Fuzzy Prokhorov metric on the set of probability measures

FUZZY PR OKHORO V METRIC ON THE SET OF PR OBABILITY MEASURES DU ˇ SAN REPOV ˇ S, ALEKSA NDR SA VCHENK O, AN D MYKHAILO ZA RICHNY I Abstra ct. W e introduce a fuzzy metric on th e set of probability measures on a fuzzy metric space. The construction is an analog ue, in the realm of fuzzy metric spaces, of the Prokhorov metric on the set of probability measures on compact metric spaces. 1. Introduction The n otion of fuzzy metric space fi rst app eared in [10] and it wa s later mo dified in [5]. The v ersion from [5], despite b eing more restrictiv e, determines the class of spaces that are closely connecte d with the class of metrizable top ological spaces. This notion wa s wid ely used in v arious p ap ers dev oted to fu zzy top ology and it has found n um erous applications – in particular to color image pro cessing (see e.g. [6] and the references therein). Differen t notions and results of the theory of metric spaces ha v e their analogues for fuzzy metric sp aces. At the same time, there are p henomena in the realm of fuzzy metric spaces that hav e no immediate analogue for metric spaces. The completeness and existence of non-completable fuzzy metric spaces can serve as an example. This demonstrates that the fuzzy metric seems to b e a structure that leads to a theory whic h app ears to b e richer than that of metric sp aces. In the theory of f uzzy metric spaces, there are analogues of v a rious constructions from theory of m etric spaces. In particular, a f uzzy Hausd orff metric w as defined in [15]. The fu zzy metrics on (fi nite and coun table) p o w ers and G -symmetric p ow ers w ere defined, in p articular in [14, 18 ]. In this p ap er, w e consider a f uzzy analogue of the Prokhoro v metric defined on th e set of all p robabilit y measures of a compact fuzzy metric sp ace. W e p ro ve that this metric ind uces the we ak* con verge nce of probability measures on compact metrizable space. Our aim of studying the sp ace of probability measures on fuzzy metric spaces is t wofol d . First, w e u se the p rop erty o f spaces of p robabilit y measur es to b e absolute extensors and this allo ws us to solv e the p roblem of (con tinuous) extension of fuzzy metric defined on a closed subsp ace. Note that the problem o f extension of structures is of a fun damen tal characte r and it arises in v arious a reas of mathemati cs: differentia l equations (extensions of solutio ns ), Date : A pril 9, 2018 . 2010 Mathematics Subje ct Classific ation. 54E70, 54A40, 60B05. Key wor ds and phr ases. F uzzy metric, Prokhorov metric, probabilit y measure, extension of fuzzy metrics. 1 2 D. REPOV ˇ S, A. SA VCHENK O, AND M. ZARICHNYI functional analysis (extensions of functionals), top ology (extensions of con tin uous maps), etc. The cla ssical p roblem of extensions of metrics, first solv ed b y Hausdorff in the 1930’s, is also of this type. Our th eorem on con tinuous extension of fuzzy metrics r elies on the extensional prop erties of the s paces of probabilit y measures. These prop erties are kno wn to hold only for th e class of metrizable spaces. Th us the reason w h y we are dealing with the fuzzy metric spaces in the sense of [5] is the fact that the top ology induced by an y fuzzy metric in this sense is metrizable (see [7]). Our second r eason to inv estigat e the spaces of p robabilit y measures is their ap - plicabilit y to the theory of pr obabilistic systems (see e.g. [3, 20, 21]). In the fuzzy metric setting, we come to the problem of fuzzy metrization of the sets of probabilit y measures. Note also that a fu zzy ultrametric on th e set of p robabilit y measures with compact s upp orts on fuzzy ultrametric spaces w as d efined in [19]. One o f the main results is that the construction of the Prokhoro v metric dete rm ines a fu nctor in the category of fuzzy metric sp aces and nonexpandin g maps. As wa s remark ed abov e, we also apply the construction of Prokhoro v m etric to the problem of extensions of f uzzy metrics. 2. Preliminaries 2.1. F uzzy metric spaces. The notion of fuzzy metric space, in one of its forms, is in tro d uced b y Kramosil and Mic halek [10]. In the present p ap er w e use the ve rs ion of this concept giv en b y George and V eeramani [5]. W e start with some necessary definitions. Definition 2.1. A b inary op eration ∗ : [0 , 1] × [0 , 1] → [0 , 1] is a con tin uous t-norm if ∗ satisfies the follo wing cond itions: (i) ∗ is comm utativ e and asso ciativ e; (ii) ∗ is con tin uous; (iii) a ∗ 1 = a for all a ∈ [0 , 1]; and (iv) a ∗ b ≤ c ∗ d wh enev er a ≤ c and b ≤ d , and a, b, c, d ∈ [0 , 1]. The follo wing are examples of t-norms: a ∗ b = ab ; a ∗ b = min { a, b } . Definition 2.2. A 3-tuple ( X, M , ∗ ) is said to b e a fuzzy metric space if X is an arbitrary set, ∗ is a con tinuous t-norm and M is a f uzzy set on X 2 × (0 , ∞ ) satisfying the f ollo wing conditions for all x, y , z ∈ X and s, t > 0: (i) M ( x, y , t ) > 0, (ii) M ( x, y , t ) = 1 if and only if x = y , (iii) M ( x, y , t ) = M ( y , x, t ), (iv) M ( x, y , t ) ∗ M ( y , z , s ) ≤ M ( x, z , t + s ), (v) th e function M ( x, y , − ) : (0 , ∞ ) → (0 , 1] is cont inuous. It was p ro v ed in [5] that in a fuzzy metric s pace X , the fu nction M ( x, y , − ) is n on- decreasing for all x, y ∈ X . The follo wing notion w as int ro duced in [5] (see Definition 2.6 th erein). FUZZY PROKHOR OV METRIC ON THE SET OF PR OBABILITY M EASURES 3 Definition 2.3. Let ( X, M , ∗ ) b e a fuzzy metric space and let r ∈ (0 , 1), t > 0 and x ∈ X . The set B ( x, r , t ) = { y ∈ X | M ( x, y , t ) > 1 − r } is called the op en b al l with cen ter x and radius r with resp ect to t . The family of all op en balls in a fuzzy metric sp ace ( X, M , ∗ ) forms a base of a top ology in X ; this top ology is denoted by τ M and is kn o wn to b e metrizable (see [5]). In the sequel, if we sp eak on a fuzzy metric on a topological space, we assume that this metric generates the initial top ology on the space. Note that B ( x, r, t 1 ) ⊂ B ( x, r , t 2 ), wheneve r t 1 ≤ t 2 . If ( X, M , ∗ ) is a fuzzy metric space and Y ⊂ X , then, clearly , M Y = M | ( Y × Y × (0 , ∞ )) : Y × Y × (0 , ∞ ) → [0 , 1] is a fuzzy metric on the set Y . W e sa y that the f uzzy m etric M Y is induc e d on Y by M . If ( X, M , ∗ ) is a fuzzy metric space, then the family { U n = { ( x, y ) ∈ X × X | M ( x, y , 1 n ) > 1 − 1 n } | n ∈ N } is a base of a u niform structure on X . This uniform structure is known to generate the top olog y τ M on X . Let ( X , M , ∗ ) and ( X ′ , M ′ , ∗ ) b e fuzzy metric spaces. A map f : X → X ′ is called nonexp anding if M ′ ( f ( x ) , f ( y ) , t ) ≥ M ( x, y , t ), for all x, y ∈ X and t > 0. F or our purp oses, it is su fficien t to consider the class of fuzzy metric spaces with the same fixed n orm (e.g. ∗ ). The f uzzy metric spaces (with the norm ∗ ) and nonexpanding m aps f orm a c ategory , whic h we denote by F MS ( ∗ ). By C F MS ( ∗ ) w e d enote its su b category consisting of compact f uzzy metric spaces. 2.2. Spaces of probabilit y measures. Let X b e a metrizable space. W e denote t h e space of p robabilit y measur es with compact supp ort in X b y P ( X ) (see e.g. [12] for the necessary d efinitions th at concern probabilit y measures). Recall that the supp ort of a probability measure µ ∈ P ( X ) is the m inimal (with resp ect to the inclusion) closed set su pp( µ ) suc h that µ ( X \ su pp( µ )) = 0. F or any x ∈ X , b y δ x w e d enote the Dirac measure concentrat ed at x . An y probability measure µ of fi nite su pp ort can b e r epresen ted as follo ws: µ = Σ n i =1 α i δ x i , where α 1 , . . . , α n ≥ 0 and Σ n i =1 α i = 1. By P ω ( X ) w e denote the set of all probabilit y m easures with finite supp orts in X . The set P ( X ) is endow ed with the w eak* top olo gy , i.e., the top ology ind uced by the weak* conv ergence. A sequence ( µ i ) in P ( X ) w eakly* con ve rges to µ ∈ P ( X ) if lim i →∞ R X ϕdµ i = R X ϕdµ , f or ev ery ϕ ∈ C ( X ). Equiv alen tly , lim i →∞ µ i ( C ) ≤ µ ( C ), for ev ery closed su bset C of X . If X is a compact Hausdorff sp ace, then there exists a n atural map ψ X : P 2 ( X ) = P ( P ( X )) defined by the formula: Z X ϕdψ X ( M ) = Z P ( X ) ¯ ϕdM , 4 D. REPOV ˇ S, A. SA VCHENK O, AND M. ZARICHNYI where ¯ ϕ : P ( X ) → R is defined by the form ula: ¯ ϕ ( µ ) = R X ϕdµ . Let X , Y b e metrizable spaces. Every conti nuous map f : X → Y generates a map P ( f ) : P ( X ) → P ( Y ) defined by the follo wing condition: P ( f )( µ )( A ) = µ ( f − 1 ( A )), for ev ery Borel sub set of Y . It is kno wn th at the map P ( f ) is also con tinuous. 3. Lukasiewicz norm a nd fuzz y me tric on th e set of p robability measure s Recall that the Luk asiewicz t-norm is d efined by th e form u la x ∗ y = max { x + y − 1 , 0 } , x, y ∈ [0 , 1] . In th e sequel, ∗ stands for the Luk asiewicz t-norm. In th e sequel, let ( X , M , ∗ ) b e a compact fu zzy metric space. F or ev ery A ⊂ X , r ∈ (0 , 1), t ∈ (0 , ∞ ) defin e: A r,t = ∪{ B ( x, r, t ) | x ∈ A } ⊂ X. Lemma 3.1. F or every A ⊂ X , every r,  ∈ (0 , 1) such that r ∗  ∈ (0 , 1) , and every t, s ∈ (0 , ∞ ) we have: ( A r,t ) ,s ⊂ A r + ,t + s . Pr o of. Let x ∈ ( A r,t ) ,s , then there exist y , z ∈ X s uc h that z ∈ A , M ( z , y , t ) > 1 − r , M ( y , x, s ) > 1 −  . Whence M ( x, z , t + s ) ≥ M ( x, y , s ) ∗ M ( y , z , t ) = max { M ( x, y , s ) + M ( y , z , t ) + 1 , 0 } > max { 1 − r + 1 −  − 1 , 0 } = 1 − r − , i.e. x ∈ A r + ,t + s .  The follo wing is obvious. Lemma 3.2. If t 1 ≤ t 2 , then, for every A ⊂ X , we have A r,t 1 ⊂ A r,t 2 . Lemma 3.3. L et t 0 ∈ (0 , ∞ ) . F or every ε > 0 , ther e exists η > 0 such that, for every x, y ∈ X , if | t − t 0 | < η , then | M ( x, y , t ) − M ( x, y , t 0 ) | < ε . Pr o of. Let J b e a closed interv al suc h that t 0 is its in terior p oint. Since the map M : X × X × (0 , ∞ ) is c ontin u ous (see [15, Prop ositio n 1]), its restriction on to X × X × J is uniformly con tin uous and th e result follo ws.  Lemma 3.4. L et t 0 ∈ (0 , ∞ ) , r 0 ∈ (0 , 1) . F or every ε > 0 , ther e exists η > 0 such that, B ( x, r 0 + ε, t ) ⊃ B ( x, r 0 , t 0 ) , whenever x ∈ X and | t − t 0 | < η . Pr o of. T his is a consequence of the previous lemma.  Define the function ˆ M : P ( X ) × P ( X ) × (0 , ∞ ) → [0 , 1] by the formula ˆ M ( µ, ν, t ) =1 − inf { r ∈ (0 , 1) | µ ( A ) ≤ ν ( A r,t ) + r and ν ( A ) ≤ µ ( A r,t ) + r for ev ery Borel sub set A ⊂ X } . W e will see later that the fu nction ˆ M is wel l defined. FUZZY PROKHOR OV METRIC ON THE SET OF PR OBABILITY M EASURES 5 Theorem 3.5. The function ˆ M is a fu zzy metric on the set P ( X ) . Pr o of. W e are goi n g to v erify th e prop erties from the definition of the fuzzy metric. First, remark that ˆ M ( µ, ν, t ) > 0. Indeed, since the set supp( µ ) ∪ supp( ν ) is compact, there exists r > 0 such that, for an y x ∈ s upp( µ ) ∪ supp( ν ), we ha v e B ( x, r , t ) ⊃ sup p( µ ) ∪ sup p( ν ) (see [15]). Then, for an y Borel set A ⊂ X , suc h that µ ( A ) > 0, we obtain A r,t ⊃ supp( ν ) and therefore ν ( A r,t ) = 1. Then clearly , ˆ M ( µ, ν, t ) > 1 − r > 0. Note that at the same time we ha ve pro ven that the function ˆ M is w ell defined. Clearly , ˆ M ( µ, µ, t ) = 1. Let now ˆ M ( µ, ν, t ) = 1. Then it is easy to see that, for an y r ∈ (0 , 1) and t > 0, w e see that µ ( A ) ≤ ν ( A r,t ) + r, ν ( A ) ≤ µ ( A r,t ) + r, whence µ ( A ) = ν ( A ), for any Borel A . F rom the element ary prop erties of p robabilit y measures it follo ws that µ ( A ) = ν ( A ). Clearly , ˆ M ( µ, ν, t ) = ˆ M ( ν, µ, t ). Let µ, ν, τ ∈ P ( X ). Supp ose that ˆ M ( µ, ν, t ) > a , ˆ M ( ν, τ , s ) > b for some a, b ∈ (0 , 1). Then there exist r ∈ (0 , 1 − a ),  ∈ (0 , 1 − b ) such that µ ( A r,t ) ≤ ν ( A r,t ) + r, ν ( A r,t ) ≤ µ ( A r,t ) + r, ν ( A ,s ) ≤ τ ( A ,s ) + , τ ( A ,s ) ≤ µ ( A ,s ) + , for ev ery Borel sub set A ⊂ X . Then µ ( A ) ≤ ν ( A r,t ) + r ≤ τ (( A r,t ) ,s ) + r +  ≤ τ ( A r + ,t + s ) + r + , τ ( A ) ≤ ν ( A r,t ) + r ≤ µ (( A r,t ) ,s ) + r +  ≤ µ ( A r + ,t + s ) + r + , hence it follo w s th at ˆ M ( µ, τ , r + s ) ≥ 1 − ( r +  ) = 1 − ( r +  ) = 1 − r + 1 −  − 1 = a + b − 1 = a ∗ b. This p ro ve s prop ert y (iv) fr om th e definition of the fuzzy metric. It remains to prov e that, for ev ery µ, ν ∈ P ( X ), the map t 7→ ˆ M ( µ, ν, t ) : (0 , ∞ ) → [0 , 1] is con tin uous. First note th at this map is n ondecreasing. Indeed, supp ose that ˆ M ( µ, ν, t 1 ) = 1 − r 1 and t 1 ≤ t 2 . Then, for every r > r 1 , and every Borel subset A of X , we ha v e µ ( A ) ≤ ν ( A r,t 1 ) + r ≤ ν ( A r,t 1 ) + r, ν ( A ) ≤ µ ( A r,t 1 ) + r ≤ µ ( A r,t 1 ) + r, whence ˆ M ( µ, ν, t 2 ) = 1 − inf { r | r > r 1 } ≥ 1 − r 1 = ˆ M ( µ, ν, t 1 ). Supp ose no w that ˆ M ( µ, ν, t 0 ) > 1 − r 0 . Then there is r < r 0 suc h that µ ( A ) ≤ ν ( A r,t 0 ) + r, ν ( A ) ≤ µ ( A r,t 0 ) + r, for ev ery Borel sub set A of X . 6 D. REPOV ˇ S, A. SA VCHENK O, AND M. ZARICHNYI Let ε > 0 b e suc h th at r + ε < r 0 . T here is η > 0 suc h that, for ev ery t ∈ ( t 0 − η, t 0 ) and ev ery x ∈ X , we ha ve B ( x, r + ε, t ) ⊃ B ( x, r , t 0 ). Then , for ev ery Borel subset A of X , w e h a v e µ ( A ) ≤ ν ( A r + ε,t ) + r + ε, ν ( A ) ≤ µ ( A r + ε,t 0 ) + r + ε, whence 1 − ˆ M ( µ, ν, t ) ≥ 1 − ( r + ε ) > 1 − r 0 . This p ro ve s the left-con tinuit y of the fun ction M ( µ, ν, − ). T o pr o v e the righ t-con tinuit y at t 0 , let ε > 0. By Lemma 3.4 , there exists η > 0 suc h th at, for ev ery Borel subs et A of X and every r ∈ (0 , 1), we ha ve A r + ε,t 0 ⊃ A r,t , whenev er | t 0 − t | < η . Let t ∈ ( t 0 , t 0 + η ). Supp ose that ˆ M ( µ, ν, t 0 ) = 1 − r 0 , ˆ M ( µ, ν, t ) = 1 − r . By th e d efinition of ˆ M , there exists r ′ ∈ ( r , r + ε ) s uc h that for ev ery Borel subset A of X , w e h a v e µ ( A ) ≤ ν ( A r ′ ,t ) + r ′ , ν ( A ) ≤ µ ( A r ′ ,t ) + r ′ . Then µ ( A ) ≤ ν ( A r ′ + ε,t ) + r ′ , ν ( A ) ≤ µ ( A r ′ + ε,t ) + r ′ and th erefore r 0 ≤ r ′ + ε ≤ r + 2 ε . S ince ˆ M ( µ, ν, t ) is nondecreasing, we conclude that r ≤ r 0 , w hence | r − r 0 | < 2 ε .  Prop osition 3.6. L et a se que nc e ( µ i ) in P ( X ) c onver ge to µ ∈ P ( X ) with r esp e ct to the top olo gy i nduc e d by the fuzzy metric ˆ M . Then ( µ i ) we akly* c onver ges to µ . Pr o of. T here exist ( r i , t i ) ∈ (0 , 1) × (0 , ∞ ), i ∈ N , su c h that the family { ˆ U i = { ( µ, ν ) ∈ P ( X ) × P ( X ) | ˆ M ( µ, ν, t i ) > 1 − r i } | i ∈ N } forms a counta b le decreasing base of the uniform structure in P ( X ) generated b y the fuzzy metric ˆ M . Without loss the generalit y , one ma y assum e that th e family { U i = { ( x, y ) ∈ X × X | M ( x, y , t i ) > 1 − r i } | i ∈ N } forms a decreasing base of the uniform structure in X generated by the fuzzy metric M (if necessary , w e d ecrease t i and/or r i ). W e also assum e that r i → 0 whenever i → ∞ . Let A b e a Borel su bset of X . Then µ i ( A ) ≤ µ ( A r i ,t i ) + r i , µ ( A ) ≤ µ i ( A r i ,t i ) + r i and ther efore lim i →∞ µ i ( A ) ≤ lim i →∞ µ ( A r i ,t i ) + r i = lim i →∞ µ ( A r i ,t i ) = µ ( ¯ A ) , whence lim i →∞ µ i ( C ) ≤ µ ( C ), for an y closed C ⊂ X . Hence, ( µ i ) weakl y* conv erges to µ .  Lemma 3.7. F or every δ ∈ (0 , 1) and every t > 0 ther e exists a c ountable family op en b al ls { B ( x i , r i , t ) | i ∈ N ) } such that the fol lowing ar e satisfie d: FUZZY PROKHOR OV METRIC ON THE SET OF PR OBABILITY M EASURES 7 (1) r i < δ for al l i ; (2) ∪ ∞ i =1 B ( x i , r i , t ) = X ; (3) µ ( ∂ B ( x i , r i , t )) = 0 for al l i . Pr o of. Let D = { x i | i ∈ N } b e a coun table den se subset of X . F or every x ∈ D , r ∈ (0 , 1), t ∈ (0 , ∞ ), let S ( x, r , t ) = { y ∈ X | M ( x, y , t ) = 1 − r } . Then, b ecause of the contin u it y of M , we see that S ( x, r, t ) ⊂ ∂ B ( x, r , t ). Since the family { S ( x i , r, t ) | r ∈ ( δ / 2 , δ ) } is disjoint, we see that there exists r i ∈ ( δ / 2 , δ ) suc h that µ ( S ( x i , r i , t )) = 0. Then also µ ( ∂ B ( x i , r i , t )) = 0. Note that, for ev ery t 0 ∈ (0 , ∞ ) the f amily { B ( x, r , t ′ ) | x ∈ X, r ∈ (0 , δ / 2) , t ′ < t } forms a base of the top ology in X . Thus, since D is dens e in X , for an y x ∈ X there is i ∈ N , r ∈ (0 , δ / 2), t ′ < t such that x i ∈ B ( x, r , t ′ ). Then also x ∈ B ( x i , r, t ′ ) ⊂ B ( x i , r i , t ). Therefore, (2) holds.  Theorem 3.8. Supp ose that ( µ i ) we ak* c onver ges to µ , for µ, µ i ∈ P ( X ) . Then ( µ i ) c onver ges to µ in the top olo gy induc e d by the fuzzy metric ˆ M . Pr o of. Let t, ε > 0. W e w ant to sho w that there exists N ∈ N su c h that, for ev ery i ≥ N , ˆ M ( µ i , µ, t ) < ε . The latter means that µ ( B ) ≤ µ i ( B ε,t ) + ε, µ i ( B ) ≤ µ ( B ε,t ) + ε, for ev ery Borel sub set A of X . Let δ ∈ (0 , ε/ 3). By Lemma 3.7, ther e exists a colle ction of op en balls { B i = B ( x i , r i , t/ 2) | i ∈ N } suc h that r i ∈ (0 , δ / 2) such that ∪ ∞ j =1 B j = X and µ ( ∂ B j ) = 0, for ev ery j . Clearly , there exists k ∈ N such that µ ( ∪ k j =1 B j ) > 1 − δ . Consider the family A = {∪ j ∈ J B j | J ⊂ { 1 , . . . , k }} . Note that, for ev ery A ∈ A , since ∂ A ⊂ ∪ k j =1 ∂ B j , we conclude that µ ( ∂ A ) = 0. Since A is op en and ( µ i ) w eakly* conv erges to µ , w e see that lim i →∞ µ i ( A ) = µ ( A ). There exists N ∈ N su c h that | µ i ( A ) − µ ( A ) | < δ , for all i ≥ N and for all A ∈ A . Note th at then µ i ( ∪ k j =1 B j ) ≥ 1 − 2 δ , for all i ≥ N . Giv en a Borel set B in X , let A = { B j | B j ∩ A 6 = ∅ , j = 1 , . . . , k } . Then the follo wing holds: (1) A ⊂ B δ,t ; (2) B ⊂ A ∪  X \ ∪ k j =1 B j  ; (3) | µ i ( A ) − µ ( A ) | < δ ; (4) µ  X \ ∪ k j =1 ) B j  ≤ δ , µ i  X \ ∪ k j =1 ) B j  ≤ 2 δ , for all i > N . 8 D. REPOV ˇ S, A. SA VCHENK O, AND M. ZARICHNYI Indeed, one has only to c hec k (1). Su pp ose that x ∈ A , th en x ∈ B j = B ( x j , r j , t/ 2), for some j , and there exists y ∈ B j ∩ B . W e obtain M ( x, y , t ) ≥ M ( x, x j , t/ 2) ∗ M ( x j , y , t/ 2) > (1 − r ) ∗ (1 − r ) ≥ 1 − 2 r, whence x ∈ B δ,t . Therefore, for ev ery i ≥ N , we hav e µ ( B ) ≤ µ ( A ) + δ ≤ µ i ( A ) + 2 δ ≤ µ i ( B δ,t ) + 2 δ ≤ µ i ( B ε,t ) + ε ; µ i ( B ) ≤ µ i ( A ) + 2 δ ≤ µ ( A ) + 3 δ ≤ µ i ( B δ,t ) + 3 δ ≤ µ i ( B ε,t ) + ε. Since B is an arbitrary Borel set, w e conclude th at ˆ M ( µ, µ i , t ) ≥ 1 − ε , for all i ≥ N .  Prop osition 3.9. L et f : X → Y b e a nonexp anding map of c omp act fuzzy metric sp ac es ( X , M , ∗ ) and X ′ , M ′ , ∗ ) . Then the induc e d map P ( f ) : P ( X ) → P ( Y ) is also nonexp anding. Pr o of. Note that, since th e map f is n onexpanding, B ( x, r , t ) ⊂ f − 1 ( B ′ ( f ( x ) , r , t )), for ev ery x ∈ X , r ∈ (0 , 1) and t ∈ (0 , ∞ ). Therefore, for an y A ⊂ X ′ w e ha v e f − 1 ( A ) r,t ⊂ f − 1 ( A r,t ). Let µ, ν ∈ P ( X ) and let A b e a Borel subset of X ′ . If ˆ M ( µ, ν, t ) > 1 − r , then P ( f )( µ )( A ) = µ ( f − 1 ( A )) ≤ ν ( f − 1 ( A ) r,t ) + r ≤ ν ( f − 1 ( A r,t )) + r = P ( f )( ν )( A r,t ) + r and similarly P ( f )( ν )( A ) ≤ P ( f )( µ )( A r,t ) + r , when ce ˆ M ′ ( P ( f )( µ ) , P ( f )( ν ) , t ) > 1 − r and the map P ( f ) is nonexpanding.  Therefore we obtain the probabilit y measur e functor P acting in the category C F MS ( ∗ ). Prop osition 3.10. F or any c omp act fuzzy metric sp ac e X , the map x 7→ δ x : X → P ( X ) is an isometric emb e dding. Pr o of. Let x, y ∈ X , t ∈ (0 , ∞ ) and M ( x, y , t ) = 1 − r 0 . Note that th en y / ∈ B ( x, r 0 , t ) = { x } r 0 ,t . Then for ev ery r ∈ (0 , 1), r < r 0 , we hav e y / ∈ B ( x, r , t ), whence δ x ( { x } ) = 1 > δ y ( { x } r,t ) + r = δ y ( B ( x, r , t )) + r = 0 + r = r . Therefore, ˆ M ( δ x , δ y , t ) ≥ 1 − r 0 . Let r ∈ (0 , 1), r > r 0 . If A 6 = ∅ is a Borel subs et of X w ith x ∈ A , then y ∈ B ( x, r , t ) ⊂ A r,t and 1 = δ x ( A ) ≤ δ y ( A r,t ) + r. Similarly , δ y ( A ) ≤ δ x ( A r,t ) + r . W e conclude th at ˆ M ( δ x , δ y , t ) ≤ 1 − r 0 .  Th u s, the identit y functor on the categ ory C F MS ( ∗ ) is a sub functor of the pr ob- abilit y measure f unctor P . FUZZY PROKHOR OV METRIC ON THE SET OF PR OBABILITY M EASURES 9 4. Extension of fu zzy metrics In this s ection w e provide an application of the Prokhoro v fuzzy metric to the problem of extensions of fu zzy metrics. Theorem 4.1. L et Y b e a nonempty close d subset of a c omp act metrizable sp ac e X . Then any fuzzy metric on Y c an b e extende d over X . Pr o of. Let M b e a fuzzy metric o n Y . without loss of generalit y , one ma y assume that Y is infinite (otherwise, one can m ultiply Y by an infinite c ompact fuzzy metric space, sa y [0 , 1] and attac h X to Y × [0 , 1] by ident ifyin g ev ery y ∈ Y and ( y , 0) ∈ Y × [0 , 1].) Then the space P ( Y ) is homeomorphic to the Hilb ert cub e and from the results of infinite-dimensional top ology of th e Hilb ert cub e it easily follo w s that there exists an em b ed ding F : X → P ( Y ) suc h that F ( y ) = δ y , for ev ery y ∈ Y (see e.g. [4]). Define M ′ : X × X × (0 , ∞ ) → R by the f orm ula M ′ ( x, y , t ) = ˆ M ( F ( x ) , F ( y ) , t ). It follo w s from Prop osition 3.10 that M ′ is an extension of M .  Epilogue A measur e µ on X is said to b e a subpr ob ability me asur e if ther e is a p robabilit y measure µ ′ on X suc h that µ ( A ) ≤ µ ′ ( A ), for ev ery Borel set A of X . The set of all subpr obabilit y measures on X can b e iden tified with the s ubspace of X ∪ { 1 } , where 1 = { 0 } stands for a terminal ob ject in th e catego ry of m etrizable spaces. Giv en a fuzzy metric M on a compact metrizable space X , there exists the uniqu e fuzzy metric M ′ : ( X ∪ { 1 } ) × ( X ∪ { 1 } ) × (0 , ∞ ) → R that extends M and such that M ′ ( x, 0 , t ) = 1 2 , f or ev ery x ∈ X . (Indeed, s ince, for ev ery x, y ∈ X and ev ery t, s ∈ (0 , ∞ ), w e hav e M ( x, y , t ) ∗ M ( y , 0 , s ) = M ( x, y , t ) ∗ 1 2 = max { M ( x, y , t ) − 1 2 , 0 } ≤ 1 2 = M ( x, 0 , t + s ) , the f uzzy metric M ′ is we ll-defin ed.) The set P ′ ( X ) of subpr obabilit y measures on X th en can b e interpreted as the set P ( X ∪ { 0 } ). The follo wing q uestions remains op en. Question 4.2. Are there analogues of the ab ov e resu lts for fuzzy metric spaces whic h are n ot necessarily compact? Question 4.3. Is there a fuzzy analogue of the K an toro vic h metric in the set of probabilit y m easures? See e.g. [13] for the d efinitions and prop erties of the Kant orovic h metric. Question 4.4. Is there a fuzzy analogue of the Prokhoro v metric in the set of prob- abilit y measures f or another choice of t-norm? Question 4.5. Is the canonical map ψ : ( P 2 ( X ) , ˆ ˆ M ) → ( P ( X ) , ˆ M ) nonexpand ing? The corresp on ding q uestion for the metric sp aces and nonexpandin g maps is dis- cussed in [20]. 10 D. REPOV ˇ S, A. SA VCHENK O, AND M. ZARICHNYI A cknowledgement s This researc h was supp orted by the Slov enian Researc h Agency grants P1-0292- 0101 and M-2057-010 1. W e thank the referees f or commen ts and suggestions. Referen ces [1] M. Barr and C. W ells, T op oses, T riples and The ori es , S p ringer-V erlag, Berlin, 1985 . [2] J. W. de Bakker an d J. I. Zuck er, Pr o c esses and denotational semantics of c oncurr ency , In for- mation and Con trol 54 (1982), 70–120. [3] E. P . de Vink and J. J. M. M. Ru t ten, Bisim ulation for pr ob abilistic tr ansition systems: a c o algebr aic appr o ach , ICALP ’97 (Bolog na), Theor. Comput. Sci. 221 :1-2 (1999), 271-293. [4] V. V. F edorch uk, Pr ob ability me asur es i n top olo gy , Usp ekh i Mat. Nauk 46:1 (1991), 41-80 (in Russian); Engl. transl. Russian Math. Surveys 46:1 (1991), 45–9 3. [5] A. George and P . V eeramani, On some r esults of analysis for f uzzy metric sp ac es , F uzzy Sets and Systems 90 (1997), 365– 368. [6] V. Gregori, S. Morillas and A. Sapena, Examples of f uzzy metrics and applic ations , F uzzy Sets and Systems (in p ress). [7] V. Gregori and S. Romaguera, Some pr op erties of fuzzy metric sp ac es , F uzzy Sets and Systems 115 (200 0), 485–489. [8] V. Gregori, S. R omaguera and P . V eeramani, A note on i ntuitionistic fuzzy metric sp ac es , Chaos, Solitons & F ractals 28 : 4 (2006), 902–9 05. [9] O. Hu b al and M. Zaric hnyi, Idemp otent pr ob ability me asur es on ultr ametric sp ac es , J. Math. Anal. Ap pl. 343 (2008), 1052–1060 . [10] I. Kramosil and J. Mic halek, F uzzy metric and statist i c al m etric sp ac es , Kybernetica 11 (1975 ), 326–334 . [11] D. Mihet ¸, F uzzy ψ -c ontr active mappings i n non-A r chime de an fuzzy metric sp ac es , F uzzy Sets and Systems 159 :6 (2008), 739– 744. [12] K. R. Parthasa rathy , Pr ob ability Me asur es on Metric Sp ac es , Academic Press, New Y ork, 1967. [13] S. T. R ac hev , Pr ob ability Metrics and the Stability of Sto chastic Mo dels , Wiley , Chichester, 1991. [14] M. R. S. Rahmat and M. S. M. No orani, Pr o duct of fuzzy metric sp ac es and fixe d p oint the or ems , Int. J. Contemp. Math. S ci. 3 :15 (2008), 703–712. [15] J. Ro dr ´ ıguez-L´ opez and S . Romaguera, The Hausdorff f uzzy metric on c omp act sets , F u zzy S ets and Systems 147 :2 (2004), 273– 283. [16] S. R omaguera , A. Sap ena and P . Tiradoi, The Banach fixe d p oint the or em in fuzzy quasi-metric sp ac es with applic ation to the domain of wor ds , T opology Appl. 154 (2007), 2196–2203 . [17] A. Sapena, A c ontribution to the study of fuzzy metric sp ac es, Appl. Gen. T op ology 2 :1 (2001), 63–75. [18] A. Sa vc henko, F uzzy hyp ersp ac e monad , preprin t. [19] A. Sav chenko and M. Zaric hnyi, F uzzy ultr am etrics on the set of pr ob ability me asur es , Proc. Infi- nite Dim. A nal. and T op ol. (I DA T) Conf. 2009, (Ed. by R. M. Aron, P . Galindo, A. Zagorod nyuk and M. Zaric hnyi), T op ology 48 :2-4 (2010), 130 –136. [20] F. v an Breugel, The metric monad for pr ob abilistic nondeterminism , preprin t. [21] F. v an Breugel and J. W orrell, A b ehaviour al pseudometric f or pr ob abilistic tr ansition syst ems , Theor. Comput . Sci. 331 :1 (2005), 115-142. [22] P . V eeramani, Best appr oxim ation in fuzzy metric sp ac es , J. F uzzy Math. 9 (2001), 75–80. FUZZY PROKHOR OV METRIC ON THE SET OF PR OBABILITY M EASURES 11 F acul ty of Ma thema tics, Physics and Mechani cs, and F acul ty of Educa tion, Univ er- sity of Ljubljana, P.O.B. 2964, 1001 Ljubljana, Slovenia E-mail addr ess : dusan.repo vs@guest.arnes.si Kherson Agrarian University, Kherson, Ukraine E-mail addr ess : savchenko1 960@rambler.ru L viv Na tional University, 79000 L viv, Ukrai ne and Institute of Ma thema tics, Physics and Mechanics, P.O.B. 2964, 1001 Ljubljana, Slovenia E-mail addr ess : mzar@litec h.lviv.ua