follows. In Section 2, we recall classical results andnotions about probabilistic metric spaces, triangle functions, the L´evy distance andthe weak convergence. In Section 3, we recal the notion of probabilistic Lipschitzmaps and probabilistic continuous functions introduced in [1]. We also give someproperties related to these notions. In Section 4, we prove the Theorem 1, which isour main result.2.
Classical Notions of Probabilistic Metric Spaces.
In this section, we recall some general well know definitions and concepts aboutprobabilistic metric spaces, as they can be found in [4], [12], [6] and [7].A (cumulative) distribution function is a function F : [ −∞ , + ∞ ] −→ [0 , F ( −∞ ) = 0; F (+ ∞ ) = 1. The set of alldistribution functions satisfying F (0) = 0 will be denoted by ∆ + . For F, G ∈ ∆ + ,the relation F ≤ G is understood as F ( t ) ≤ G ( t ), for t ∈ R . For all a ∈ R , thedistribution function H a is defined by H a ( t ) = (cid:26) t ≤ a t > a, and, for a = + ∞ , by H ∞ ( t ) = (cid:26) t ∈ [ −∞ , + ∞ [1 if t = + ∞ It is well known that (∆ + , ≤ ) is a complete lattice with respectively H ∞ and H as minimal and maximal element. Thus, for any nonempty set I and any familly( F i ) i ∈ I of distributions in ∆ + , the function F = sup i ∈ I F i is also an element of ∆ + .2.1. Triangle function and probabilistic metric space.Definition 1.
A binary operation τ on ∆ + is called a triangle function if andonly if it is commutative, associative, non-decreasing in each place, and has H asneutral element. In other words: (i) τ ( F, L ) ∈ ∆ + for all F, L ∈ ∆ + . (ii) τ ( F, L ) = τ ( L, F ) for all F, L ∈ ∆ + . (iii) τ ( F, τ ( L, K )) = τ ( τ ( F, L ) , K ) , for all F, L, K ∈ ∆ + . (iv) τ ( F, H ) = F for all F ∈ ∆ + . (v) F ≤ L = ⇒ τ ( F, K ) ≤ τ ( L, K ) for all F, L, K ∈ ∆ + . ROBABILISTIC ARZELA-ASCOLI THEOREM 3
Definition 2.
A triangle function τ is said to be sup-continuous (see for instance [3] ) if for all nonempty set I and all familly ( F i ) i ∈ I of distributions in ∆ + and all L ∈ ∆ + , we have sup i ∈ I τ ( F i , L ) = τ (sup i ∈ I ( F i ) , L ) . For simplicity of notations, in all what follows, the triangle function τ will bedenoted by the binary operation ⋆ as follows: τ ( L, K ) :=
L ⋆ K.
It follows from the axioms ( i )-( iv ) that (∆ + , ⋆ ) is an abelian monoid having H asa neutral element. Definition 3.
Let ⋆ be a triangle function on ∆ + . (1) A sequence ( F n ) of distributions in ∆ + converges weakly to a function F in ∆ + if ( F n ( t )) converges to F ( t ) at each point t of continuity of F . In this case, wewrite indifferently F n w −→ F or lim n F n = F . (2) We say that the law ⋆ is continuous at ( F, L ) ∈ ∆ + × ∆ + if we have F n ⋆L n w −→ F ⋆ L , whenever F n w −→ F and L n w −→ L . A classical class of triangle functions, that are both continuous and sup-continuous,is provided by operations taking the form : for all
F, L ∈ ∆ + and for all t ∈ R ,( F ⋆ T L )( t ) := sup s + u = t T ( F ( s ) , L ( u )) , (1)where T : [0 , × [0 , → [0 , • T ( x, y ) = T ( y, x ) ( commutativity); • T ( x, T ( y, z )) = T ( T ( x, y ) , z ) (associativity); • T ( x, y ) ≤ T ( x, z ) whenever y ≤ z (monotonicity ); • T ( x,
1) = x (boundary condition). Definition 4.
Let G be a set and let D : G × G −→ (∆ + , ⋆, ≤ ) be a map. We saythat ( G, D, ⋆ ) is a probabilistic metric space if the following axioms hold: (i) D ( p, q ) = H iff p = q . (ii) D ( p, q ) = D ( q, p ) for all p, q ∈ G (iii) D ( p, q ) ⋆ D ( q, r ) ≤ D ( p, r ) for all p, q, r ∈ G This notion of probabilistic distance naturally leads to associated metric con-cepts, such as Cauchy sequence and completeness.
Definition 5.
In a a probabilistic metric space ( G, D, ⋆ ) , a sequence ( z n ) ⊂ G issaid to be a Cauchy sequence if for all t ∈ R , lim n,p −→ + ∞ D ( z n , z p )( t ) = H ( t ) . (Equivalently, if D ( z n , z p ) w −→ H , when n, p −→ + ∞ ). A probabilistic metric space ( G, D, ⋆ ) is said to be complete if every Cauchy sequence ( z n ) ⊂ G weakly convergesto some z ∞ ∈ G , that is lim n → + ∞ D ( z n , z ∞ )( t ) = H ( t ) for all t ∈ R , we will brieflynote lim n D ( z n , z ∞ ) = H . MOHAMMED BACHIR, BRUNO NAZARET
Examples . Every complete metric space induce a probabilistic complete metricspace. Indeed, if d is a complete metric on G and ⋆ is a triangle function on ∆ + satisfying H a ⋆ H b = H a + b for all a, b ∈ R + (see the example in the formula 1 belowand references [12] and [4]), then ( G, D, ⋆ ) where, D ( p, q ) = H d ( p,q ) , ∀ p, q ∈ G, is a probabilistic complete metric space.2.2. L´evy distance, weak convergence and compactness.Definition 6.
Let F and G be in ∆ + , let h be in (0 , , and let A ( F, G ; h ) denotethe condition ≤ G ( t ) ≤ F ( t + h ) + h for all t ∈ (0 , h − ) . The modified L´evy distance is the map d L defined on ∆ + × ∆ + as d L ( F, G ) = inf { h : both A ( F, G ; h ) and A ( G, F ; h ) hold } . Note that for any F and G in ∆ + , both A ( F, G ; 1) and A ( G, F ; 1) hold, whence d L is well-defined and d L ( F, G ) ≤ F d L ( F, H ) is non-increassing, that is F, G ∈ ∆ + , F ≤ G = ⇒ d L ( G, H ) ≤ d L ( F, H ) . We also recall the following results due to D. Sibley in [20, Theorem 1. and Theorem2].
Lemma 1. ( [20] ) The function d L is a metric on ∆ + and (∆ + , d L ) is compact. Lemma 2. ( [20] ) Let ( F n ) be a sequence of functions in ∆ + , and let F be anelement of ∆ + . Then ( F n ) converges weakly to F if and only if d L ( F n , F ) −→ ,when n −→ + ∞ . Definition 7.
Let ( G, D, ⋆ ) be a probabilistic metric space. For x ∈ G and t > ,the strong t -neighborhood of x is the set N x ( t ) = { y ∈ G : D ( x, y )( t ) > − t } , and the strong neighborhood system for G is { N x ( t ); x ∈ G, t > } . Lemma 3. ( [12] ) Let t > and x, y ∈ G . Then we have y ∈ N x ( t ) if and only if d L ( D ( x, y ) , H ) < t . Definition 8.
A complete probabilistic metric space ( K, D, ⋆ ) is called compact iffor all t > , the open cover { N x ( t ) : x ∈ K } has a finite subcover. Proposition 1. ([8, Theorem 2.2, Theorem 2.3])
Let ( K, D, ⋆ ) be a completeprobabilistic metric space. Then, we have: (1) ( K, D, ⋆ ) is compact if and only if every sequence has a convergent subse-quence. (2) If ( K, D, ⋆ ) is compact, then ( K, D, ⋆ ) is separable. ROBABILISTIC ARZELA-ASCOLI THEOREM 5 Some Properties of Probabilistic continuous and 1-Lipschitz map
We recall from [1] the notion of probabilistic Lipschitz maps and probabilisticcontinuous functions defined from a probabilistic metric space into ∆ + . Definition 9. ( [1] ) Let ( G, D, ⋆ ) be a probabilistic metric space and let f be afunction f : G −→ ∆ + . (1) We say that f is continuous at z ∈ G if f ( z n ) w −→ f ( z ) , when D ( z n , z ) w −→ H .We say that f is continuous if f is continuous at each point z ∈ G . (2) We say that f is a probabilistic -Lipschitz map if: ∀ x, y ∈ G, D ( x, y ) ⋆ f ( y ) ≤ f ( x ) . We can also define k -Lipschitz maps for any nonegative real number k ≥ f satisfying ∀ x, y ∈ G, D k ( x, y ) ⋆ f ( y ) ≤ f ( x ) , where, for all x, y ∈ G and all t ∈ R , D k ( x, y )( t ) = D ( x, y )( tk ) if k > D ( x, y )( t ) = H ( t ) if k = 0. For sake of simplicity, we shall only treat in thispaper the case of 1-Lipschitz maps, but our main result result could be easilyextended to this more general setting. Examples . Let (
G, d ) be a metric space. Assume that ⋆ is a triangle function on∆ + satisfying H a ⋆ H b = H a + b for all a, b ∈ R + (for example if ⋆ = ⋆ T where T isa lef-continuous triangular norm). Let ( G, D, ⋆ ) be the probabilistic metric spacedefined with the probabilistic metric D ( p, q ) = H d ( p,q ) . Let L : ( G, d ) −→ R + be a real-valued map. Then, L is a non-negative 1-Lipschitzmap if and only if f : ( G, D, ⋆ ) −→ ∆ + defined for all x ∈ G by f ( x ) := H L ( x ) is a probabilistic 1-Lipschitz map. This example shows that the framework ofprobabilistic 1-Lipschitz maps encompasses the classical determinist case.By C ⋆ ( G, ∆ + ) we denote the space of all (probabilistic) continuous functions f : G −→ ∆ + . We equip the space C ⋆ ( G, ∆ + ) with the following metric d ∞ ( f, g ) := sup x ∈ G d L ( f ( x ) , g ( x ))where d L denotes the modified L´evy distance on ∆ + .By Lip ⋆ ( G, ∆ + ) we denote the space of all probabilistic 1-Lipschitz maps Lip ⋆ ( G, ∆ + ) := { f : G −→ ∆ + /D ( x, y ) ⋆ f ( y ) ≤ f ( x ); ∀ x, y ∈ G } . For all x ∈ G , by δ x we denote the map δ x : G −→ ∆ + y D ( y, x ) . MOHAMMED BACHIR, BRUNO NAZARET
We set G ( G ) := { δ x /x ∈ G } and by δ , we denote the operator δ : G −→ G ( G ) ⊂ Lip ⋆ ( G, ∆ + ) x δ x . The following proposition gives a canonical way to build probabilistic Lipschitzmaps.
Proposition 2.
Let ( G, D, ⋆ ) be a probabilistic metric space such that ⋆ is sup-continuous. Let f : ( G, D, ⋆ ) −→ ∆ + be any map and A be any no-empty subset of G . Then, the map ˜ f A ( x ) := sup y ∈ A [ f ( y ) ⋆ D ( x, y )] , for all x ∈ G is a probabilistic -Lipschitz map and we have ˜ f A ( x ) ≥ f ( x ) , for all x ∈ A .Proof. The proof is similar to the standard inf-convolution construction. The factthat ˜ f A ( x ) ≥ f ( x ) for all x ∈ A is immediate from the definition of ˜ f A . Let us nowprove that it is probabilistic 1-Lipschiptz. Let x , y ∈ G . Then, for all z ∈ A , wehave˜ f A ( y ) = sup z ∈ A [ f ( z ) ⋆ D ( y, z )] ≥ f ( z ) ⋆ D ( y, z ) ≥ f ( z ) ⋆ ( D ( y, x ) ⋆ D ( x, z )) = ( f ( z ) ⋆ D ( x, z )) ⋆ D ( y, x ) . We get the conclusion by taking the supremum with respect to z ∈ A . (cid:3) Proposition 3. ( [1, Proposition 3.6] ) Let ( G, D, ⋆ ) be a probabilistic metric spacesuch that ⋆ is continuous. Then, every probabilistic -Lipschitz map defined on G is continuous. In other words, we have that Lip ⋆ ( G, ∆ + ) ⊂ C ⋆ ( G, ∆ + ) . Proposition 4.
Let ( G, D, ⋆ ) be a probabilistic metric space. Then, the space ( C ⋆ ( G, ∆ + ) , d ∞ ) is a complete metric space.Proof. Let ( f n ) be a Cauchy sequence in ( C ⋆ ( G, ∆ + ) , d ∞ ). In particular, for each x ∈ G , ( f n ( x )) is Cauchy in (∆ + , d L ) which is compact (Lemma 1). Thus, thereexists a function f : G −→ ∆ + such that the sequence ( f n ) pointwise convergesto f on G (with respect to the metric d L ). It is easy to see that in fact ( f n )uniformly converges to f , since it is Cauchy sequence in ( C ⋆ ( G, ∆ + ) , d ∞ ). We needto prove that f is continuous on G . Let x ∈ G and ( x k ) be a sequence such that d L ( D ( x k , x ) , H ) −→
0, when k −→ + ∞ . For all ε >
0, there exists N ε ∈ N suchthat n ≥ N ε = ⇒ d ∞ ( f n , f ) := sup x ∈ G d L ( f n ( x ) , f ( x )) ≤ ε (2)Using the continuity of f N ε , we have that there exists η ( ε ) > d L ( D ( x k , x ) , H ) ≤ η ( ε ) = ⇒ d L ( f N ε ( x k ) , f N ε ( x )) ≤ ε (3)Using (2) and (3), we have that d L ( f ( x k ) , f ( x )) ≤ d L ( f ( x k ) , f N ε ( x k )) + d L ( f N ε ( x k ) , f N ε ( x )) + d L ( f N ε ( x ) , f ( x )) ≤ ε This shows that f is continuous on G . Finally, we proved that every Cauchysequence ( f n ) uniformly converges to a continuous function f . In other words, thespace ( C ⋆ ( G, ∆ + ) , d ∞ ) is complete. (cid:3) ROBABILISTIC ARZELA-ASCOLI THEOREM 7 The main result: Probabilistic Arzela-Ascoli theorem
The following theorem is the main result of the paper. Its proof will be givenafter some intermediate lemmas.
Theorem 1.
Let ( K, D, ⋆ ) be a probabilistic complete metric space such that ⋆ iscontinuous and sup-continuous. Then, the following assertions are equivalent. (1) ( K, D, ⋆ ) is compact, (2) the metric space ( Lip ⋆ ( K, ∆ + ) , d ∞ ) is compact (equivalently, Lip ⋆ ( K, ∆ + ) is a compact subset of ( C ⋆ ( K, ∆ + ) , d ∞ ) ). Lemma 4.
Let ( G, D, ⋆ ) be a probabilistic metric space such that ⋆ is continuous.Then, the set Lip ⋆ ( G, ∆ + ) is uniformly equicontinuous. In other words: ∀ ε > , ∃ η ( ε ) > ∀ f ∈ Lip ⋆ ( G, ∆ + ) , ∀ x, y ∈ G ; d L ( D ( x, y ) , H ) < η ( ε ) = ⇒ d L ( f ( x ) , f ( y )) < ε. Proof.
Since (∆ + , d L ) is a compact metric space (see Lemma 1) and ⋆ is continuous,then ⋆ is uniformly continuous from ∆ + × ∆ + into ∆ + . It follows that ∀ ε > , ∃ η ( ε ) > ∀ F ∈ ∆ + , ∀ x, y ∈ G : d L ( D ( x, y ) , H ) < η ( ε ) = ⇒ d L ( D ( x, y ) ⋆ F, F ) < ε. In particular, we have for all f ∈ Lip ⋆ ( G, ∆ + ) and all x , y ∈ G ,max[ d L ( D ( x, y ) ⋆ f ( x ) , f ( x )) , d L ( D ( x, y ) ⋆ f ( y ) , f ( y ))] < ε. (4)We are going to prove that d L ( f ( x ) , f ( y )) ≤ max[ d L ( D ( x, y ) ⋆ f ( x ) , f ( x )) , d L ( D ( x, y ) ⋆ f ( y ) , f ( y ))] . Let h , h > A ( D ( x, y ) ⋆ f ( x ) , f ( x ) , h ), A ( f ( x ) , D ( x, y ) ⋆ f ( x ) , h ), A ( D ( x, y ) ⋆ f ( y ) , f ( y ) , h ) and A ( f ( y ) , D ( x, y ) ⋆ f ( y ) , h ) hold, which means thatfor all t ∈ (0 , h − ) and all t ′ ∈ (0 , h − ) we have:0 ≤ D ( x, y ) ⋆ f ( x )( t ) ≤ f ( x )( t + h ) + h ≤ f ( x )( t ) ≤ D ( x, y ) ⋆ f ( x )( t + h ) + h ≤ D ( x, y ) ⋆ f ( y )( t ′ ) ≤ f ( y )( t ′ + h ) + h ≤ f ( y )( t ′ ) ≤ D ( x, y ) ⋆ f ( y )( t ′ + h ) + h From the second, the fourth inequalities and the fact that f is 1-Lipschitz, we getthat for all t ∈ (0 , h − ) and all t ′ ∈ (0 , h − ) we have:0 ≤ f ( x )( t ) ≤ f ( y )( t + h ) + h ≤ f ( y )( t ′ ) ≤ f ( x )( t ′ + h ) + h It follows that for all s ∈ (0 , min( h − , h − )) we have:0 ≤ f ( x )( s ) ≤ f ( y )( s + max( h , h )) + max( h , h )0 ≤ f ( y )( s ) ≤ f ( x )( s + max( h , h )) + max( h , h )Thus, we have that d L ( f ( x ) , f ( y )) ≤ max( h , h ) for all h , h > A ( D ( x, y ) ⋆ f ( x ) , f ( x ) , h ), A ( f ( x ) , D ( x, y ) ⋆ f ( x ) , h ), A ( D ( x, y ) ⋆ f ( y ) , f ( y ) , h )and A ( f ( y ) , D ( x, y ) ⋆ f ( y ) , h ) hold. This implies that d L ( f ( x ) , f ( y )) ≤ max( d L ( D ( x, y ) ⋆ f ( x ) , f ( x )) , d L ( D ( x, y ) ⋆ f ( y ) , f ( y ))) . Using the above inequality and (4), we get the conclusion. (cid:3)
MOHAMMED BACHIR, BRUNO NAZARET
We recall the following useful proposition (see [1]).
Proposition 5. ( [1, Proposition 3.5] ) Let ( F n ) , ( L n ) , ( K n ) ⊂ (∆ + , ⋆ ) . Supposethat ( a ) the triangle function ⋆ is continuous, ( b ) F n w −→ F , L n w −→ L and K n w −→ K . ( c ) for all n ∈ N , F n ⋆ L n ≤ K n ,then, F ⋆ L ≤ K . Lemma 5.
Let ( G, D, ⋆ ) be a probabilistic metric space. Let ( f n ) be a sequence of -Lipschitz maps and L be a subset of G . Suppose that there exists a function f defined on L such that f n ( x ) w −→ f ( x ) , when n −→ + ∞ , for all x ∈ L . Then, f is -Lipschitz on L .Proof. Since f n is 1-Lipschitz map for each n ∈ N , then we have, for all x, y ∈ L and for all n ∈ N : D ( x, y ) ⋆ f n ( x ) ≤ f n ( y )Using Proposition 5, we get that for all x, y ∈ LD ( x, y ) ⋆ f ( x ) ≤ f ( y )In other words, f is 1-Lipschitz maps on L . (cid:3) Lemma 6.
Let ( K, D, ⋆ ) be a probabilistic compact metric space and ( f n ) be asequence of -Lipschitz maps. Suppose that there exists a -Lipschitz map f suchthat d L ( f n ( x ) , f ( x )) −→ , when n −→ + ∞ for all x ∈ K . Then, ( f n ) convergesuniformly to f , that is, d ∞ ( f n , f ) −→ , when n −→ + ∞ .Proof. Let ε > η ( ε ) be the uniform equicontinuity moduleof Lip ⋆ ( G, ∆ + ). Since ( K, D, ⋆ ) is compact, there exists a finite set A such that K = ∪ a ∈ A N a ( η ( ε )). Since d L ( f n ( a ) , f ( a )) −→
0, when n −→ + ∞ for all a ∈ A .Then, for each a ∈ A , there exists P a ∈ N such that n ≥ P a = ⇒ d L ( f n ( a ) , f ( a )) ≤ ε Since A is finite, we have that n ≥ max a ∈ A P a = ⇒ sup a ∈ A d L ( f n ( a ) , f ( a )) ≤ ε Thus, for all x ∈ K = ∪ a ∈ A N a ( η ( ε )), there exists a ∈ A such that x ∈ N a ( η ( ε ))and so we have that for all n ≥ max a ∈ A P a : d L ( f n ( x ) , f ( x )) ≤ d L ( f n ( x ) , f n ( a )) + d L ( f n ( a ) , f ( a )) + d L ( f ( a ) , f ( x )) ≤ ε. In other words, n ≥ max a ∈ A P a = ⇒ d ∞ ( f n , f ) := sup x ∈ K d L ( f n ( x ) , f ( x )) ≤ ε (cid:3) We finally need the following result from [1].
Lemma 7. (see [1, Theorem 3.7] ) Let ( G, D, ⋆ ) be a probabilistic metric space suchthat ⋆ is sup-continuous and let A be a nonempty subset of G . Let f : A −→ ∆ + bea probabilistic -Lipschitz map. Then, there exists a probabilistic -Lipschitz map ˜ f : G −→ ∆ + such that ˜ f | A = f . ROBABILISTIC ARZELA-ASCOLI THEOREM 9
Let us now give the proof of our main result.
Proof of Theorem 1. (1) = ⇒ (2) Suppose that ( K, D, ⋆ ) is compact. Let ( f n ) n ⊂ Lip ⋆ ( K, ∆ + ) be a sequence. We need to shows that there exists a subsequenceof ( f n ) that converges uniformly to some 1-Lipschitz function. Indeed, since K iscompact, it is separable, that is, there existes a sequence ( x k ) k ⊂ K which is densein K for the probabilistic metric D . Let us denote L := { x k : k ∈ N } . We knowfrom Lemma 1 and Lemma 2 that (∆ + , w) is a compact metrizable space. Thus, byTykhonov theorem, the space (∆ + ) N is a compact metrizable. Hence, the sequence( f n ) has a subsequence ( f ϕ ( n ) ) that converges pointwise on L to some function f ,necessarily 1-Lipschitz map on L by Lemma 5. By Lemma 7 and Proposition 3, f extend to a 1-Lipschitz map ˜ f on K , and this extension is unique since L is dense.Let us prove now that ( f ϕ ( n ) ) converges pointwise on K to ˜ f . Indeed, let x ∈ K and let us choose a subsequence from L denoted also by ( x k ) that converges to x ,that is D ( x k , x ) w −→ H . Since ( f n ) is a sequence of 1-Lipschitz functions, then wehave, D ( x k , x ) ⋆ f ϕ ( n ) ( x k ) ≤ f ϕ ( n ) ( x ) D ( x k , x ) ⋆ f ϕ ( n ) ( x ) ≤ f ϕ ( n ) ( x k )Since ∆ + is compact, to show that ( f ϕ ( n ) ( x )) is convergent sequence, it suffices toprove that any convergent subsequence of ( f ϕ ( n ) ( x )) converges to the same limit˜ f ( x ). Indeed, let f ψ ( n ) ( x ) be a subsequence of f ϕ ( n ) ( x ) that converges to some g ( x ) ∈ ∆ + . Using the above inequalities and Proposition 5, we get that D ( x k , x ) ⋆ ˜ f ( x k ) ≤ g ( x ) D ( x k , x ) ⋆ g ( x ) ≤ ˜ f ( x k )Using the fact D ( x k , x ) w −→ H , the continuity of ˜ f on K and Proposition 5 in theabove inequalities, we get that g ( x ) = ˜ f ( x ). Finally, we proved that there exists asubsequence ( f ϕ ( n ) ) of ( f n ) that converges pointwise on K to some 1-Lipschitz map˜ f . That is, f ϕ ( n ) ( x ) w −→ ˜ f ( x ) for all x ∈ K . Using Lemma 6, we get that ( f ϕ ( n ) )converges uniformly to ˜ f on K .(2) ⇐ = (1) Suppose that ( Lip ⋆ ( K, ∆ + ) , d ∞ ) is compact. Let ( x n ) be a sequenceof K . We need to prove that ( x n ) has a convergent subsequence. Indeed, considerthe sequence ( δ x n ) of 1-Lipschitz maps, defined by δ x n : x D ( x n , x ) for each n ∈ N . By assumption, there exists a subsequence ( δ x ϕ ( n ) ) that converges uniformlyto some 1-Lipschitz map, in particular it is a Cauchy sequence. In other words, wehave lim p,q −→ + ∞ sup x ∈ K d L ( δ x ϕ ( p ) ( x ) , δ x ϕ ( q ) ( x )) = 0 . In particular we have lim p,q −→ + ∞ d L ( δ x ϕ ( p ) ( x ϕ ( q ) ) , H ) = 0 . Equivalently, lim p,q −→ + ∞ d L ( D ( x ϕ ( p ) , x ϕ ( q ) ) , H ) = 0 . This shows that the sequence ( x ϕ ( n ) ) is Cauchy in K (see Lemma 2). Thus, thesequence ( x ϕ ( n ) ) converges to some point x ∈ K , since K is complete. (cid:3) References
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