@ Appl. Gen. Topol. 21, no. 1 (2020), 35-51 doi:10.4995/agt.2020.11865 c© AGT, UPV, 2020 On a metric on the space of idempotent probability measures Adilbek Atakhanovich Zaitov Tashkent Institute of Architecture and Civil Engineering, 13, Navoi Str, Tashkent city, 100011, Uzbekistan Chirchik State Pedagogical Institute, 104, Amir Temur Str., Chirchik town, 111700, Uzbek- istan. (adilbek zaitov@mail.ru) Communicated by D. Werner Abstract In this paper we introduce a metric on the space I(X) of idempotent probability measures on a given compact metric space (X, ρ), which extends the metric ρ. It is proven the introduced metric generates the pointwise convergence topology on I(X). 2010 MSC: 28C20; 54E35. Keywords: compact metrizable space; idempotent measure; metrization. 1. Introduction Idempotent mathematics is based on replacing the usual arithmetic opera- tions with a new set of basic operations, i. e., on replacing numerical fields by idempotent semirings and semifields. Typical example is the so-called max-plus algebra Rmax. Many authors (S. C. Kleene, S. N. N. Pandit, N. N. Vorobjev, B. A. Carré, R. A. Cuninghame-Green, K. Zimmermann, U. Zimmermann, M. Gondran, F. L. Baccelli, G. Cohen, S. Gaubert, G. J. Olsder, J.-P. Quadrat, and others) used idempotent semirings and matrices over these semirings for solving some applied problems in computer science and discrete mathematics, starting from the classical paper by S. C. Kleene [7]. Received 21 May 2019 – Accepted 26 November 2019 http://dx.doi.org/10.4995/agt.2020.11865 A. A. Zaitov The modern idempotent analysis (or idempotent calculus, or idempotent mathematics) was founded by V. P. Maslov and his collaborators [10]. Some preliminary results are due to E. Hopf and G. Choquet, see [2], [5]. Idempotent mathematics can be treated as the result of a dequantization of the traditional mathematics over numerical fields as the Planck constant h tends to zero taking imaginary values. This point of view was presented by G. L. Litvinov and V. P. Maslov [11]. In other words, idempotent mathematics is an asymptotic version of the traditional mathematics over the fields of real and complex numbers. The basic paradigm is expressed in terms of an idempotent correspondence principle. This principle is closely related to the well-known correspondence principle of N. Bohr in quantum theory. Actually, there exists a heuristic corre- spondence between important, interesting, and useful constructions and results of the traditional mathematics over fields and analogous constructions and re- sults over idempotent semirings and semifields (i. e., semirings and semifields with idempotent addition). A systematic and consistent application of the idempotent correspondence principle leads to a variety of results, often quite unexpected. As a result, in parallel with the traditional mathematics over fields, its “shadow,” idempo- tent mathematics, appears. This “shadow” stands approximately in the same relation to traditional mathematics as classical physics does to quantum theory. Recall [10] that a set S equipped with two algebraic operations: addition ⊕ and multiplication �, is said to be a semiring if the following conditions are satisfied: • the addition ⊕ and the multiplication � are associative; • the addition ⊕ is commutative; • the multiplication � is distributive with respect to the addition ⊕: x� (y ⊕z) = x�y ⊕x�z and (x⊕y) �z = x�z ⊕y �z for all x, y, z ∈ S. A unit of a semiring S is an element 1 ∈ S such that 1 � x = x � 1 = x for all x ∈ S. A zero of the semiring S is an element 0 ∈ S such that 0 6= 1 and 0 ⊕ x = x ⊕ 0 = x for all x ∈ S. A semiring S is called an idempotent semiring if x⊕x = x for all x ∈ S. A (an idempotent) semiring S with neutral elements 0 and 1 is called a (an idempotent ) semifield if every nonzero element of S is invertible. Note that diöıds, quantales and inclines are examples of idempotent semirings [10]. Let R = (−∞, +∞) be the field of real numbers and R+ = [0, +∞) be the semiring of all nonnegative real numbers (with respect to the usual addition and multiplication). Consider a map Φh : R+ → S = R∪{−∞} defined by the equality Φh(x) = h ln x, h > 0. c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 36 A metric on the space of idempotent measures Let x, y ∈ X and u = Φh(x), v = Φh(y). Put u⊕h v = Φh(x + y) and u�v = Φh(xy). The imagine Φh(0) = −∞ of the usual zero 0 is a zero 0 and the imagine Φh(1) = 0 of the usual unit 1 is a unit 1 in S with respect to these new operations. Thus S obtains the structure of a semiring R(h) isomorphic to R+. A direct check shows that u⊕h v → max{u, v} as h → 0. The convention −∞�x = −∞ allows us to extend ⊕ and � over S. It can easily be checked that S forms a semiring with respect to the addition u ⊕ v = max{u, v} and the multiplication u � v = u + v with zero 0 = −∞ and unit 1 = 0. Denote this semiring by Rmax; it is idempotent, i. e., u⊕u = u for all its elements u. The semiring Rmax is actually a semifield. The analogy with quantization is obvious; the parameter h plays the role of the Planck constant, so R+ can be viewed as a “quantum object” and Rmax as the result of its “dequantization”. This passage to Rmax is called the Maslov dequantization (for details, see [8], [9], [15]). The notion of idempotent (Maslov) measure finds important applications in different parts of mathematics, mathematical physics and economics (see the survey article [10] and the bibliography therein). Topological and categorical properties of the functor of idempotent measures were studied in [16], [17]. Although idempotent measures are not additive and the corresponding func- tionals are not linear, there are some parallels between topological properties of the functor of probability measures and the functor of idempotent mea- sures (see, for example [15], [14], [16]) which are based on existence of natural equiconnectedness structure on both functors. However, some differences appear when the problem of the metrizability of the space of idempotent probability measures is studied. The problem of the metrizability of the space of the usual probability measures was investigated in [3]. We show that the analog of the metric introduced in [3] (on the space of probability measures) is not a metric on the space of idempotent probability measures. We show the mentioned analog is only a pseudometric. It is well-known that if (X, ρ) is a compact metric space, then the space P(X) of probability measures can be endowed with the Kantorovich metric. In [17], M. Zarichnyi posed the problem of building a metric on the space of idempotent probability measures. Still the problem of existence of a natural metrization of the space I(X) has been open. In this paper we give a positive answer and introduce a metric on the space of idempotent probability measures. 2. Idempotent probability measures. Preliminaries Let X be a compact Hausdorff space, C(X) be the algebra of continuous functions on X with the usual algebraic operations. On C(X) the operations ⊕ and � are determined by ϕ⊕ψ = max{ϕ,ψ} and ϕ�ψ = ϕ + ψ where ϕ, ψ ∈ C(X). c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 37 A. A. Zaitov Recall [17] that a functional µ: C(X) → R is said to be an idempotent probability measure on X if it satisfies the following properties: (1) µ(λX) = λ for all λ ∈ R, where λX is a constant function; (2) µ(λ�ϕ) = λ�µ(ϕ) for all λ ∈ R and ϕ ∈ C(X); (3) µ(ϕ⊕ψ) = µ(ϕ) ⊕µ(ψ) for all ϕ, ψ ∈ C(X). For a compact Hausdorff space X by I(X) we denote the set of all idempotent probability measures on X. Since I(X) ⊂ RC(X), we consider I(X) as a subspace of RC(X). A family of sets of the form 〈µ; ϕ1, . . . , ϕn; ε〉 = {ν ∈ I(X) : |ν(ϕi) −µ(ϕi)| < ε, i = 1, . . . , n} is a base of open neighbourhoods of a given idempotent probability measure µ ∈ I(X) according to the induced topology, where ϕi ∈ C(X), i = 1, . . . , n, and ε > 0. It is obvious that the induced topology and the pointwise convergence topology on I(X) coincide. Let X, Y be compact Hausdorff spaces and f : X → Y be a continuous map. It is easy to check that the map I(f) : I(X) → I(Y ) determined by the formula I(f)(µ)(ψ) = µ(ψ ◦f) is continuous. The construction I is a normal functor acting in the category of compact Hausdorff spaces and their continuous maps. Therefore, for each idempotent probability measure µ ∈ I(X) one may determine its support : suppµ = ⋂{ A ⊂ X : A = A, µ ∈ I(A) } . Consider functions of the type λ : X → [−∞, 0]. On a given set X we determine a max-plus-characteristic function ⊕χA : X → Rmax of a subset A ⊂ X by the rule ⊕χA(x) = { 0 at x ∈ A, −∞ at x ∈ X \A. (2.1) For a singleton {x} we will write ⊕χx instead of ⊕χ{x}. Let F1, F2, . . . , Fn be a disjoint system of closed sets of a space X, and a1, a2, . . . , an be non-positive real numbers. A function ⊕χ a1, ...,an F1, ...,Fn (x) =   a1 at x ∈ F1, . . . , an at x ∈ Fn, −∞ at x ∈ X \ n⋃ i=1 Fn (2.2) we call the max-plus-step-function defined by the sets F1, F2, . . . , Fn and the numbers a1, a2, . . . , an. Note that ⊕χaA(x) = a� ⊕χA(x) = { 0 �a at x ∈ A, −∞ at x ∈ X \A = { a at x ∈ A, −∞ at x ∈ X \A c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 38 A metric on the space of idempotent measures for a set A in X and a non-positive number a. Consequently, for a disjoint system of closed sets F1, F2, . . . , Fn in a space X, and non-positive real numbers a1, a2, . . . , an we have ⊕χ a1, ...,an F1, ...,Fn (x) = ⊕χa1F1 (x) ⊕ ⊕χa2F2 (x) ⊕ . . . ⊕ ⊕χanFn (x). In the case when F1, F2, . . . , Fn are singletons, say Fi = {xi}, i = 1, . . . , n, we have (2.3) ⊕χ a1, ...,an {x1}, ...,{xn} = ⊕χa1{x1} ⊕ ⊕χa2{x2} ⊕ . . . ⊕ ⊕χan{xn}. The notion of density for an idempotent measure was introduced in [8], where the main result on the existence on densities for arbitrary measures was proved. A more detailed exposition is given in [9] – the first systematic monograph on the idempotent analysis. Later the paper [1] appeared, where further investigations of densities were done. Let µ ∈ I(X). Then we can define a function dµ : X → [−∞, 0] by the formula (2.4) dµ(x) = inf{µ(ϕ) : ϕ ∈ C(X) such that ϕ ≤ 0 and ϕ(x) = 0}, x ∈ X. The function dµ is upper semicontinuous and is called the density of µ. Con- versely, each upper semicontinuous function f : X → [−∞, 0] with max{f(x) : x ∈ X} = 0 determines an idempotent measure νf by the formula (2.5) νf (ϕ) = ⊕ x∈X f(x) �ϕ(x), ϕ ∈ C(X). Note that a function f : X → R is said to be upper semicontinuous if for each x ∈ X, and for every real number r which satisfies f(x) < r, there exists an open neighbourhood U ⊂ X of x such that f(x′) < r for all x′ ∈ U. It is easy to see that functions defined by (2.1) or by (2.2) are upper semicontinuous. Put US(X) = { λ: X → [−∞, 0] ∣∣ λ is upper semicontinuous and there exists a x0 ∈ X such that λ(x0) = 0 } . Then we have I(X) = {⊕ x∈X λ(x) �δx : λ ∈ US(X) } . Obviously that ⊕ x∈X ⊕χx0 (x) � δx = δx0 , i. e. for a max-plus-characteristic function ⊕χx0 formula (2.5) defines the Dirac measure δx0 supported on the singleton {x0}. A set of all Dirac measures on a Hausdorff compact space X we denote by δ(X). It is easy to notice that the space X and the subspace δ(X) ⊂ I(X) are homeomorphic. This phenomenon gives us the opportunity to consider X as subspace of I(X). Let A be a closed subset of a compact Hausdorf space X. It is easy to check that ν ∈ I(A) iff {x ∈ X : dν(x) > −∞}⊂ A. Hence, supp ν = {x ∈ X : dν(x) > −∞}. c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 39 A. A. Zaitov It is evident that supp ν = {x1, . . . , xn} if and only if the density dν of ν has the shape (2.3) for singletons {x1}, . . . , {xn} and for some non-negative numbers a1, . . . , an with ai > −∞, i = 1, . . . , n, and max{a1, . . . , an} = 0. In this case ν is said to be an idempotent probability measure with finite support. A subset of I(X) consisting of all idempotent probability measures with finite support we denote by Iω(X). Consider an idempotent probability measure µ = ⊕ x∈X λ(x)�δx ∈ I(X) and a finite system {U1, . . . , Un} of open sets Ui ⊂ X such that supp µ∩Ui 6= ∅, i = 1, . . . , n, and supp µ ⊂ n⋃ i=1 Ui. Define a set (2.6) 〈µ; U1 . . . , Un; ε〉 = { ν = ⊕ x∈X γ(x) �δx ∈ I(X) : supp ν ∩Ui 6= ∅, supp ν ⊂ n⋃ i=1 Ui, and |λ(x) −γ(y)| < ε at the points x ∈ supp µ∩Ui and y ∈ supp ν ∩Ui, i = 1, . . . , n, } . Theorem 2.1. The sets of the type (2.6) form a base of the pointwise conver- gence topology in I(X). Proof. Let 〈µ; ϕ; ε〉 be a prebase element, where ϕ ∈ C(X), ε > 0 and µ =⊕ x∈X λ(x) � δx ∈ I(X). As ϕ is continuous, for each point x ∈ supp µ there is its open neighbourhood Ux in X such that for any point y ∈ Ux the inequality |ϕ(x)−ϕ(y)| < ε 2 holds. From the open cover {Ux : x ∈ supp µ} in X of supp µ by owing to compactness of supp µ one can choose a finite subcover {Ui : i = 1, . . . , n}. Further, for every ν = ⊕ x∈X γ(x) � δx ∈ 〈µ; U1, . . . , Un; ε2〉 we have |λ(x) −γ(y)| < ε 2 at x ∈ supp µ∩Ui and y ∈ supp ν ∩Ui. Let us estimate the following absolute value |µ(ϕ)−ν(ϕ)| = ∣∣∣∣ ⊕ x∈X λ(x) �ϕ(x) − ⊕ x∈X γ(x) �ϕ(x) ∣∣∣∣ = a. Two cases are possible: Case 1 : ⊕ x∈X λ(x)�ϕ(x) ≥ ⊕ x∈X γ(x)�ϕ(x). Let ⊕ x∈X λ(x)�ϕ(x) = λ(x′)� ϕ(x′). Then x′ ∈ Ui for some i, and a = ⊕ x∈X λ(x) �ϕ(x) − ⊕ x∈X γ(x) �ϕ(x) = λ(x′) �ϕ(x′) − ⊕ x∈X γ(x) �ϕ(x) ≤ ≤ (for every y ∈ supp ν ∩Ui) ≤ ≤ λ(x′) �ϕ(x′) −γ(y) �ϕ(y) = |λ(x′) �ϕ(x′) −γ(y) �ϕ(y)| ≤ ≤ |λ(x′) −γ(y)| + |ϕ(x′) −ϕ(y)| < ε 2 + ε 2 = ε. c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 40 A metric on the space of idempotent measures Case 2 : ⊕ x∈X λ(x)�ϕ(x) ≤ ⊕ x∈X γ(x)�ϕ(x). Let ⊕ x∈X γ(x)�ϕ(x) = γ(x′)� ϕ(x′). Then x′ ∈ Ui for some i, and a = ⊕ x∈X γ(x) �ϕ(x) − ⊕ x∈X λ(x) �ϕ(x) = γ(x′) �ϕ(x′) − ⊕ x∈X λ(x) �ϕ(x) ≤ ≤ (for every y ∈ supp µ∩Ui) ≤ ≤ γ(x′) �ϕ(x′) −λ(y) �ϕ(y) = |γ(x′) �ϕ(x′) −λ(y) �ϕ(y)| ≤ ≤ |λ(x′) −γ(y)| + |ϕ(x′) −ϕ(y)| < ε 2 + ε 2 = ε. So, |µ(ϕ) −ν(ϕ)| < ε. From here ν ∈ 〈µ; ϕ; ε〉, in other words,〈 µ; U1, . . . , Un; ε 2 〉 ⊂〈µ; ϕ; ε〉. � Theorem 2.1 immediately yields the following statement. Corollary 2.2. The subset Iω(X) is everywhere dense in I(X) with respect to the pointwise convergence topology. We recall some concepts from [13] and if necessary, modify them for the max-plus case . Let X and Y be compact Hausdorff spaces, f : X → Y be a map, f◦ : C(Y ) → C(X) be the induced operator defined by equality f◦(ϕ) = ϕ ◦ f, ϕ ∈ C(Y ). We say that an operator u: C(X) → C(Y ) is a max-plus-linear operator provided u(α � ϕ ⊕ β � ψ) = α � u(ϕ) ⊕ β � u(ψ) for every pair of functions ϕ, ψ ∈ C(X), where −∞ ≤ α, β ≤ 0, α ⊕ β = 0. A max-plus-linear operator u: C(X) → C(Y ) is max-plus-regular provided ‖u‖ = sup{‖u(ϕ)‖ : ϕ ∈ C(X), ‖ϕ‖ ≤ 1} = 1 and u(1X) = 1Y . A max-plus- linear operator u: C(X) → C(Y ) is said to be a max-plus-linear exave for f provided f◦◦u is the identity on f◦(C(Y )) or equivalently f◦◦u◦f◦ = f◦. A max-plus-regular exave is a max-plus-linear exave which is a regular operator. If f is a homeomorphic embedding, then a max-plus-linear exave (max-plus- regular exave) for f is called max-plus-linear extension operator (max-plus- regular extension operator ). If f is a surjective map, then a max-plus-linear exave (max-plus-regular exave) for f is called max-plus-linear averaging oper- ator (max-plus-regular averaging operator ). Remind, in category theory a monomorphism (an epimorphism) is a left- cancellative (respectively, right-cancellative) morphism, that is, a morphism f : Z → X (respectively, f : X → Y ) such that, for each pair of morphisms g1, g2 : Y → Z the following implication holds f ◦g1 = f ◦g2 ⇒ g1 = g2 (respectively, g1 ◦f = g2 ◦f ⇒ g1 = g2). If u is an exave for f : X → Y and y ∈ f(X), then for every function ϕ ∈ C(Y ) we have (u◦f◦)(ϕ)(y) = ϕ(y). c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 41 A. A. Zaitov Proposition 2.3. Let f : X → Y be a map. A max-plus-regular operator u: C(X) → C(Y ) is a max-plus-regular extension (respectively, averaging) op- erator if and only if f◦ ◦u = idC(X) (respectively, u◦f◦ = idC(Y )). Proof. Let u be a max-plus-regular extension (respectively, averaging) opera- tor. Then the induced operator f◦ : C(Y ) → C(X) is an epimorphism (respec- tively, monomorphism). That is why equalities f◦ ◦u◦f◦ = f◦ = idC(X) ◦f◦ imply f◦ ◦ u = idC(X) (respectively, f◦ ◦ u ◦ f◦ = f◦ = f◦ ◦ idC(Y ) imply u◦f◦ = idC(Y )). Let u be a max-plus-regular operator and f◦ ◦ u = idC(X). It requires to show f : X → Y is an embedding. Suppose f(x1) = f(x2), x1, x2 ∈ X. Assume there exists a function ϕ ∈ C(X) such that ϕ(x1) 6= ϕ(x2). Conversely, we have ϕ(x1) = f ◦ ◦u(ϕ)(x1) = u(ϕ)(f(x1)) = u(ϕ)(f(x2)) = f◦ ◦u(ϕ)(x2) = ϕ(x2). We get a contradiction. So, x1 = x2. Let u be a max-plus-regular operator and u◦f◦ = idC(Y ). We should show that f : X → Y is a surjective map. Suppose f is not so. Then Y \f(X) 6= ∅ and for every y ∈ Y \f(X), since the image f(X) is a compact Hausdorff space, any ϕ: f(X) → R has different extensions ϕ1, ϕ2 : Y → R such ϕ1(y) 6= ϕ2(y). Hence, ϕ1 6= ϕ2. On the other hand ϕ1 = u◦f◦(ϕ1) = u◦f◦(ϕ2) = ϕ2. The obtained contradiction finishes the proof. � An epimorphism f : X → Y is said to be a max-plus-Milutin epimorphism provided it permits a max-plus-regular averaging operator. A compact Haus- dorff space X is a max-plus-Milutin space if there exists a max-plus-Milutin epimorphism f : Dτ → X [13]. Every compact metrizable space is a Milutin space ([4], Corollary VIII.4.6.). Analogously, every compact metrizable space is a max-plus-Milutin space. 3. An analog of the Kantorovich metric It is well-known (see, for example [4]) that every zero-dimensional space of the weight m ≥ ℵ0 embeds into Cantor cube Dm. Consequently, a zero- dimensional compact metrizable space is a max-plus-Milutin space. Let µi = ⊕ x∈X λi(x) � δx ∈ I(X), i = 1, 2. Put Λ1 2 = Λ(µ1, µ2) = {ξ ∈ I(X2) : I(πi)(ξ) = µi, i = 1, 2}, where πi : X × X → X is the projection onto i-th factor, i = 1, 2. We will show the set Λ(µ1, µ2) is nonempty. Let xi 0 ∈ supp µi be points such that λi(xi 0) = 0, i = 1, 2. Then the direct checking shows that I(πi)(ξ) = µi, i = 1, 2, for all ξ ∈ I(X2) of the form ξ = ξ0 ⊕R(µ1,µ2). Here ξ0 = 0 � δ(x1 0,x2 0) ⊕ x∈X\{x1 0} λ2(x) � δ(x1 0,x) ⊕ ⊕ x∈X\{x2 0} λ1(x) � δ(x,x2 0) c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 42 A metric on the space of idempotent measures is an idempotent probability measure on X2, and R(µ1, µ2) = ⊕ x∈X\{x1 0} y∈X\{x2 0} γ(x, y) �δ(x,y) is some functional on C(X) where −∞≤ γ(x, y) ≤ min{λ1(x), λ2(y)}, x ∈ M, y ∈ N, M ⊂ X \{x1 0}, N ⊂ X \{x2 0}. Thus ξ ∈ Λ(µ1, µ2), i. e. Λ(µ1, µ2) 6= ∅. In fact, here more is proved: it is easy to see if |X| ≥ 2 and |Y | ≥ 2 then quantity of the numbers γ(x, y) is uncountable. From here one concludes that the potency of the set Λ(µ1, µ2) is no less than continuum potency as soon as each of the supports supp µi, i = 1, 2, contains no less than two points. Note that ξ = ξ0 if one takes empty set as K and M. Idempotent probability measures ξ ∈ I(X2) with I(πi)(ξ) = µi, i = 1, 2 we will call as a coupling of µ1 and µ2. The following statement is rather evident. Proposition 3.1. Let µi = ⊕ x∈X λi(x) �δx, i = 1, 2, be idempotent probability measures. Then every their coupling ξ = ⊕ (x,y)∈X2 λ1 2(x, y) � δ(x,y) ∈ I(X2) satisfies the following equalities: λ1(x) = ⊕ y∈X λ1 2(x, y), x ∈ X, and λ2(y) = ⊕ x∈X λ1 2(x, y), y ∈ X. Consider a compact metrizable space (X, ρ). We define a function ρ0 : I(X)× I(X) → R by the formula ρ0(µ1, µ2) = inf{ξ(ρ) : ξ ∈ Λ1 2}. This function was offered by V. V. Uspenskii and in [3] it was proved that it is a metric on the space P(X) of probability measures. Its analog for idempotent probability measures is not a metric on the space of idempotent probability measures. L. V. Kantorovich, G. Sh. Rubinshtein offer another metric on the space of all measures [6]. For the space of probability measures their metric has the form ρK(µ1, µ2) = inf{ξ(ρ) : ξ ∈ P(X ×X), P(π1)(ξ) −P(π2)(ξ) = µ1 −µ2}. In [12] it was shown that on the space of all probability measures the above metrics ρ0 and ρK coincide. Proposition 3.2. For every pair µ1, µ2 ∈ I(X) there exists a coupling ξ ∈ I(X2) of µ1 and µ2 such that ρ0(µ1, µ2) = ξ(ρ). c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 43 A. A. Zaitov Proof. Consider a sequence {ξn} of couplings of µ1 and µ2 such that ξn(ρ) −→ ρ0(µ1, µ2). Passing in the case of need to a subsequence, owing to the compact- ness of I(X2), it is possible to assume that {ξn} tends to some ξ ∈ I(X2). Since the projections I(πi) are continuous, ξ is a coupling of µ1 and µ2. Further, for an arbitrary ε > 0 there exists n0 such that ξn ∈ 〈ξ; ρ; ε〉 for all n ≥ n0, where 〈ξ; ρ; ε〉 is a prebase neighbourhood of ξ in the pointwise convergence topology on I(X2). So, |ξ(ρ) − ξn(ρ)| < ε. Consequently, ρ0(µ1, µ2) = ξ(ρ). � Proposition 3.3. The function ρ0 is a pseudometric on I(X). Proof. Since each ξ ∈ I(X2) is order-preserving then the inequality ρ ≥ 0 immediately implies ρ0 ≥ 0. So, ρ0 is nonnegative. Obviously, ρ0 is symmetric. Let µ1 = µ2 = µ. There exists λ ∈ US(X) such that µ = ⊕ x∈X λ(x) � δx. Then ξµ = ⊕ x∈X λ(x) � δ(x,x) is a coupling of µ1 and µ2, and 0 ≤ ρ0(µ1, µ2) = inf{ξ(ρ) : ξ ∈ Λ1 2}≤ ξµ(ρ) = ⊕ x∈X λ(x) = 0, i. e. ρ0(µ1, µ2) = 0. Let us show that the triangle inequality is true as well. Take arbitrary triple µi ∈ I(X), i = 1, 2, 3. Let µ1 2, µ2 3 ∈ I(X2) be couplings of µ1 and µ2, and µ2 and µ3, respectively, such that ρ0(µ1, µ2) = µ1 2(ρ) and ρ0(µ2, µ3) = µ2 3(ρ), respectively. For a compact Hausdorff space X we put X1 = X2 = X3 = X, X1 2 3 = X 3 = X1 ×X2 ×X3, Xij = X 2 = Xi ×Xj, and let π1 2 3ij : X1 2 3 → Xij, π ij k : Xij → Xk, 1 ≤ i < j ≤ 3, k ∈{i, j}, be corresponding projections. According to Corollary 4.3 [17] the functor I is bicommutative. Using this fact one can similarly to Lemma 4 [3] show that for idempotent probability measures µ2 ∈ I(X2), µ1 2 ∈ I(X1 2), µ2 3 ∈ I(X2 3) such that I(π1 22 )(µ1 2) = µ2 = I(π 2 3 2 )(µ2 3), there exists µ1 2 3 ∈ I(X1 2 3) which satisfies the equalities I(π1 2 31 2 )(µ1 2 3) = µ1 2 and I(π 1 2 3 2 3 )(µ1 2 3) = µ2 3. Put (3.1) µ1 3 = I(π 1 2 3 1 3 )(µ1 2 3). c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 44 A metric on the space of idempotent measures Then according to Proposition 3.1 µ1 3 is a coupling of µ1 and µ3. Using Proposition 3.1, we obtain ρ0(µ1, µ2) + ρ0(µ2, µ3) = µ1 2(ρ) + µ2 3(ρ) = = ⊕ (x1,x2)∈X1 2 dµ1 2 (x1, x2) �ρ(x1, x2) + ⊕ (x2,x3)∈X2 3 dµ2 3 (x2, x3) �ρ(x2, x3) = = ⊕ (x1,x2,x3)∈X1 2 3 dµ1 2 3 (x1, x2, x3) �ρ(x1, x2)+ + ⊕ (x1,x2,x3)∈X1 2 3 dµ1 2 3 (x1, x2, x3) �ρ(x2, x3) ≥ ≥ ⊕ (x1,x2,x3)∈X1 2 3 (dµ1 2 3 (x1, x2, x3) �ρ(x1, x2)+ +dµ1 2 3 (x1, x2, x3) �ρ(x2, x3)) = = ⊕ (x1,x2,x3)∈X1 2 3 dµ1 2 3 (x1, x2, x3) � (ρ(x1, x2) + ρ(x2, x3)) ≥ ≥ ⊕ (x1,x2,x3)∈X1 2 3 dµ1 2 3 (x1, x2, x3) �ρ(x1, x3) = = ⊕ (x1,x3)∈X1 3 dµ1 3 (x1, x3) �ρ(x1, x3) = µ1 3(ρ) ≥ ρ0(µ1, µ3), i. e. ρ0(µ1, µ3) ≤ ρ0(µ1, µ2) + ρ0(µ2, µ3). Here dν is the density function of the corresponding measure ν ((2.4), see page 39). � Unlike usual probability measures, the function ρ0 is not a metric on I(X). Example 3.4. Let (X, ρ) be a metric space, x, y ∈ X be points such that ρ(x, y) = 1. Consider idempotent probability measures µ1 = 0�δx⊕(−2)�δy and µ2 = 0 � δx ⊕ (−4) � δy. One can directly check that the idempotent probability measure ξ = 0�δ(x,x) ⊕(−2)�δ(y,x) ⊕(−4)�δ(x,y) is a coupling of µ1 and µ2, and ξ(ρ) = 0. That is why ρ0(µ1, µ2) = 0, though µ1 6= µ2. Example 3.4 shows that the functors P of probability measures and I of idempotent probability measures are not isomorphic. 4. On a metric on the space of idempotent probability measures Let (X, ρ) be a metric compact space. We define distance functions ρ1 : I(X)× I(X) → R and ρ2 : Iω(X) × Iω(X) → R as follows ρ1(µ1, µ2) = inf{sup{ρ(x, y) : (x, y) ∈ supp ξ} : ξ ∈ Λ1 2}, c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 45 A. A. Zaitov where µ1,µ2 ∈ I(X), and ρ2(µ1, µ2) = inf   ∑ (x,y)∈supp ξ eλ1(x)+λ2(y) ·ρ(x, y)∑ x∈supp µ1 eλ1(x) · ∑ y∈supp µ2 eλ2(y) : ξ ∈ Λ1 2   , where µi = ⊕ x∈X λi(x) �δx ∈ Iω(X), i = 1, 2. It is easy to notice that ρ2 ≤ ρ1 on Iω(X). The following statement has technical character and its proof consists of labour-intensive calculations (similarly calculations were done in [16]). Lemma 4.1. For every pair µi = ⊕ x∈X λi(x) �δx ∈ Iω(X), i = 1, 2, and for a coupling ξ ∈ Iω(X2) of µ1 and µ2 we have ρ2(µ1, µ2) = ∑ (x,y)∈supp ξ eλ1(x)+λ2(y) ·ρ(x, y)∑ x∈supp µ1 eλ1(x) · ∑ y∈supp µ2 eλ2(y) if and only if ρ0(µ1, µ2) = ξ(ρ). Theorem 4.2. The function ρ1 is a metric on I(X) which is an extension of the metric ρ. Proof. Obviously, ρ1 is nonnegative and symmetric. If µ1 = µ2 then similarly to the proof of Proposition 3.3 one can show that ρ1(µ1,µ2) = 0. Inversely, let ρ1(µ1,µ2) = 0. Then there exists a coupling ξ ∈ Λ1 2 such that ρ(x, y) = 0 for all (x, y) ∈ supp ξ. Consequently supp ξ must lie in the diagonal ∆(X) = {(x, x) : x ∈ X}. Applying Proposition 3.1, we have dµ1 = dµ2 , which implies µ1 = µ2. Now, it remains to check the triangle axiom. But the checking consists only of the repeating of procedure at the proof of Proposition 3.3. For every pair of Dirac measures δx, δy, x, y ∈ X, the uniqueness of a coupling ξ ∈ I(X2) of δx and δy, ξ = 0 � δ(x,y), implies that ρ1(δx, δy) = ξ(ρ) ⊕ρ(x, y) = 0 � δ(x,y)(ρ) ⊕ρ(x, y) = ρ(x, y). From here we get that ρ1 is an extension of ρ. � Lemma 4.3. diam(I(X), ρ1) = diam(X, ρ). Proof. Indeed, since we may consider X as a subspace of I(X) we get diam(X, ρ) ≤ diam(I(X), ρ1). On the other hand, by construction we have ρ1(µ1, µ2) = inf{sup{ρ(x, y) : (x, y) ∈ supp ξ} : ξ ∈ Λ1 2}≤ ≤ sup{ρ(x, y) : (x, y) ∈ supp ξ}≤ sup{ρ(x, y) : (x, y) ∈ X×X} = diam(X, ρ) for an arbitrary pair µ1,µ2 ∈ I(X). Consequently, diam(I(X), ρ1) ≤ diam(X, ρ). � c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 46 A metric on the space of idempotent measures Theorem 4.4. The function ρ2 is a metric on Iω(X) which is an extension of the metric ρ. Proof. By construction ρ2 is non-negative. It is clear that ρ2 is symmetric. The above noticed inequality ρ2 ≤ ρ1 on Iω(X) implies that ρ2 satisfies the identity axiom, i. e. ρ2(µ, ν) = 0 if and only if µ = ν. By definition we have ρ2(δx, δy) = ρ(x, y). For a triple µi ∈ Iω(X), i = 1, 2, 3, let µ1 2, µ2 3 ∈ Iω(X2) be couplings of µ1 and µ2, and µ2 and µ3, respectively, satisfying Proposition 3.2. Let µ1 3 ∈ Iω(X2) be an idempotent probability measures, defined by (3.1). Then Proposition 3.1 yields that µ1 3 is a coupling of µ1 and µ3. Applying Proposition 3.1, Lemma 4.1 and Theorem 2, we have ρ2(µ1, µ2) + ρ2(µ2, µ3) ≥ ρ2(µ1, µ3). � Let µ, ν ∈ I(X).Corollary 2.2 implies the existence of sequences {µn}, {νn}⊂ Iω(X) converging to µ and ν respectively. We have 0 ≤ ρ2(µn,νn) ≤ρ1(µn,νn) ≤ diam(X, ρ). Therefore there exists a limit of the sequence {ρ2(µn,νn)}. Put ρI(µ, ν) = lim n→∞ ρ2(µn,νn). Now Theorem 4.4 gives the following result. Corollary 4.5. The function ρI is a metric on I(X) which is an extension of the metric ρ. Note that ρI ≤ ρ1. For this reason from Lemma 4.3 we obtain the following statement. Corollary 4.6. diam(I(X), ρI) = diam(X, ρ). Proposition 4.7. Let X be a compact metrizable space and a sequence {µn}⊂ I(X) converges to µ0 ∈ I(X) with respect to point-wise convergence topology. Then for every open neighbourhood U of the diagonal ∆(X) = {(x, x) : x ∈ X} there exist a positive integer n and a coupling µ0 n ∈ I(X2) of µ0 and µn such that (4.1) ⊕ (x,y)∈X2\U dµ0 n (x, y) �ρ(x, y) = −∞. Proof. At first we consider the case of zero-dimensional compact metrizable space X. There exists a disjoint clopen cover {V1, . . . , Vn} of X (i. e. a cover, which consists of open-closed sets of X) such that Vi × Vi ⊂ U for each i = 1, . . . , n. As µn → µ there exists n such that µn ∈ 〈µ; ⊕χV1, ⊕χV2, . . . , ⊕χVn ; ε〉. We will construct a coupling µ0 n ∈ I(X2) of µ0 and µn. There exists a base of the compact metrizable space X consisting of clopen sets V ε1ε2...εk i , 1 ≤ i ≤ s, εk ∈{0, 1}, 1 ≤ k < ∞, such that c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 47 A. A. Zaitov 1) V 0i ∪V 1 i = Vi; 2) V 0i ∩V 1 i = ∅; 3) V ε1ε2...εk0 i ∪V ε1ε2...εk1 i = V ε1ε2...εk i ; 4) V ε1ε2...εk0 i ∩V ε1ε2...εk1 i = ∅. The sets V ε1ε2...εk i ×V ε′1ε ′ 2...ε ′ k i′ form a base of the compact metrizable space X1 2. To determine µ0 n it is enough to construct its density function. Let µ0 = ⊕ x∈X λ0(x) � δx, µn = ⊕ x∈X λn(x) �δx. We set λ ε1...εk,ε ′ 1...ε ′ k ii′ = ⊕ (x,y)∈X×X (λ0(x) �λn(y)) �δ(x,y)(⊕χ V ε1...εk i ×V ε′ 1 ...ε′ k i′ ), i. e. λ ε1...εk,ε ′ 1...ε ′ k ii′ = ⊕ (x,y)∈V ε1...εk i ×V ε′ 1 ...ε′ k i′ λ0(x) �λn(y). It is clear that λ ε′1...ε ′ k i′ = s⊕ i=1 λ ε1...εk,ε ′ 1...ε ′ k ii′ and λ ε1...εk i = s⊕ i′=1 λ ε1...εk,ε ′ 1...ε ′ k ii′ , where λ ε1...εk i = ⊕ x∈X λ0(x) � δx(⊕χV ε1...εk i ) = ⊕ x∈V ε1...εk i λ0(x) and λ ε′1...ε ′ k i′ = ⊕ x∈X λn(x) � δx(⊕χ V ε′ 1 ...ε′ k i ) = ⊕ x∈V ε′ 1 ...ε′ k i′ λn(x). Put dµ0 n = lim s→∞ s⊕ i, i′=1 ⊕χ λ ε1...εk, ε ′ 1...ε ′ k i i′ V ε1...εk i ×V ε′ 1 ...ε′ k i′ . Then dµ0 n is an upper semicontinuous function on X 2 and µ0,n =⊕ (x,y)∈X2 dµ0 n (x, y) � δ(x,y) is a coupling of µ0 and µn with supp µ0,n ⊂ U. Consequently, ⊕ (x,y)∈X2\U dµ0 n (x, y) = −∞ and, the equation (4.1) is proved for the zero-dimensional case. Now let X be an arbitrary compact metrizable space. There exists a zero- dimensional compact metrizable space Z, a max-plus-Milutin epimorphism f : Z → X and a max-plus-regular averaging operator u: C(Z) → C(X) cor- responding to this epimorphism. The dual max-plus-map u⊕ which we de- fine by the equality u⊕(µ)(ϕ) = µ(u(ϕ)), ϕ ∈ C(Z), generates an embedding u⊕ : I(X) → I(Z). c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 48 A metric on the space of idempotent measures For idempotent probability measures µ′0 = u ⊕(µ0) and µ ′ n = u ⊕(µn) there exists a coupling µ′0,n = ⊕ (x′,y′)∈Z2 dµ′0 n (x ′, y′) � δ(x′,y′) ∈ I(Z ×Z) of µ′0 and µ′n such that ⊕ (x′,y′)∈Z2\(f×f)−1(U) dµ′0 n (x ′, y′) �ρ(x′, y′) = −∞. Put µ0,n = I(f ×f)(µ′0 n). Then for every ϕ ∈ C(X2) we have µ0,n(ϕ) = I(f ×f)(µ′0 n)(ϕ) = µ ′ 0 n(ϕ◦ (f ×f)) = = ⊕ (x′,y′)∈Z2 dµ′0 n (x ′, y′) �ϕ◦ (f ×f)(x′, y′) = = ⊕ (x′,y′)∈Z2 dµ′0 n (x ′, y′) �ϕ(f(x′), f(y′)) = = ⊕ (x,y)∈X2 dµ′0 n (x, y)) �δ(x,y)(ϕ), i. e. µ0,n = ⊕ (x,y)∈X2 dµ′0 n (x, y)) � δ(x,y). Here dµ′0 n (x, y) = ⊕ (x′,y′)∈(f×f)−1(x,y) dµ′0 n (x ′, y′). That is why ⊕ (x,y)∈X2\U dµ′0 n (x, y) �ρ(x, y) = −∞. So, µ0,n = I(f × f)(µ′0 n) satisfies (4.1). It remains to show that µ0,n is a coupling of µ0 and µn. A diagram (4.2) Z ×Z f×f−−−−→ X ×Xyθ121 yπ121 Z f−−−−→ X is commutative, where θ121 , π 12 1 are projections onto the first corresponding factors. Then I(π121 )(µ0 n) = I(π 12 1 ) ◦ I(f ×f)(µ ′ 0 n) = I(π 12 1 ◦ (f ×f))(µ ′ 0 n) = = (owing to commutativity of the diagram (4.2)) = = I(f ◦θ121 )(µ ′ 0 n) = I(f) ◦ I(θ 12 1 )(µ ′ 0,n) = I(f)(µ ′ 0) = I(f)(u ⊕(µ0)), i. e. for every ϕ ∈ C(X) we have I(π121 )(µ0 n)(ϕ) = I(f)(u ⊕(µ0))(ϕ) = u ⊕(µ0)(ϕ◦f) = u⊕(µ0)(f◦(ϕ)) = = µ0(u◦f◦(ϕ)) = (with respect to Proposition 2.3) = µ0(ϕ). c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 49 A. A. Zaitov Thus, I(π121 )(µ0 n) = µ0. Similarly, I(π 12 2 )(µ0 n) = µn. The Proposition is proved. � Theorem 4.8. The metric ρI generates pointwise convergence topology on I(X). Proof. Let {µn} ⊂ I(X) be a sequence and µ0 ∈ I(X). Suppose the sequence converges to µ0 with respect to the pointwise convergence topology but not by ρI. Passing in the case of need to a subsequence, it is possible to regard that ρI(µn, µ0) ≥ a > 0 for all positive integer n. Consider an open neighbourhood of the diagonal ∆(X): U = { (x, y) ∈ X2 : ρ(x, y) < a 2 } . By virtue of Proposition 4.7 there exist a positive integer n and a coupling µ0 n ∈ I(X2) of µ0 and µn such that⊕ (x,y)∈X2\U dµ0 n (x, y) �ρ(x, y) = −∞. Therefore, supp µ0 n ⊂ U, and ρI(µn, µ0) ≤ ρ1(µn, µ0) ≤ sup (z,t)∈supp µ0 n {ρ(z, t)} = sup (z,t)∈supp µ0 n {µ0 n(ρ)⊕ρ(z, t)} = = sup (z,t)∈supp µ0 n {( ⊕ (x,y)∈X2 dµ0 n (x, y) �ρ(x, y) ) ⊕ρ(z, t) } = = sup (z,t)∈supp µ0 n {( ⊕ (x,y)∈X2\U dµ0 n (x, y)�ρ(x, y)⊕ sup (x,y)∈U dµ0 n (x, y)�ρ(x, y) ) ⊕ ⊕ρ(z, t) } = sup (z,t)∈supp µ0 n {( sup (x,y)∈U dµ0 n (x, y) �ρ(x, y) ) ⊕ρ(z, t) } ≤ ≤ sup (z,t)∈U {( sup (x,y)∈U dµ0 n (x, y)�ρ(x, y) ) ⊕ρ(z, t) } = sup (z,t)∈U {ρ(z, t)}≤ a 2 . The obtained contradiction finishes the proof. � Acknowledgements. 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