On C -properties of the space of idempotent robability measures
aa r X i v : . [ m a t h . GN ] M a y ON C -PROPERTIES OF THE SPACE OF IDEMPOTENT PROBABILITYMEASURESIshmetov, Azad Yangibayevich Tashkent Institute of Architecture and Civil Engineering,Department of Mathematics and Natural Disciplinesishmetov [email protected]
Abstract
In the work it is shown that the space of idempotent probability measures with compactsupports is kappa-metrizable if the given Tychonoff space is kappa-metrizable. It is con-structed a series of max-plus-convex subfunctors of the functor of idempotent probabilitymeasures with compact supports. Further, it is established that the functor of idempotentprobability measures with the compact supports preserves openness of continuous maps.
Keywords and phrases:
Idempotent measure; open map; kappa-metric.idempotentmeasure, open map.2010
Mathematics Subject Classification:
The theory of idempotent measures belongs to idempotent mathematics, i. e. the fieldof the mathematics based on replacement of usual arithmetic operations with idempotent (as,for example, x ⊕ y = max { x, y } ). The idempotent mathematics intensively develops at thistime (see, for example, [1], survey article [2] and the bibliography in it). Its communicationwith traditional mathematics is described by the informal principle according to which there isa heuristic compliance between important, interesting and useful designs the last and similarresults of idempotent mathematics.In the present article we investigate a functor I β which is an extension of the functor ofidempotent probability measures from the category of compact Hausdorff spaces onto the cat-egory of Tychonoff spaces and their continuous maps. In traditional mathematics to it therecorresponds the functor P β of probability measures. The concept of an idempotent measure(Maslovs measure) finds numerous applications in various field of mathematics, mathematicalphysics and economy. In particular, such measures arise in problems of dynamic optimization;the analogy between Maslov’s integration and optimization is noted also in [1]. It is well-knownthat use of measures of Maslov for modeling of uncertainty in mathematical economy can be sorelevant as far as also use of classical probability theory.Unlike a case of probability measures to which consideration extensive literature is devotedtopological properties of the spaces of idempotent measures were practically not investigated. Inwork [2] M.Zarichnyi gave a number of appendices of idempotent measures in various branchesof modern sciences.Let R = ( −∞ , + ∞ ) be the real line. On the set R S {−∞} we define operations ⊕ and ⊙ by the rules: u ⊕ v = max { u, v } and u ⊙ v = u + v . It is easy to see −∞ is the zero , andthe usual zero 0 is the unit on R S {−∞} . The collection ( R S {−∞} , ⊕ , ⊙ , , ), forms the‘max-plus’ semi-field which we denote by R max .Let X be a compact (i. e. Hausdorff compact space, plural : compacts. Note that a com-pactum is a metrizable compact space, plural : compacta), C ( X ) be the algebra of all continuousfunctions defined on X . C ( X ) is endowed with the usual pointwise algebraic operations andsup-norm. We introduce the following operations:1) ⊙ : R × C ( X ) → C ( X ) by a rule ⊙ ( λ, ϕ ) = λ ⊙ ϕ = ϕ + λ X , where ϕ ∈ C ( X ) and λ X isconstant function accepting everywhere on X the value λ ∈ R ;2) ⊕ : C ( X ) × C ( X ) → C ( X ) by a rule ⊕ ( ϕ, ψ ) = ϕ ⊕ ψ = max { ϕ, ψ } , where ϕ, ψ ∈ C ( X ).1 efinition 1[2]. A functional µ : C ( X ) → R is called an idempotent probability measure on X if it satisfies the following properties:(i) µ ( λ X ) = λ for any λ ∈ R ( norm axiom );(ii) µ ( λ ⊙ ϕ ) = λ ⊙ µ ( ϕ ) for any λ ∈ R and ϕ ∈ C ( X ) ( homogeneity axiom );(iii) µ ( ϕ ⊕ ψ ) = µ ( ϕ ) ⊕ µ ( ψ ) for any ϕ , ψ ∈ C ( X ) ( additivity axiom ).The number µ ( ϕ ) is called the Maslov’s integral corresponding to µ . The set of all idempotentprobability measure on X we denoted by I ( X ). We have I ( X ) ⊂ R C ( X ) . Consider I ( X ) withinduced from R C ( X ) topology. The sets of the look h µ ; ϕ , ..., ϕ k ; ε i = { ν ∈ I ( X ) : | ν ( ϕ i ) − µ ( ϕ i ) | < ε, i = 1 , ..., k } form a base of neighborhoods of an idempotent probability measure µ ∈ I ( X ) concerning tothis topology. form a base of neighborhoods of an idempotent probability measure concerningto this topology. Here ϕ , ..., ϕ k ∈ C ( X ) and ε >
0. It is well known that for any compact X the space I ( X ) is also a compact. Let f : X → Y be a continuous map of compacts. Then theequality I ( f )( µ )( ϕ ) = µ ( ϕ ◦ f ) , µ ∈ I ( X ) , ϕ ∈ C ( Y ) , defines a map I ( f ) : I ( X ) → I ( Y ) which is continuous. For an idempotent probability measure µ ∈ I ( X ) we define its support:supp µ = \ { F ⊂ X : F is a closed subset of X and µ ∈ I ( F ) } . For a compact X and positive integer n we put I n ( X ) = { µ ∈ I ( X ) : | supp µ | ≤ n } . Further I ω ( X ) = ∞ [ n =1 I n ( X ) . Let X is Tychonoff space, βX be the Stone- ˇCech compact extension of X . We define [11,15] a subspace I β ( X ) = { µ ∈ I ( βX ) : supp µ ⊂ X } , which elements we call as idempotent probability measures with compact support . Let βf : βX → βY , where be the maximal extension of a continuous map f : X → Y of Tychonoff spaces. Then I ( βf )( I β ( X )) ⊂ I β ( Y ). Put I β ( f ) = I ( βf ) | I β ( X ) . Thus, the operation I β is a functor acting in the category T ych
Tychonoff spaces and theircontinuous maps.For positive integer n put I β,n ( X ) = { µ ∈ I β ( X ) : | supp µ | ≤ n } . Put I β,ω ( X ) = ∞ S n =1 I β,n ( X ). Proposition 1. If Y is everywhere dense in a compact X , then I β,ω ( Y ) is everywhere densein I ( X ). Proof.
It is well-known [11] that I ω ( X ) is everywhere dense in I ( X ). Therefore it isenough to establish that I β,ω ( Y ) is everywhere dense in I ω ( X ). Take a measure µ ∈ I ω ( X )and its basic neighbourhood h µ ; ϕ , ..., ϕ k ; ε i . Let µ = λ ⊙ δ x ⊕ λ ⊙ δ x ⊕ ... ⊕ λ s ⊙ δ x s . As Y is everywhere dense in X , there are points y , ..., y s such that | ϕ i ( x j ) − ϕ i ( y j ) | < ε for all i = 1 , ..., k ; j = 1 , ..., s . There exist λ ′ , ..., λ ′ s , that | λ i − λ ′ j | < ε for all j = 1 , ..., s . That is why ν = λ ′ ⊙ δ y ⊕ λ ′ ⊙ δ y ⊕ ... ⊕ λ ′ s ⊙ δ y s ∈ h µ ; ϕ , ..., ϕ k ; ε i ∩ I β,ω . Proposition 1 is proved.2 orollary 1. If Y is an everywhere dense subspace of a compact X , then I ω ( Y ) and I β ( Y )are everywhere dense subspaces of I ( X ). Proposition 2. If Y is everywhere dense subspace of a Tychonoff space X , then I β,ω ( Y ) isan everywhere dense subset of I β ( X ). Proof.
Let bX be a compact extension of X . Then Y , being everywhere dense in X , iseverywhere dense in bX . According to the proposition 1 the set I β,ω ( Y ) is everywhere densein I ( bX ). But I β,ω ( X ) ⊂ I β ( X ) ⊂ I ( bX ). Therefore, I β,ω ( Y ) is everywhere dense in I β ( X ) .Proposition 2 is proved. Corollary 2.
For every Tychonoff space X the set I β,ω ( X ) is everywhere dense in I β ( X ). Definition 2[3].
Let P be some topological property. A Tychonoff space X is called C − P -space if it has a compact extension bX , satisfying the property P .Objects of our attention are C -dyadic spaces, C -Milyutin spaces, C -Dugundji spaces, C -absolute retracts or C - AR -spaces (see [3]). To research of the specified classes of spaces we needthe following auxiliary statement: if X is a Tychonoff space of weight ≤ τ and bX is its compactextension which is a dyadic compact, then wbX ≤ τ [3].Note that for a topological space X its weight (i. e. the smallest power of bases of X ) isdenoted by wX . Theorem 1. If X is a C -dyadic space of the weight ≤ ω then I β is also a C -dyadic space. Proof.
Let bX be a compact extension of X which is dyadic. Then the weight of bX is notmore than ω . Then there is an epimorphism f : D ω → bX . But D ω is a Dugundji compact.Therefore, I β ( D ω ) is an absolute retract according to [18]. But every AR -compact is dyadic.Therefore, the compact I β ( bX ) is also dyadic, being image of a dyadic compact I β ( D ω ) rathercontinuous map. At last, by a Corollary 1 the space I ( bX ) is a compact extension of the space I β ( bX ). Hence I β ( X ) is C -dyadic. Theorem 1 is proved. Proposition 3[3].
For Tychonoff space X of the weight ≤ ω the following conditions areequivalent:1) X is C -Milyutin space;2) X is C -Dugundji space. Theorem 2.
Let X be a C -Milyutin space of weight ≤ ω . Then I β ( X ) is C -absoluteretract. Proof.
According to Proposition 3 X is a C -Dugundji space. Take a compact extension bX of X , which is a Dugundji compact. Therefore wbX ≤ ω . Then the compact I ( bX ) is anabsolute retract according to [18]. But by Corollary 1 I ( bX ) is a compact extension of I β ( X ),from here follows that I β ( X ) is a C - AR -space. Theorem 2 is proved.Since every AR -compact is a Dugundji space, Theorem 2 implies Corollary 3.
Functor I β translates the class of C -absolute retracts of weight ≤ ω into theclass of C -absolute retracts. Definition 3[3]. κ -metric (kappa-metric) on a Tychonoff space X is a non-negative function ρ ( x, C ) of two variables: points x ∈ X and canonically closed sets C = [ < C > ] ⊂ X , satisfyingto the following axioms:1) ( belongings axiom ). ρ ( x, C ) = 0 if and only if x ∈ C ;2) ( monotonicity axiom ).If C ′ ⊂ C then ρ ( x, C ) ≤ ρ ( x, C ′ );3) ( continuity axiom ). At fixed C the function ρ ( x, C ) is continuous by x ;4) ( union axiom ). ρ ( x, [ S α C α ]) = inf α ρ ( x, C α ) for any increasing well-ordered sequence ofcanonically closed sets C α ⊂ X . Theorem 3.
If a Tychonoff space X is C - κ -metrizable, then I β ( X ) is also C - κ -metrizable. Proof.
Let bX be a κ -metrizable compact extension of X . By E. V. Shchepin’s theorem3 class of the κ -metrizable compacts coincides with a class of the open generated compacts [3].Here, a compact is open generated if it is homeomorphic to the limit space of some countable-directed continuous inverse spectrum S consisting of compacta and open projections. Let bX =lim ← S be the above stated representation of the open generated compact bX . Then owing to thecontinuity [3] of the functor I we have . I ( bX ) = lim ← I ( S ) . (1)On the same reason the inverse spectrum I ( S ) is continuous. Projections of the spectrum I ( S )are open [7, 8]. Thus, equality (1) gives that the compact I ( bX ) is open generated, i. e. I ( bX )is kappa-metrizable, and as it was noted above (see Corollary 1), it is a compact extension of I β ( X ). Thus, I β ( X ) is C - kappa-metrizable. Theorem 3 is proved.While the proof of the main result we should use the following two lemmas proved in [4 – 6]. Lemma 1.
Let f : X → Y be a continuous map, y ∈ Y and ϕ ∈ C b ( Y ). Then there is afunction ψ ∈ C b ( Y ) such that ψ ◦ f ≤ ϕ and ψ ( y ) = inf (cid:8) ϕ ( x ) : x ∈ f − ( y ) (cid:9) . Lemma 2.
Let f : X → Y be a continuous map, y ∈ Y and ν ∈ I β such that I β ( f )( ν ) = δ y .Then for any ϕ ∈ C b ( X ) such that ϕ ( x ) ≥ c ( ϕ ( x ) ≤ c ) at all x ∈ f − ( y ) we have ν ( ϕ ) ≥ c (respectively, ν ( ϕ ) ≤ c ).Remind that a subset A of I ( X ) is max-plus-convex if α ⊙ µ ⊕ β ⊙ ν ∈ I ( X ) for every pairof measures µ, ν ∈ I ( X ) where α, β ∈ R max and α ⊕ β = (= 0). Proposition 3.
For a map f : X → Y of compacts and every measure ν ∈ I ( X ) thepreimage I ( f ) − ( ν ) is a max-plus-convex set in I ( X ). Proof.
Let µ , µ ∈ I ( f ) − ( ν ). Then for all α, β ∈ R max with α ⊕ β = we have I ( f )( α ⊙ µ ⊕ β ⊙ µ )( ψ ) = ( α ⊙ µ ⊕ β ⊙ µ )( ψ ◦ f ) = α ⊙ µ ( ψ ◦ f ) ⊕ β ⊙ µ ( ψ ◦ f ) == α ⊙ I ( f )( µ )( ψ ) ⊕ β ⊙ I ( f )( µ )( ψ ) = α ⊙ ν ( ψ ) ⊕ β ⊙ ν ( ψ ) = ν ( ψ ) ,ψ ∈ C ( Y ). So, I ( f )( α ⊙ µ ⊕ β ⊙ µ ) = ν . Proposition 3 is proved. Proposition 4 [16].
The set I β is a max-plus-convex subset of R C b ( X ) . Definition 4.
A subfunctor F of the functor I β , acting in the category T ych , is calledmax-plus-convex if for any Tychonoff space X the space F ( X ) is a max-plus-convex subset of I β ( X ).Equivalent definition looks as follows. A subfunctor F of I β is max-plus if for any Tychonoffspace X , for each pair µ , µ ∈ F ( X ) and for all α, β ∈ R max , α ⊕ β = we have α ⊙ µ ⊕ β ⊙ µ ∈ F ( X ).For a cardinal number τ we denote by I τ the operation which puts in compliance to everyTychonoff space X the set I τ ( X ) of all measures µ ∈ I β ( X ) which supports power less then τ ,and to each continuous map f : X → Y a map I τ ( f ) which is the restriction of I β ( f ) on I τ ( X ).The following statement gives a large class of max-plus-convex subfunctors of I β . Theorem 4.
Let τ be an infinite cardinal number. Then I τ is a max-plus-convex subfunctorof I β . Proof.
For a cardinal number τ and a Tychonoff space X by definition we have I τ ( X ) = { µ ∈ I β ( X ) : | supp µ | < τ } . For every continuous map f : X → Y we have I τ ( f ) = I β ( f ) | I τ ( X ) . Sincesupp I τ ( f )( µ ) = supp I β ( f )( µ ) = f (supp µ ) , µ ∈ I τ ( X ) , | supp I τ ( f )( µ ) | = | f (supp µ ) | ≤ | supp µ | < τ we have I τ ( f )( I τ ( X )) ⊂ I τ ( Y ), i. e. I τ ( f ) is a map from I τ ( X ) into I τ ( Y ). The preservationof compositions of maps and identical map by the operation I τ is obvious. Therefore, I τ is asubfunctor of I β .Let’s check its convexity. Let X be a Tychonoff space, µ , µ ∈ I β ( X ), α, β ∈ R max , α ⊕ β = and µ = α ⊙ µ ⊕ β ⊙ µ . If β or α equals to −∞ then µ coincides with a measure µ or µ ,respectively. If α > −∞ (in this case β = 0 ), or β > −∞ (in this case α = 0 ) thensupp µ = supp µ ∪ supp µ . Anyway supp µ ⊂ supp µ ∪ supp µ . Therefore, | supp µ | ≤ | (supp µ ) | + | supp µ | . From here taking into account the infinity of the cardinal number τ we receive | supp µ | < τ , i.e. µ ∈ I τ ( X ). Theorem 4 is proved.Let f : X → Y be a continuous map and ϕ ∈ C b ( X ). By ϕ ∗ (respectively, ϕ ∗ ) wedenote a function ϕ ∗ : Y → R (respectively, ϕ ∗ : Y → R ) defined by the rule ϕ ∗ ( y ) =sup (cid:8) ϕ ( x ) : x ∈ f − ( y ) (cid:9) (respectively, ϕ ∗ ( y ) = inf (cid:8) ϕ ( x ) : x ∈ f − ( y ) (cid:9) ). It is known that if f is an open map then the functions ϕ ∗ and ϕ ∗ are continuous. Theorem 5.
Let f : X → Y be a continuous map from a Tychonoff space X to a Tychonoffspace Y . Then the map I β ( f ) : I β ( X ) → I β ( Y ) is open if and only if f is open. Proof.
Let f : X → Y be such a map that the map I β ( X ) → I β ( Y ) is open. Fix a point x ∈ X . Let y = f ( x ). Take such ϕ ∈ C b ( X ), that ϕ ( x ) = 0. Put V = { x ∈ X : − < ϕ ( x ) < } . As the sets of the view V form a base of neighbourhood of the point x it is sufficient toshow that f ( V ) is an open neighbourhood of y . Consider an open neighbourhood W = { µ ∈ I β ( X ) : − < µ ( ϕ ) < } of the idempotent probability measure δ x ∈ I β ( X ). Then I β ( f )( W )is an open neighbourhood of the idempotent probability measure δ y ∈ I β ( Y ). There are func-tions ψ , ψ , ..., ψ k ∈ C b ( Y ) with ψ i ( y ) and ε with 0 < ε <
1, such that H = h δ y , ψ , ψ , ..., ψ k ; ε i ⊂ I β ( f )( W ).Put G = n T i =1 { y ∈ Y : − ε < ψ i ( y ) < ε } . Then G is an open neighbourhood of the point y .Let y ∈ G be an arbitrary point. Then δ y ∈ H . Consequently, there exists µ ∈ I β ( X ) such that µ ∈ W and I β ( f )( µ ) = δ y . By the condition − < µ ( ϕ ) <
1. Since every idempotent probabilitymeasure is an order-preserving functional, then µ is an order-preserving functional, that is whythere exists x ∈ f − ( y ) such that − < ϕ ( x ) <
1. Thus, G ⊂ f ( V ) and therefore f is open.Let now f : X → Y be an open map. Assume I β ( f ) is not open. Then there exist:1) an idempotent probability measure µ ∈ I β ( X ),2) a net of idempotent probability measures { ν α } ⊂ I β ( X ) converging to ν = I β ( f )( µ ) and3) a neighbourhood W of µ such that I β ( f ) − ( ν α ) T W = ⊘ for every α .As I ω ( Y ) is everywhere dense in I β ( Y ) one can assume that all of ν α are idempotent prob-ability measures with finite support. Put A α = I β ( X ) − ( ν α ) and [ A α ] = (cid:2) I β ( f ) − ( ν α ) (cid:3) I ( βX ) .As I ( βX ) is a compact and the function I ( βf ) is continuous the net [ A α ] converges to [ A ] =5 I β ( f ) − ( ν ) (cid:3) I ( βX ) according to Vietoris topology in exp I ( βX ). Besides, owing to continu-ity of the map I ( βf ) we have [ A ] ⊂ I ( βf ) − ( ν ) and µ / ∈ [ A ] (otherwise condition 3) isviolated). As µ / ∈ [ A ] for every µ ∈ [ A ] there exists ϕ µ ∈ C b ( X ) ∼ = C ( βX ) such that µ ( ϕ µ ) = µ ( ϕ µ ). According to the assumption for every α there exists a finite set { y α , ..., y αn α } such that ν α ∈ I ( { y α , ..., y αn α } ). For every y αi choose x αi ∈ X such that f ( x αi ) = y αi and ϕ ( x αi ) = ϕ ∗ ( y αi ), µ ∈ [ A ]. Define an embedding j α : { y α , ..., y αn α } → X by the rule j α ( y αi ) = x αi and put µ α = I ( j α )( ν α ). It is easy to see that ϕ = ϕ ∗ ◦ f = ϕ ∗ ◦ βf on every { y α , ..., y αn α } . That is why µ α ( ϕ ) = µ α ( ϕ ∗ ◦ f ) = µ α ( ϕ ∗ ◦ βf ) = I ( βf )( µ α )( ϕ ∗ ) = ν α ( ϕ ∗ )for every α . Let µ be a limit of the net ( µ α ). Then µ ∈ [ A ] and µ ( ϕ ) = lim α µ α ( ϕ ) = lim α µ α ( ϕ ∗ ◦ βf ) = lim α I ( βf )( µ α )( ϕ ∗ ) = lim α ν α ( ϕ ∗ ) = ν ( ϕ ∗ ) , ϕ ∈ C ( X ) . On the other hand ν ( ϕ ∗ ) = I ( βf )( µ )( ϕ ∗ ) = µ ( ϕ ∗ ◦ βf ). Thus, µ ( ϕ ) = µ ( ϕ ∗ ◦ βf ) (2)for every ϕ ∈ C ( X ). Similarly, µ ( ϕ ) = µ ( ϕ ∗ ◦ βf ) (3)for every ϕ ∈ C ( X ). Let ϕ µ ∈ C ( X ) be a function such that µ ( ϕ µ ) = µ ( ϕ µ ). Suppose µ ( ϕ µ ) > µ ( ϕ µ ). Since ϕ ∗ βf ≥ ϕ owing to (2) we have µ ( ϕ µ ) = µ ( ϕ ∗ µ ◦ βf ) ≥ µ ( ϕ µ ).Analogously one can show the assumption µ ( ϕ µ ) < µ ( ϕ µ ) also is false. Thus, we get acontradiction, which shows that I β ( f ) is open. Theorem 5 is proved. Acknowledgement.
The author would like to thank to professor Adilbek Zaitov the headof the department of Mathematics and Natural Disciplines of Tashkent institute of architectureand civil engineering for comprehensive support and attention.