@ Appl. Gen. Topol. 22, no. 2 (2021), 399-415doi:10.4995/agt.2021.15101 © AGT, UPV, 2021 Geometrical properties of the space of idempotent probability measures Kholsaid Fayzullayevich Kholturaev Tashkent Institute of Irrigation and Agricultural Mechanization Engineers, 39, Kori Niyoziy Str., 100000, Tashkent, Uzbekistan (xolsaid 81@mail.ru) Communicated by S. Romaguera Abstract Although traditional and idempotent mathematics are “parallel”, by an application of the category theory we show that objects obtained the similar rules over traditional and idempotent mathematics must not be “parallel”. At first we establish for a compact metric space X the spaces P(X) of probability measures and I(X) idempotent probability measures are homeomorphic (“parallelism”). Then we construct an ex- ample which shows that the constructions P and I form distinguished functors from each other (“parallelism” negation). Further for a com- pact Hausdorff space X we establish that the hereditary normality of I3(X)\X implies the metrizability of X. 2010 MSC: 52A30; 54E35; 54B30; 18B30. Keywords: category; functor; compact Hausdorff space; idempotent mea- sure. 1. Introduction Idempotent mathematics is a new branch of mathematical sciences, rapidly developing and gaining popularity over the last two decades. It is closely re- lated to mathematical physics. The literature on the subject is vast and in- cludes numerous books and an all but innumerable body of journal papers. An important stage of development of the subject was presented in the book Received 13 February 2021 – Accepted 06 May 2021 http://dx.doi.org/10.4995/agt.2021.15101 K. F. Kholturaev “Idempotency” edited by J. Gunawardena [6]. This book arose out of the well- known international workshop that was held in Bristol, England, in October 1994. The next stage of development of idempotent and tropical mathematics was presented in the book Idempotent Mathematics and Mathematical Physics edited by G. L. Litvinov and V. P. Maslov [13]. The book arose out of the international workshop that was held in Vienna, Austria, in February 2003. In [14] it was delivered the proceedings of the International Workshop on Idem- potent and Tropical Mathematics and Problems of Mathematical Physics, held at the Independent University of Moscow, Russia, on August 25-30, 2007. 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 [8], [19]. The modern idempotent analysis (or idempotent calculus, or idempotent mathematics) was founded by V. P. Maslov and his collaborators [11], [12], [10], [14]. Some preliminary results are due to E. Hopf and G. Choquet, see [1], [7]. 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 in [13]. 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. The notion of idempotent (Maslov) measure finds important applications in different parts of mathematics, mathematical physics and economics (see the survey article [14] and the bibliography therein). Topological and categorical properties of the functor of idempotent measures were studied in [20], [21]. 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 [16], [17], [20]) which are based on existence of natural equiconnectedness structure on both functors. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 400 Geometrical properties of idempotent measures A notion of central importance in categorical topology is that of topological functor. Various applications of topological functors described in [5]. In the present paper we show that for a compact metric space X the spaces P(X) of probability measures and I(X) idempotent probability measures are homeomorphic. Further we construct an example which shows that the con- structions P and I form distinguished functors from each other. This phenom- enon shows that the category theory finds out such subtle moments of relations between topological spaces which against common sense. In other words, we get such a conclusion: Although traditional and idempotent mathematics are parallel, an appli- cation of the category theory shows, objects obtained the similar rules over traditional and idempotent mathematics must not be “parallel”. Further, for a compact Hausdorff space X we establish that the hereditary normality of I3(X)\X implies the metrizability of X. 2. Preliminaries Recall [14] 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 with neutral elements 0 and 1 is called a semifield if every nonzero element of S is invertible. A semiring S is called an idempotent semiring if x ⊕ x = x for all x ∈ S. An idempotent semiring S is called an idempotent semifield if it is a semi- field. Note that diöıds, quantales and inclines are examples of idempotent semirings [14]. 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, V. N. Kolokoltsov and others) used idempotent semirings and matrices over these semirings for solving some applied problems in computer science and discrete mathematics. 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+ → R (h) = R ∪ {−∞} © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 401 K. F. Kholturaev defined by the equality Φh(x) = h ln x, h > 0. 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. The convention −∞ ⊙ x = −∞ allows us to extend ⊕h and ⊙ over R(h). Thus we obtained the structure of a semiring (R(h), ⊕h, ⊙) which is isomorphic to (R+, +, ·). A direct check shows that u ⊕h v → max{u, v} as h → 0. It can eas- ily be checked that R ∪ {−∞} forms a semiring with respect to the addi- tion 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, ⊕, ⊙) generates the semi- field (Rmax, ⊕, ⊙, 0, 1). The analogy with quantization is obvious; the param- eter 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”. The described passage (R+, +, ·, 0, 1) Φh ≃ (R(h), ⊕h, ⊙, −∞, 0) h→0 −−−→ (Rmax, ⊕, ⊙, 0, 1) is called the Maslov dequantization. Let X be a compact Hausdorff space, C(X) be the algebra of continuous functions on X with the usual algebraic operations (i. e. with the addition “+” and the multiplication “·”). On C(X) the operations ⊕ and ⊙ are determined by ϕ ⊕ ψ = max{ϕ,ψ} and ϕ ⊙ ψ = ϕ + ψ where ϕ, ψ ∈ C(X). Recall [21] that a functional µ: C(X) → R is said to be an idempotent probability measure on X if it has the following properties: (I1) µ(λX) = λ for all λ ∈ R, where λX is a constant function; (I2) µ(λ ⊙ ϕ) = λ ⊙ µ(ϕ) for all λ ∈ R and ϕ ∈ C(X); (I3) µ(ϕ ⊕ ψ) = µ(ϕ) ⊕ µ(ψ) for all ϕ, ψ ∈ C(X). Let I(X) denote the set of all idempotent probability measures on a com- pact Hausdorff space X, and RC(X) be a set of all maps C(X) → R. Obviously I(X) ⊂ RC(X). One can treat RC(X) = ∏ ϕ∈C(X) Rϕ where Rϕ = R, ϕ ∈ C(X). We consider RC(X) with the product topology and consider I(X) as its sub- space. 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. So, we get a topological space I(X), equipped with the pointwise convergence topology. In [21] it was shown that for each compact Hausdorff space X the space I(X) is also a compact Hausdorff space. 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 © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 402 Geometrical properties of idempotent measures acting in the category Comp of compact Hausdorff spaces and their continuous maps. Remind that a functor F : Comp → Comp on the category of compact Haus- dorff spaces and continuous maps is said to be normal (see [15], Definition 14) if it satisfies the following conditions: (F1) F is continuous (i. e., F(lim S) = lim F(S)); (F2) F preserves weight (i. e., wX = wF(X)); (F3) F is monomorphic (i. e., preserves the injectivity of maps); (F4) F is epimorphic (i. e., preserves the surjectivity of maps); (F5) F preserves intersections (i. e., F(∩ α Xα) = ∩ α F(Xα)); (F6) F preserves preimages (i. e., F(f−1) = F(f)−1); (F7) F preserves singletons and the empty space (i. e., F(1) = 1 and F(∅) = ∅). Let us decipher this definition. Let S = {Xα, p β α; A} be an inverse system of compact Hausdorff spaces, and let lim S = lim ← S be its limit. According to the Kurosh theorem, the limit of any inverse system of nonempty compact Hausdorff spaces is nonempty (see [2], Theorem 3.13) and compact Hausdorff space (see [2], Proposition 3.12). The action of the functor F on the compact Hausdorff spaces Xα and the maps p β α, where α, β ∈ A and α ≺ β, produces the inverse system F(S) = {F(Xα), F(p β α); A}. Let lim F(S) be the limit of this system. By virtue of condition (F1), we have F(lim S) = lim F(S). Given a topological space X, let wX denote its weight, i. e., the minimum cardinality of a base of X. By condition (F2), the weights of the compact spaces X and F(X) are equal. Since the functor F is monomorphic (by condition (F3)), we can assume F(A) to be a subspace of F(X) for a closed A ⊂ X. The space F(A) is identified with a subspace of F(X) by means of the embedding F(iA), where iA : A → X is the identity embedding. According to condition (F4), if f : X → Y is a continuous map “onto” then so is F(f): F(X) → F(Y ). For a monomorphic functor F , con- ditions (F5) and (F6) mean that, for any family {Aα} of closed subsets of a compact Hausdorff space X, we have F(∩ α Aα) = ∩ α F(Aα) (this is condition (F5)), and for any continuous map f : X → Y and any closed B in Y , we have F(f−1(B)) = F(f)−1(F(B)) (this is condition (F6)). The singleton preservation condition means that F takes any one-point space to a one-point space. The intersection preservation condition makes it possible to define an im- portant notion of the support of a monomorphic functor F . The support of a point x ∈ F(X) is a closed set supp x ⊂ X such that, for any closed A ⊂ X, © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 403 K. F. Kholturaev we have A ⊃ supp x if and only if x ∈ F(A) ([15], Definition 18). Given an intersection-preserving functor F , each point x ∈ F(X) has support, which is defined by supp x = ∩{A ⊂ X : A = A, x ∈ F(A)}, where A denotes the closure of A. As it was mentioned above, the functor I is normal; therefore, for each compact Hausdorff space X and any idempotent probability measure µ ∈ I(X), the support of µ is defined as: suppµ = ∩ { A ⊂ X : A = A, µ ∈ I(A) } . For a positive integer n we define the following set In(X) = {µ ∈ I(X) : |suppµ| ≤ n} . Put Iω(X) = ∞ ∪ n=1 In(X). The set Iω(X) is everywhere dense in I(X) [18], [21]. An idempotent probability measure µ ∈ Iω(X) is called an idempotent probability measure with finite support. Note that if µ is an idempotent probability measure with the finite support suppµ = {x1, x2, . . . , xk} then µ can be represented as µ = λ1 ⊙ δx1 ⊕ λ2 ⊙ δx2 ⊕ . . . , ⊕λk ⊙ δxk uniquely, where −∞ < λi ≤ 0, i = 1, . . . k, λ1 ⊕λ2 ⊕ . . . , ⊕λk = 0. Here, as usual, for x ∈ X by δx we denote a functional on C(X) defined by the formula δx(ϕ) = ϕ(x), ϕ ∈ C(X), and called the Dirac measure. It is supported at the point x. Let X be a compact Hausdorff space. A continuous linear functional µ: C(X) → R is said to be a measure on X. The Riesz theorem about isomorphism between the normalized space (C(X))∗ dual to C(X) (i. e. the space of all continuous functional on C(X)) and the space M(X) of all finite regular measures on X is substantiation of the above definition (see [4], page 192, paragraph 3.1). A measure µ ∈ M(X) is positive (µ ≥ 0) if µ(ϕ) ≥ 0 for each ϕ ∈ C(X), ϕ ≥ 0. A measure µ is positive if and only if ‖µ‖ = µ(1X). Really, let µ ≥ 0 and ‖ϕ‖ ≤ 1. Then µ(1X − ϕ) ≥ 0. Consequently, µ(1X) ≥ µ(ϕ). From here ‖µ‖ = sup{|µ(ϕ)| : ϕ ∈ C(X), ‖ϕ‖ ≤ 1} = µ(1X). Contrary, let now µ(1X) = ‖µ‖ and ϕ ≥ 0. Put ψ = 1X − ϕ ‖ϕ‖ . Since ‖ψ‖ ≤ 1 we have µ(ψ) ≤ ‖µ‖ = µ(1X), i. e. µ(1X)− 1 ‖ϕ‖ ·µ(ϕ) ≤ µ(1X). Hence µ(ϕ) ≥ 0. A measure µ is normed, if ‖µ‖ = 1. A positive, normed measure is said to be a probability measure. Thus, we can define the notion of probability measure as the following. A probability measure on a given compact Hausdorff space X is a functional µ: C(X) → R satisfying the conditions: (P1) µ(λX) = λ for all λ ∈ R, where λX – constant function; (P2) µ(λϕ) = λµ(ϕ) for all λ ∈ R and ϕ ∈ C(X); © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 404 Geometrical properties of idempotent measures (P3) µ(ϕ + ψ) = µ(ϕ) + µ(ψ) for all ϕ, ψ ∈ C(X). The set of all probability measures on a compact Hausdorff space X is de- noted by P(X). The set P(X) is endowed with the pointwise convergence topology, i. e. we consider P(X) as a subspace of RC(X). It is well known the topological spaces P(X) and I(X) equipped with the pointwise convergence topology are compact Hausdorff spaces. It is easy to sea that, the conditions of normality (P1) and (I1) are the same, and the conditions of homogeneity (P2) and (I2), and the conditions of additivity (P3) and (I3) are mutually similar, just operations are different. In other words, the definition of the idempotent probability measure is “parallel” to the traditional one. In section 3 we will show that the spaces P(X) and I(X) are homeomorphic, i. e. constructions P and I generate “parallel” objects. At the same time, in section 4 we will show that functors P and I are not isomorphic, i. e. the constructions P and I themselves are not “parallel”. Note that idempotent probability measures were investigated in [21]. Unlike this work in the present paper we establish our results constructively, while in [21] the results were gotten descriptively. 3. Spaces P(X) and I(X) are homeomorphic Theorem 3.1. For an arbitrary Hausdorff finite space X the spaces P(X) and I(X) are homeomorphic. Proof. We determine the map z P I : P(X) −→ I(X), by the following equality z P I ( n∑ i=1 αiδxi ) = n⊕ i=1    ln αi − n⊕ j=1 ln αj   ⊙ δxi   , n∑ i=1 αiδxi ∈ P(X), here n∑ i=1 αi = 1, αi > 0 for all i = 1, . . . , n, and the map z I P : I(X) −→ P(X), by the rule z I P ( n⊕ i=1 λi ⊙ δxi ) = n∑ i=1 eλi n∑ j=1 eλj · δxi, n⊕ i=1 λi ⊙ δxi ∈ I(X), here n ⊕ i=1 λi = 0, λi > −∞, i = 1, . . . , n. We will show that the maps zPI and z I P are continuous and mutually inverse. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 405 K. F. Kholturaev 1) For each probability measure n∑ i=1 αiδxi ∈ P(X) the following equalities hold z I P ( z P I ( n∑ i=1 αiδxi )) = zIP   n⊕ i=1  ln αi − n⊕ j=1 ln αj   ⊙ δxi   = = n∑ i=1 e ln αi− n⊕ j=1 ln αj n∑ l=1 e ln αl− n⊕ j=1 ln αj · δxi = n∑ i=1 eln αi : e n⊕ j=1 ln αj ( n∑ l=1 eln αl ) : e n⊕ j=1 ln αj · δxi = = n∑ i=1 αiδxi n∑ l=1 αl = n∑ i=1 αiδxi; 2) For each idempotent probability measure ⊕n i=1 λi ⊙ δxi ∈ I(X) we have zPI ( zIP ( n⊕ i=1 λi ⊙ δxi )) = zPI   n∑ i=1 eλi n∑ j=1 eλj δxi   = = n⊕ i=1  ln eλi n∑ j=1 eλj − n⊕ l=1 ln eλl n∑ j=1 eλj   ⊙ δxi = = n⊕ i=1  ln eλi − ln n∑ j=1 eλj − n⊕ l=1  lneλl − ln n∑ j=1 eλj     ⊙ δxi = = n⊕ i=1  λi − ln n∑ j=1 eλj − n⊕ l=1 λl + ln n∑ j=1 eλj   ⊙ δxi = n⊕ i=1 λi ⊙ δxi. Consequently, the compositions zPI z I P : I(X) → I(X) and z I P z P I : P(X) → P(X) are the identical maps. Now we will show that the maps zPI and z I P are continuous. Since they are mutually inverse maps between compact Hausdorff spaces, it suffices to show the continuity only of one of them. We show that the map zPI : P(X) → I(X) is continuous. Let µ0 = n0∑ i(0)=1 α i(0) 0 δxi(0)0 ∈ P(X) be a probability measure, {µt} ∞ t=1 = { nt∑ i(t)=1 α i(t) t δxi(t) t }∞ t=1 ⊂ P(X) be a sequence converging to µ0 in the pointwise convergence topol- ogy (symbolically lim t→∞ µt = µ0). It means that lim t→∞ µt(ϕ) = µ0(ϕ) for all ϕ ∈ C(X). © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 406 Geometrical properties of idempotent measures For each point x i(0) 0 ∈ supp µ0 = {x 1 0, x 2 0, . . . , x n0 0 } consider a characteristic function χi(0) = χ{ x i(0) 0 } : X → R, i(0) = 1, . . . , n0. These functions are continuous, i. e. χi(0) ∈ C(X) since X is provided with the discrete topology. Evidently, lim t→∞ µt(χi(0)) = µ0(χi(0)) = α i(0) 0 , i(0) = 1, . . . , n0.(⋆) (⋆) implies the following two conclusions: (Output1) Since each α i(0) 0 > 0, we have x i(0) 0 ∈ supp µt for all t greater than or equals to some ti(0). Hence supp µ0 ⊂ supp µt for all t ≥ max{t1, . . . , tn0}; (Output2) Let α i(0) t be the barycentre mass of µt at x i(0) 0 ∈ supp µt, t = 1, 2 . . . . Then lim t→∞ α i(0) t = α i(0) 0 . On the other hand, the continuity of the logarithm function ln and the operation ⊕ implies the equality lim t→∞ ln α i(0) t = ln α i(0) 0 . Hence, lim t→∞  ln αi(0)t − n0⊕ j(0)=1 ln α j(0) t   = ln αi(0)0 − n0⊕ j(0)=1 ln α j(0) 0 . Therefore, lim t→∞ zPI (µt) = z P I (µ0), i. e. the map z P I is continuous. Theorem 3.1 is proved. � Corollary 3.2. For an arbitrary metrizable compact space X the spaces P(X) and I(X) are homomorphic. Proof. As well-known that a metrizable compact space has a dense countable subset. Let M be a dense countable set in X. For each n let Mn be a n- point subset of M, n = 1, 2, . . . , such that M1 ⊂ · · · ⊂ Mn ⊂ Mn+1 ⊂ . . . , and ∞ ∪ n=1 Mn = M. One can directly verified that ∞ ∪ n=1 P(Mn) and ∞ ∪ n=1 I(Mn) are dense in P(X) and I(X) respectively. Let z∞ : ∞ ∪ n=1 P(Mn) → ∞ ∪ n=1 I(Mn) be such a map that z∞|P (Mn) = z P I for each n = 1, 2, . . . . Then z∞ is a homeomorphism and continued over all P(X) uniquely. Let z : P(X) → I(X) be this continuation. It is clear z is a homeomorphism. Corollary 3.2 is proved. � 4. Functors P and I are not isomorphic A subset L of the space C(X) is called [21] a max-plus-linear subspace in C(X), if: 1) λX ∈ L for each λ ∈ R; 2) λ ⊙ ϕ ∈ L for each λ ∈ R and ϕ ∈ L; 3) ϕ ⊕ ψ ∈ L for each ϕ,ψ ∈ L. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 407 K. F. Kholturaev Lemma 4.1 ([21] The max-plus variant of the Hahn-Banach theorem). Let L be a max-plus-linear subspace in C(X). Let µ : L −→ R be a functional sat- isfying the conditions of normality (I1), homogeneity (I2) and additivity (I3) (with C(X) replaced by L). For an arbitrary ϕ0 ∈ C(X)\L there exists an ex- tension of the functional µ satisfying the conditions of normality, homogeneity and additivity on the minimal max-plus-linear subspace L ′ containing L∪{ϕ0}. Consider the following subset in C(X × Y ): C0 = { n⊕ i=1 ϕi ⊙ ψi : ϕi ∈ C(X) and ψi ∈ C(Y ), i = 1, . . . , n; n ∈ N } . It is obvious that C0 is a max-plus-linear subspace in C(X). For every pair (µ,ν) ∈ I(X) × I(Y ) we put (µ⊗̃ν) ( n⊕ i=1 ϕi ⊙ ψi ) = n⊕ i=1 µ(ϕi) ⊙ ν(ψi). Proposition 4.2. The constructed functional µ⊗̃ν satisfies the conditions of normality, homogeneity and additivity on C0. Proof. Each c ∈ R can be represented as cX×Y = aX ⊙bY , where a, b ∈ R and a+b = c. Therefore, (µ⊗̃ν)(cX×Y ) = (µ⊗̃ν)(aX ⊙bY ) = µ(a)⊙ν(b) = a⊙b = c. Let λ ∈ R and ⊕n i=1 ϕi ⊙ ψi ∈ C0. Then (µ⊗̃ν) ( λ ⊙ n⊕ i=1 ϕi ⊙ ψi ) = (µ⊗̃ν) ( n⊕ i=1 (λ ⊙ ϕi) ⊙ ψi ) = = n⊕ i=1 µ(λ ⊙ ϕi) ⊙ ν(ψi) = n⊕ i=1 λ ⊙ µ(ϕi) ⊙ ν(ψi) = = λ ⊙ n⊕ i=1 µ(ϕi) ⊙ ν(ψi) = λ ⊙ (µ⊗̃ν) ( n⊕ i=1 ϕi ⊙ ψi ) . Finally, let ⊕n i=1 ϕ1 i ⊙ ψ1 i ∈ C0 and ⊕m j=1 ϕ2 j ⊙ ψ2 j ∈ C0. Then (µ⊗̃ν)   n⊕ i=1 ϕ1 i ⊙ ψ1 i ⊕ m⊕ j=1 ϕ2 j ⊙ ψ2 j   = = (µ⊗̃ν) (⊕ ϕk l ⊙ ψk l ) = ⊕ µ(ϕk l) ⊙ ν(ψk l) = = n⊕ i=1 µ(ϕ1 i) ⊙ ν(ψ1 i) ⊕ m⊕ j=1 µ(ϕ2 j) ⊙ ν(ψ2 j) = = (µ⊗̃ν) ( n⊕ i=1 ϕ1 i ⊙ ψ1 i ) ⊕ (µ⊗̃ν)   m⊕ j=1 ϕ2 j ⊙ ψ2 j   . © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 408 Geometrical properties of idempotent measures Proposition 4.2 is proved. � Since C0 is a max-plus-linear subspace in C(X × Y ), according to Lemma 1 for the idempotent probability measure µ⊗̃ν there exists its extension ξ all over C(X × Y ) which satisfies the conditions of normality, homogeneity and additivity on C(X × Y ). So, we have proved the following max-plus variant of the Fubini theorem. Theorem 4.3. For every pair (µ, ν) ∈ I(X)×I(Y ) there exists an idempotent probability measure ξ ∈ I(X ×Y ) such that ξ(ϕ⊙ψ) = µ(ϕ)⊙ν(ψ), ϕ ∈ C(X), ψ ∈ C(Y ). From the results of work [18] (see section 3) it follows that if |X| ≥ 2, |Y | ≥ 2, then µ⊗̃ν has uncountable many extensions on C(X × Y ). Put µ ⊗ ν = ⊕{ ξ ∈ I(X × Y ) : ξ|C0 = µ⊗̃ν } . Similarly to the traditional case, µ ⊗ ν we call as a “tensor” product of idempotent probability measure µ and ν. Further, to distinguish the tensor products we will use symbols ⊗I and ⊗P for the idempotent and traditional cases, respectively. Let us give the classical option of the Fubini theorem. Theorem 4.4 ([4]). For every pair (µ, ν) ∈ P(X)×P(Y ) there exists a unique probability measure µ⊗P ν ∈ P(X ×Y ) such that (µ⊗P ν)(ϕ·ψ) = µ(ϕ)·ν(ψ), ϕ ∈ C(X), ψ ∈ C(Y ). Now we need some concepts from the category theory [4], [15]. Let Fi : C → C ′, i = 1, 2, be to functors from the category C = (O, M) to the category C ′ = (O ′, M ′). A family of morphisms Φ = {ϕX : F1(X) → F2(X)|X ∈ O} ⊂ M ′ is said to be a natural transformation of the functor F1 to the functor F2, if for each morphism f : X → Y of the category C a diagram F1(X) F1(f) −−−−→ F1(Y ) ϕX y ϕY y F2(X) F2(f) −−−−→ F2(Y ) is commutative, i. e. F2(f) ◦ ϕX = ϕY ◦ F1(f). If, for every object X in C, the morphism fX is an isomorphism in C ′, then Φ = {fX} is said to be a natural isomorphism (or sometimes natural equivalence or isomorphism of functors). Two functors Fi : C → C ′, i = 1, 2, are called naturally isomorphic or simply isomorphic if there exists a natural isomorphism from F1 to F2. Our goal is to show that the functors P and I are not isomorphic. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 409 K. F. Kholturaev Example 4.5. Consider the sets X = {a,b,c}, Y = {a,b}, Z = {a,c}, where a, b, c are different points (these sets are supplied with discrete topologies). Define the following maps: f : X −→ Y, f(a) = f(c) = a, f(b) = b, g : X −→ Z, g(a) = g(b) = a, g(c) = c. Consider compact Hausdorff spaces X, Y × Z, and the map (f, g): X −→ Y × Z. It suffices to show the map (P(f), P(g)) has a property that the map (I(f), I(g)) does not possess it. At first we show the map (P(f), P(g)): P(X) −→ P(Y ) × P(Z) is an embedding. In fact, for any pair of probability measures µ = α1δa + α2δb + α3δc, ν = β1δa + β2δb + β3δc, with positive α1, α2, α3, β1, β2, β3, α1 + α2 + α3 = 1, β1 + β2 + β3 = 1, the following equalities take place P(f)(µ) = (α1 + α3)δa + α2δb, P(f)(ν) = (β1 + β3)δa + β2δb, P(g)(µ) = (α1 + α2)δa + α3δc, P(g)(ν) = (β1 + β2)δa + β3δc. Therefore, (P(f), P(g))(µ) = (P(f), P(g))(ν) if and only if    α1 + α3 = β1 + β3, α2 = β2, α1 + α2 = β1 + β2, α3 = β3. (1.P) System (1.P) has a unique solution α1 = β1, α2 = β2 and α3 = β3. Hence, µ = ν. Thus, (P(f), P(g))(µ) = (P(f), P(g))(ν) if and only if µ = ν, i. e. (P(f), P(g)): P(X) → P(Y ) × P(Z) is an embedding. Consequently, the diagram P(X) (P (f), P (g)) // P ((f, g)) �� P(Y ) × P(Z) ⊗P ww♦♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ P(Y × Z) holds, i. e. P((f, g)) = ⊗P ◦ (P(f), P(g)),(2.P) where the map ⊗P : P(Y ) × P(Z) → P(Y × Z) acts as ⊗P (µ, ν) = µ ⊗P ν. Remind, the uniqueness of the solution of system (1.P) and equality (2.P) may be considered as corollaries of Theorem 4.4. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 410 Geometrical properties of idempotent measures Now we show that the map (I(f), I(g)): I(X) → I(Y ) × I(Z) is not an embedding. Really, for idempotent probability measures µ = λ1 ⊙ δa ⊕ λ2 ⊙ δb ⊕ λ3 ⊙ δc, ν = γ1 ⊙ δa ⊕ γ2 ⊙ δb ⊕ γ3 ⊙ δc, with −∞ < λ1, λ2, λ3, γ1, γ2, γ3 ≤ 0 and λ1 ⊕ λ2 ⊕ λ3 = γ1 ⊕ γ2 ⊕ γ3 = 0, the following equalities hold I(f)(µ) = (λ1 ⊕ λ3) ⊙ δa ⊕ λ2 ⊙ δb, I(f)(ν) = (γ1 ⊕ γ3) ⊙ δa ⊕ γ2 ⊙ δb, I(g)(µ) = (λ1 ⊕ λ2) ⊙ δa ⊕ λ3 ⊙ δc, I(g)(ν) = (γ1 ⊕ γ2) ⊙ δa ⊕ γ3 ⊙ δc. The equality (I(f), I(g))(µ) = (I(f), I(g))(ν) is true if and only if    λ1 ⊕ λ3 = γ1 ⊕ γ3, λ2 = γ2, λ1 ⊕ λ2 = γ1 ⊕ γ2, λ3 = γ3. (1.I) System (1.I) has infinitely many solutions. For example, for every pair of λ1 and γ1 with −∞ < λ1 ≤ 0, −∞ < γ1 ≤ 0 a 6-tuple (λ1, γ1, 0, 0, 0, 0) is its solution. The equality (I(f), I(g))(µ) = (I(f), I(g))(ν) is true for this 6-tuple although λ1 6= γ1. This means that the map (I(f), I(g)) is not an embedding. That means that the following diagram I(X) (I(f), I(g)) // I((f, g)) �� I(Y ) × I(Z) ⊗I ww♣♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ I(Y × Z) does not hold, i. e. the equality I((f, g)) = ⊗I ◦ (I(f), I(g))(2.I) is wrong. To present the existence of infinitely many solutions of system (1.I), or re- lation (2.I) consider idempotent probability measures µ = −1 ⊙ δa ⊕ 0 ⊙ δb ⊕ 0 ⊙ δc, ν = −2 ⊙ δa ⊕ 0 ⊙ δb ⊕ 0 ⊙ δc. Then I(f)(µ) = 0 ⊙ δa ⊕ 0 ⊙ δb, I(f)(ν) = 0 ⊙ δa ⊕ 0 ⊙ δb, I(g)(µ) = 0 ⊙ δa ⊕ 0 ⊙ δc, I(g)(ν) = 0 ⊙ δa ⊕ 0 ⊙ δc, and (I(f), I(g))(µ) = (0 ⊙ δa ⊕ 0 ⊙ δb, 0 ⊙ δa ⊕ 0 ⊙ δc) = (I(f), I(g))(ν), which yields I(f)(µ) ⊗I I(g)(µ) = 0 ⊙ δ(a, a) ⊕ 0 ⊙ δ(b, c) = I(f)(ν) ⊗I I(g)(ν). © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 411 K. F. Kholturaev On the other hand since (f, g)(a) = (a, a), (f, g)(b) = (b, a), (f, g)(c) = (a, c), we have I((f, g))(µ) = −1 ⊙ δ(a, a) ⊕ 0 ⊙ δ(b, a) ⊕ 0 ⊙ δ(a, c), I((f, g))(ν) = −2 ⊙ δ(a, a) ⊕ 0 ⊙ δ(b, a) ⊕ 0 ⊙ δ(a, c). So, I((f, g))(µ) 6= I((f, g))(ν). Moreover, I((f, g))(µ) 6= ⊗I ◦ (I(f), I(g))(µ) and I((f, g))(ν) 6= ⊗I ◦ (I(f), I(g))(ν). Thus, (1) The map (P(f), P(g)): P(X) → P(Y )×P(Z) is an embedding, while the map (I(f), I(g)): I(X) → I(Y ) × I(Z) is not an embedding. (2) For the embedding (f, g): X → Y ×Z the embedding P((f, g)): PX) → P(Y × Z) might be defined by the rule (2.P), while the embedding I((f, g)): I(X) → I(Y × Z) does not have such a decomposition. (3) Regardless of the natural transformation Φ = {ϕX : P(X) → I(X) : X ∈ Comp} a diagram P(X) (P (f), P (g)) −−−−−−−−→ P(Y ) × P(Z) ϕX y yϕY ×ϕZ I(X) (I(f), I(g)) −−−−−−−→ I(Y ) × I(Z) can not be commutative. So, we came to the following important Conclusion: although for a metriz- able Hausdorff compact X the spaces I(X) and P(X) are homeomorphic, Ex- ample 4.5 shows, the difference between the constructions P and I appears even on finite sets. Thus, the functors P and I are not isomorphic. Note this conclusion was proclaimed in [21] as Proposition 2.15. But it was not equipped with detailed proof. 5. On a metricise criterion of the compact Hausdorff spaces The well-known M. Katětov’s theorem states [9] that the hereditary nor- mality of the cube X3 of a Hausdorff compact space X follows metrizability of X. In 1989, V. V. Fedorchuk generalized [3] Katětov’s theorem for a normal functors of the degree ≥ 3, acting in the category Comp. Many publications in the field of general topology are devoted to the issues of KatětovÆs theorem and problem. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 412 Geometrical properties of idempotent measures The set of all nonempty closed subsets of the topological space X is denoted by exp X. For open subsets U1, . . . , Un ⊂ X a family of the sets of the type O〈U1, . . . , Un〉 = {F : F ∈ exp X, F ⊂ n⋃ i=1 Un, F ∩ Ui 6= ∅, i = . . . , n} forms a base of a topology on exp X. This topology is called the Vietoris topol- ogy, the set exp X equipped with the Vietoris topology is called a hyperspace of the topological space X. For a compact X its hyperspace exp X is a compact. For the compact X, the natural number n, the functor F we put Fn = {a ∈ F(X) : | supp a| ≤ n}, F0n = Fn(X) \ Fn−1(X), where supp a = ⋂{ A ⊂ X : A = A, a ∈ F(A) } is a support of the element a ∈ F(X). In particular, expn X = {K ∈ exp X : |K| ≤ n}, exp0n = expn X \ expn−1 X, and In = {µ ∈ I(X) : | supp µ| ≤ n}, I0n = In(X) \ In−1(X). Let τ be an uncountable cardinal number. Put Nτ = {x : x < τ}. Provide Nτ with the discrete topology. Then it becomes a local compact Hausdorff space (which is not compact, since |Nτ| = τ > ℵ0). By αNτ = Nτ ∪ {p} we denote its one-point compactification, where p 6∈ Nτ. In [3] (Proposition 1) it was shown that if τ is an uncountable cardinal then exp2 αNτ is not hereditary normal. We claim the following statement. Proposition 5.1. For every uncountable cardinal number τ the space exp03 αNτ is not normal. Proof. Obviously, there exist disjoint subsets F1 and F2 of Nτ such that F1 is uncountable and F2 is countable. Take a point x0 ∈ F1 ∪ F2. Choose subsets A1 and A2 of the space exp 0 3 αNτ, assuming A1 = { {p,x,x0} : x ∈ F1 \ {x0} } , A2 = { {p,x ′ ,x0} : x ′ ∈ F2 \ {x0} } , Obviously, A1 ∩ A2 = ∅. Let F = {x1,x2,x3} ∈ exp 0 3 αNτ \ A1. The set O({x1},{x2},{x3}) is an open neighbourhood of the set F wich does not intersect A1. Hence, the set A1 is closed in exp03 αNτ. Similarly one can check that A2 is closed in exp 0 3 αNτ. For each x ∈ Nτ we put Ux = O〈αNτ \ {x0,x},{x0},{x}〉 ∩ exp 0 3 αNτ. It is easy to see that the smallest by inclusion neighbourhoods of the sets A1 and A2 in exp 0 3 (αNτ) are the sets OA1 = ⋃ x∈F1 Ux and OA2 = ⋃ x∈F2 Ux, respectively. For the set {a,b,x0}, where a ∈ F1, b ∈ F2, we have {a,b,x0} ∈ O (αNτ \ {x0,a},{x0},{a}) ⊂ OA1, {a,b,x0} ∈ O (αNτ \ {x0,b},{x0},{b}) ⊂ OA2. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 413 K. F. Kholturaev This means, OA1 ∩ OA2 6= ∅. Proposition 5.1 is proved. � Proposition 5.2. Let τ is an uncountable cardinal number. Then I03 (αNτ) is not normal space. Proof. For each compact X the set exp03 X is closed in I 0 3 (X). Really, a corre- spondence {a, b, c} 7→ 0⊙δa ⊕0⊙δb ⊕0⊙δc establishes an identical embedding of exp03 X into I 0 3 (X). This embedding is continuous and it is closed map. On the other hand the normality is a hereditary property for the closed subsets of the space. Therefore according to Proposition 5.1 the space I03 (αNτ) is not normal. Proposition 5.2 is proved. � In [3] (Theorem 2) it was established that if for a normal functor F of degree ≥ 3 the Hausdorff compact space F(X) is hereditary normal then a Hausdorff compact space X is metrizable. We get modified shape of this result for the functor I, and it might be considered as a metricize criterion of compact Hausdorff spaces. Theorem 5.3. Let X be a compact Hausdorff space. If I3(X) \ X is a hered- itarily normal space then X is metrizable. Proof. Suppose the compact X is non-metrizable. If X has a unique noniso- lated point then X is homeomorphic to αNτ for τ = |X| > ω. Proposition 5.2 implies that I03 (X) is not normal. But according to the condition I 0 3 (X) must be normal as a subset of the hereditarily normal space I3(X) \ X. We get a contradiction. Now let a and b be distinguished nonisolated points of the compact X. There are open neighbourhoods U and V of points a and b, respectively, such that U ⋂ V = ∅. We consider set Z = U × exp2 V and by the formula λ(x,y,z) = 0 ⊙ δx ⊕ 0 ⊙ δy ⊕ 0 ⊙ δz we define the topological embedding λ : Z → I3(X) \ X. The result of M. Katetov [9, Corollary 1] (which asserts the perfectly normality of the factor X under the condition of hereditarily normality of the product of X × Y ) implies that the factor exp2 V of the product Z = U × exp2 V is perfectly normal. Further, applying the result of V. V. Fedorchuk [3] (which asserts the metrizability of the compact X if for a normal functor F of degree ≤ 2 the space F(X) is perfectly normal) we conclude that V is metrizable. Similarly, one can show that U is metrizable. Therefore each nonisolated point of the compact X has a metrizable closed neighbourhood. Hence the compact X is locally metrizable. Therefore it is metrizable. Theorem 5.3 is proved. � Corollary 5.4. Let X be a compact Hausdorff space and n ≥ 3. If In(X) \ X is a hereditarily normal space then X is metrizable. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 414 Geometrical properties of idempotent measures Acknowledgements. The author expresses deep gratitude to the editor, pro- fessor Salvador Romaguera and the referee for suggestions and useful advice. Also the author would like to express thanks to professor Adilbek Zaitov for the revealed shortcomings, the specified remarks. References [1] G. Choquet, Theory of capacities, Ann. Inst. Fourier 5 (1955), 131–295. [2] V. V. Fedorchuk, Fully closed mappings and their applications, Fundam. Prikl.Mat. 9, no. 4 (2003), 105–235. [J .Math. Sci. (New York) 136:5, 4201–4292 (2006)]. [3] V. V. Fedorchuk, On Katetov’s theorem on the cube, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 1989, no. 4, 93–96. [4] V. V. Fedorchuk and V. V. Filippov, General Topology. Basic Constructions (in Russian). – Moscow. 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