A functional representation of the capacity multiplication monad
aa r X i v : . [ m a t h . GN ] M a r A FUNCTIONAL REPRESENTATION OF THE CAPACITYMULTIPLICATION MONAD
TARAS RADUL
Institute of Mathematics, Casimirus the Great University of Bydgoszcz, Poland;Department of Mechanics and Mathematics, Ivan Franko National University ofLviv, Universytettska st., 1. 79000 Lviv, Ukraine.e-mail: [email protected]
Key words and phrases:
Monad, capacity, fuzzy integral, triangular norm.[MSC 2010]18B30, 18C15, 28E10, 54B30
Abstract.
Functional representations of the capacity monad based on themax and min operations were considered in [10] and [7]. Nykyforchyn con-sidered in [8] some alternative monad structure for the possibility capacityfunctor based on the max and usual multiplication operations. We show thatsuch capacity monad (which we call the capacity multiplication monad) hasa functional representation, i.e. the space of capacities on a compactum X can be naturally embedded (with preserving of the monad structure) in somespace of functionals on C ( X, I ). We also describe this space of functionals interms of properties of functionals. Introduction
Functional representations of monads (i.e. natural embeddings into R C ( X,S ) which preserves a monad structure where S is a subset of R ) were considered in [11]and [12]. Some functional representations of hyperspace monad were constructedin [13] and [14].Capacities (non-additive measures, fuzzy measures) were introduced by Choquetin [1] as a natural generalization of additive measures. They found numerous ap-plications (see for example [2],[4],[16]). Categorical and topological properties ofspaces of upper-semicontinuous capacities on compact Hausdorff spaces were in-vestigated in [9]. In particular, there was built the capacity functor which is afunctorial part of a capacity monad M based on the max and min operations.Well known is the Choquet integral, which is, in fact, some functional representa-tion of the functor M , i.e., the space of capacities M X can be naturally embeddedin R C ( X ) . But this representation does not preserve the monad structure. Nyky-forchyn using the Sugeno integral provided a functional representation of capacitiesas functionals on the space C ( X, I ) which preserves the monad structure [7]. Somemodification of the Sugeno integral yields a functional representation of capacitiesas functionals on the space C ( X ) [10].Let us remark that the min operation is a triangular norm on the unit interval I .Another important triangular norm is the multiplication operation. Nykyforchynbuild in [8] a capacity monad based on the max and multiplication operations. (Letus remark that recently Zarichnyi proposed to use triangular norms to constructmonads [20]). The main aim of this paper is to find a representation of the monadfrom [8]. We use a fuzzy integral based on the max and multiplication operationsfor this purpose. Capacities and monads By Comp we denote the category of compact Hausdorff spaces (compacta) andcontinuous maps. For each compactum X we denote by C ( X ) the Banach space ofall continuous functions φ : X → R with the usual sup-norm: k φ k = sup {| φ ( x ) | | x ∈ X } . We also consider on C ( X ) the natural partial order.In what follows, all spaces and maps are assumed to be in Comp except for R ,the spaces C ( X ) and functionals defined on C ( X ) with X compact Hausdorff.We recall some categorical notions (see [15] and [17] for more details). We definethem only for the category Comp . The central notion is the notion of monad (ortriple) in the sense of S.Eilenberg and J.Moore.A monad [3] T = ( T, η, µ ) in the category
Comp consists of an endofunctor T : Comp → Comp and natural transformations η : Id Comp → T (unity), µ : T → T (multiplication) satisfying the relations µ ◦ T η = µ ◦ ηT = T and µ ◦ µT = µ ◦ T µ .(By Id
Comp we denote the identity functor on the category
Comp and T is thesuperposition T ◦ T of T .)Let T = ( T, η, µ ) be a monad in the category
Comp . The pair (
X, ξ ) where ξ : T X → X is a map is called a T - algebra if ξ ◦ ηX = id X and ξ ◦ µX = ξ ◦ T ξ .Let (
X, ξ ), (
Y, ξ ′ ) be two T -algebras. A map f : X → Y is called a T -algebrasmorphism if ξ ′ ◦ T f = f ◦ ξ .A natural transformation ψ : T → T ′ is called a morphism from a monad T =( T, η, µ ) into a monad T ′ = ( T ′ , η ′ , µ ′ ) if ψ ◦ η = η ′ and ψ ◦ µ = µ ′ ◦ ηT ′ ◦ T ψ . If all ofthe components of ψ are monomorphisms then the monad T is called a submonad of T ′ and ψ is called a monad embedding .Let A be a subset of X . By F ( X ) we denote the family of all closed subsets of X . Put I = [0 , ν : F ( X ) → I is called an upper-semicontinuous capacity on X if the three following properties hold for each closedsubsets F and G of X :1. ν ( X ) = 1, ν ( ∅ ) = 0,2. if F ⊂ G , then ν ( F ) ≤ ν ( G ),3. if ν ( F ) < a , then there exists an open set O ⊃ F such that ν ( B ) < a for eachcompactum B ⊂ O .A capacity ν is extended in [9] to all open subsets U ⊂ X by the formula ν ( U ) = sup { ν ( K ) | K is a closed subset of X such that K ⊂ U } .It was proved in [9] that the space M X of all upper-semicontinuous capacitieson a compactum X is a compactum as well, if a topology on M X is defined bya subbase that consists of all sets of the form O − ( F, a ) = { c ∈ M X | c ( F ) < a } ,where F is a closed subset of X , a ∈ [0 , O + ( U, a ) = { c ∈ M X | c ( U ) > a } ,where U is an open subset of X , a ∈ [0 , M X simply capacities.A capacity ν ∈ M X for a compactum X is called a necessity (possibility) capac-ity if for each family { A t } t ∈ T of closed subsets of X (such that S t ∈ T A t is a closedsubset of X ) we have ν ( T t ∈ T A t ) = inf t ∈ T ν ( A t ) ( ν ( S t ∈ T A t ) = sup t ∈ T ν ( A t )).(See [19] for more details.) We denote by M ∩ X ( M ∪ X ) a subspace of M X con-sisting of all necessity (possibility) capacities. Since X is compact and ν is upper-semicontinuous, ν ∈ M ∩ X iff ν satisfy the simpler requirement that ν ( A ∩ B ) =min { ν ( A ) , ν ( B ) } .If ν is a capacity on a compactum X , then the function κX ( ν ), that is definedon the family F ( X ) by the formula κX ( ν )( F ) = 1 − ν ( X \ F ), is a capacity aswell. It is called the dual capacity (or conjugate capacity ) to ν . The mapping κX : M X → M X is a homeomorphism and an involution [9]. Moreover, ν is anecessity capacity if and only if κX ( ν ) is a possibility capacity. This implies in FUNCTIONAL REPRESENTATION OF THE CAPACITY MULTIPLICATION MONAD 3 particular that ν ∈ M ∪ X iff ν satisfy the simpler requirement that ν ( A ∪ B ) =max { ν ( A ) , ν ( B ) } . It is easy to check that M ∩ X and M ∪ X are closed subsets of M X .The assignment M extends to the capacity functor M in the category of com-pacta, if the map M f : M X → M Y for a continuous map of compacta f : X → Y is defined by the formula M f ( c )( F ) = c ( f − ( F )) where c ∈ M X and F is a closedsubset of X . This functor was completed to the monad M = ( M, η, µ ) [9], where thecomponents of the natural transformations are defined as follows: ηX ( x )( F ) = 1if x ∈ F and ηX ( x )( F ) = 0 if x / ∈ F ; µX ( C )( F ) = sup { t ∈ [0 , | C ( { c ∈ M X | c ( F ) ≥ t } ) ≥ t } , where x ∈ X , F is a closed subset of X and C ∈ M ( X ) (see [9]for more details).It was shown in [5] that M ∪ and M ∩ are subfunctors of M and if we takecorresponding restrictions of the functions µX , we obtain submonads M ∪ and M ∩ of the monad M .The semicontinuity of capacities yields that we can change sup for max in thedefinition of the map µX . More precisely, existing of max follows from Lemma 3.7[9]. For a closed set F ⊂ X and for t ∈ I put F t = { c ∈ M X | c ( F ) ≥ t } . We canrewrite the definition of the map µX as follows µX ( C )( F ) = max {C ( F t ) ∧ t | t ∈ (0 , } .Let us remark that the operation ∧ is a triangular norm. It seems naturally toconsider instead ∧ another triangular norm. Define the map µ • X : M X → M X by the formula µ • X ( C )( F ) = max {C ( F t ) · t | t ∈ (0 , } . (Existing of max as wellfollows from Lemma 3.7 [9].) Proposition 1.
The natural transformation µ • does not satisfy the property µ • ◦ µ • M = µ • ◦ M µ • .Proof. Consider X = { a, b } , where { a, b } is a two-point discrete space. Define A ∈ M X as follows A ( α ) = 1 iff α ⊃ { a } and A ( α ) = 0 otherwise for α ∈ F ( M X ). Define A ∈ M X as follows A ( α ) = 1 iff α = M X , A ( α ) = iff α ⊃ { a } and A ( α ) = 0 otherwise for α ∈ F ( M X ). Now, define ג ∈ M ( X ) bythe formula ג (Λ) = ηM X ( A )(Λ) + ηM X ( A )(Λ) for Λ ∈ F ( M X ).We have µ • X ◦ M ( µ • X )( ג )( { a } ) = max { ג (( µ • X ) − ( { a } t )) · t | t ∈ (0 , } . It iseasy to see that µ • X ( A ) = µ • X ( A ) = . Then ג (( µ • X ) − ( { a } )) · = 1 · = .Hence we obtain µ • X ◦ µ • M X ( ג )( { a } ) ≥ .On the other hand µ • X ◦ µ • M X ( ג )( { a } ) = max { µ • M X ( ג )( { a } t )) · t | t ∈ (0 , } = max { max { ג (( { a } t ) s ) · s | s ∈ (0 , } · t | t ∈ (0 , } . The function δ ( s, t ) = ג (( { a } t ) s ) is nonincreasing on both variables. We have δ ( s, t ) = 0 foreach ( s, t ) such that s > and t > . Moreover δ (1 , ) = δ ( ,
1) = . Hence µ • X ◦ µ • M X ( ג )( { a } ) = max { max { ג (( { a } t ) s ) · s | s ∈ (0 , } · t | t ∈ (0 , } = . (cid:3) Remark 1.
Since the triple M • = ( M, η, µ • ) does not form a monad, the problemof uniqueness of the monad M stated in [9] is still open. But things may turn out differently if we restrict the map µ • X to the set M ∪ ( M ∪ X ) ⊂ M ( M X ). It is easy to see that for such restriction we can con-sider the sets A t in the definition of the map µ • X as subsets of M ∪ X . It wasdeduced from some general facts that the triple M •∪ = ( M ∪ , η, µ • ) is a monad [8].For sake a completeness we give here a direct proof. Lemma 1.
We have µ • X ( M ∪ ( M ∪ X )) ⊂ M ∪ X for each compactum X .Proof. Consider any
A ∈ M ∪ ( M ∪ X ) and B , C ∈ F ( X ). Since B t and C t aresubsets of M ∪ X , we have ( C ∪ B ) t = C t ∪ B t . Then µ • X ( A )( B ∪ C ) = max {A (( C ∪ B ) t ) · t | t ∈ (0 , } = max {A ( C t ∪ B t ) · t | t ∈ (0 , } = max { max {A ( C t ) · t | t ∈ (0 , } , max {A ( B t ) · t | t ∈ (0 , } = max { µ • X ( A )( B ) , µ • X ( A )( C ) } . (cid:3) TARAS RADUL
We will use the notation µ • X also for the restriction µ • X | M ∪ X . Theorem 1.
The triple M •∪ = ( M ∪ , η, µ • ) is a monad.Proof. It is easy to check that η and µ • are well-defined natural transformations ofcorresponding functors. Let us check two monad properties.Take any compactum X , ν ∈ M ∪ X and A ∈ F ( X ). Then we have µ • X ◦ ηM ∪ X ( ν )( A ) = max { η M ∪ X ( ν )( A t ) · t | t ∈ (0 , } = ν ( A ) and µ • X ◦ M ∪ ( ηX )( ν )( A ) =max { M ∪ ( ηX )( ν )( A t ) · t | t ∈ (0 , } = max { ν (( ηX ) − ( A t )) · t | t ∈ (0 , } =max { ν ( A ) · t | t ∈ (0 , } = ν ( A ). We obtain the equality µ • ◦ M ∪ η = µ • ◦ ηM ∪ = M ∪ .Now, consider any ג ∈ M ∪ ( X ) and A ∈ F ( X ). Put a = µ • X ◦ M ∪ ( µ • X )( ג )( A ) =max { ג (( µ • X ) − ( A t )) · t | t ∈ (0 , } and b = µ • X ◦ µ • M ∪ X ( ג )( { a } ) == max { µ • M ∪ X ( ג )( A t )) · t | t ∈ (0 , } = max { max { ג (( A t ) s ) · s | s ∈ (0 , } · t | t ∈ (0 , } .There exists t ∈ (0 ,
1] such that a = ג (( µ • X ) − ( A t )) · t . We have ( µ • X ) − ( A t ) = {A ∈ M ∪ ( X ) | µ • X ( A ) ≥ t } = {A ∈ M ∪ ( X ) | there exists c ∈ (0 ,
1] such that A ( A c ) · c ≥ t } = {A ∈ M ∪ ( X ) | there exists c ∈ (0 ,
1] such that A ( A c ) ≥ t c } .Since ג is a possibility capacity, there exists A ∈ M ∪ ( X ) and c ∈ (0 ,
1] suchthat A ( A c ) ≥ t c and ג (( µ • X ) − ( A t )) = ג ( {A } ). But then we have a ≤ ג (( A c ) t c ) · t = ג (( A c ) t c ) · t c · c ≤ b .On the other hand choose p , z ∈ (0 ,
1] such that b = ג (( A p ) z ) · p · z .Since ג is a possibility capacity, there exists B ∈ ( A p ) z such that ג (( A p ) z ) = ג ( {B } ). We have B ( A p ) ≥ z , hence µ • X ( B )( A ) ≥ z · p . Then we obtain b = ג ( {B } ) · p · z ≤ ג (( µ • X ) − ( A p · z )) · p · z ≤ a . (cid:3) Functional representation of the monad M •∪ A monad F = ( F, η, µ ) is called an
IL-monad if there exists a map ξ : F I → I such that the pair ( I, ξ ) is an F -algebra and for each X ∈ Comp there exists apoint-separating family of F -algebras morphisms { f α : ( F X, µX ) → ( I, ξ ) | α ∈ A } [12].There was defined a monad V I in [12], which is universal in the class of IL-monads. By V I X we denote the power I C ( X,I ) . For a map φ ∈ C ( X, I ) we denoteby π φ or π ( φ ) the corresponding projection π φ : V I X → I . For each map f : X → Y we define the map V I f : V I X → V I Y by the formula π φ ◦ V I f = π φ ◦ f for φ ∈ C ( Y, I ).For a compactum X we define components hX and mX of natural transformationsby π φ ◦ hX = φ and π φ ◦ mX = π ( π φ ) for all φ ∈ C ( X, I )). The triple V I = ( V I , h, m )forms a monad in the category Comp and for each monad F there exists a monadembedding l : F → V I if and only if F is IL-monad [12]. Moreover, for a compactum X the map lX : F X → V I X is defined by the conditions π φ ◦ lX = ξ ◦ F φ for each ψ ∈ C ( X, I ). Theorem 2.
The monad M •∪ is an IL-monad.Proof. Define the map ξ : M ∪ I → I by the formula ξ ( ν ) = max { ν ([ t, · t | t ∈ (0 , } . We can check that the pair ( I, ξ ) is an M •∪ -algebra by the same but simplerarguments as in the proof of Theorem 1.Consider any compactum X and two distinct capacities ν , β ∈ M ∪ X . Thenthere exists A ∈ F ( X ) such that ν ( A ) = β ( A ). We can suppose that ν ( A ) <β ( A ). Since ν and β are possibility capacities, there exist a , b ∈ A such that ν ( { a } ) = ν ( A ) and β ( { b } ) = β ( A ). Choose a point t ∈ ( ν ( A ) , β ( A )). Put B = { x ∈ X | ν ( { x } ) ≥ t } . Since ν is a possibility capacity and ν ( X ) = 1, B isnot empty. Since ν is upper semicontinuous, B is closed. Evidently, B ∩ A = ∅ .Choose a function ϕ ∈ C ( X, I ) such that ϕ ( B ) ⊂ { } and ϕ ( A ) ⊂ { } . Then FUNCTIONAL REPRESENTATION OF THE CAPACITY MULTIPLICATION MONAD 5 π ϕ ◦ lX ( ν ) = ξ ◦ M ∪ ϕ ( ν ) = max { M ∪ ϕ ( ν )([ s, · s | s ∈ (0 , } = max { ν ( ϕ − [ s, · s | s ∈ (0 , } ≤ t < β ( A ) ≤ β ( ϕ − { } ) · ≤ π ϕ ◦ lX ( β ). It is easy to check that π φ ◦ lX = ξ ◦ M ∪ φ : M ∪ X → I is an M •∪ -algebras morphism. (cid:3) Hence we obtain an monad embedding l : M •∪ → V I such that π ϕ ◦ lX ( ν ) =max { ν ( ϕ − [ s, · s | s ∈ (0 , } for each compactum X , ν ∈ M ∪ X and ϕ ∈ C ( X, I ).Let X be any compactum. For any c ∈ I we shall denote by c X the constantfunction on X taking the value c . Following the notations of idempotent mathe-matics (see e.g., [6]) we use the notation ⊕ in I and C ( X, I ) as an alternative formax. We will use the notation ν ( ϕ ) = π ϕ ◦ lX ( ν ) for ν ∈ V I X and ϕ ∈ C ( X, I ).Consider the subset SX ⊂ V I X consisting of all functionals ν satisfying thefollowing conditions(1) ν (1 X ) = 1;(2) ν ( λ · ϕ ) = λ · ν ( ϕ ) for each λ ∈ I and ϕ ∈ C ( X, I );(3) ν ( ψ ⊕ ϕ ) = ν ( ψ ) ⊕ ν ( ϕ ) for each ψ , ϕ ∈ C ( X, I ).Let us remark that properties 1 and 2 yield that ν ( c X ) = c for each ν ∈ SX and c ∈ I . Theorem 3. lX ( M ∪ X ) = SX .Proof. Consider any ν ∈ M ∪ X . Put υ = lX ( ν ). Then we have υ (1 X ) == max { ν ((1 X ) − [ s, · s | s ∈ (0 , } = max { ν ( X ) · s | s ∈ (0 , } = 1.Take any c ∈ I and ϕ ∈ C ( X, I ). For c = 0 the Property 2 is trivial. For c > υ ( cϕ ) = max { ν (( cϕ ) − [ s, · s | s ∈ (0 , } = max { ν ( ϕ − [ sc , · sc | s ∈ (0 , } · c = c · υ ( ϕ ).Consider any ψ and ϕ ∈ C ( X, I ). We have υ ( ψ ⊕ ϕ ) = max { ν (( ψ ⊕ ϕ ) − [ s, · s | s ∈ (0 , } = max { ν ( ψ − [ s, ∪ ϕ − [ s, · s | s ∈ (0 , } = max { ( ν ( ψ − [ s, ⊕ ν ( ϕ − [ s, · s | s ∈ (0 , } = υ ( ψ ) ⊕ υ ( ϕ ). We obtained lX ( M ∪ X ) ⊂ SX .Take any υ ∈ SX . For A ∈ F ( X ) put Υ A = { ϕ ∈ C ( X, I ) | ϕ ( a ) = 1 for each a ∈ A } . Define ν : F ( X ) → I as follows ν ( A ) = inf { υ ( ϕ ) | ϕ ∈ Υ A } if A = ∅ and ν ( ∅ ) = 0. It is easy to see that ν satisfies Conditions 1 and 2 from the definition ofcapacity.Let ν ( A ) < η for some η ∈ I and A ∈ F ( X ). Then there exists ϕ ∈ Υ A suchthat υ ( ϕ ) = χ < η . Choose ε > ε ) χ < η . Put δ = ε and ψ = min { δ X , ϕ } . Then υ ( ψ ) ≤ υ ( ϕ ) = χ and υ ((1 + ε ) ψ ) ≤ (1 + ε ) χ < η . Put U = ϕ − ( δ, U is an open set and U ⊃ A . But for each compact K ⊂ U we have (1 + ε ) ψ ∈ Υ K . Hence ν ( K ) < η .Finally take any A , B ∈ F ( X ). Evidently ν ( A ∪ B ) ≥ ν ( A ) ⊕ ν ( B ). Suppose ν ( A ∪ B ) > ν ( A ) ⊕ ν ( B ). Then there exists ϕ ∈ Υ A and ψ ∈ Υ B such that ν ( A ∪ B ) > υ ( ϕ ) ⊕ υ ( ψ ) = υ ( ϕ ⊕ ψ ). But ϕ ⊕ ψ ∈ Υ A ∪ B and we obtain acontradiction. Hence ν ∈ M ∪ X .Let us show that lX ( ν ) = υ . Take any ϕ ∈ C ( X, I ). Denote ϕ t = ϕ − [ t, lX ( ν )( ϕ ) = max { inf { υ ( χ ) | χ ∈ Υ ϕ t } · t | t ∈ (0 , } = max { inf { υ ( tχ ) | χ ∈ Υ ϕ t } | t ∈ (0 , } . For each t ∈ (0 ,
1] put χ t = min { t ϕ, X } ∈ Υ ϕ t . We have tχ ≤ ϕ , hence υ ( tχ ) ≤ υ ( ϕ ). Then we have inf { υ ( tχ ) | χ ∈ Υ ϕ t } ≤ υ ( ϕ ) for each t ∈ (0 , lX ( ν )( ϕ ) ≤ υ ( ϕ ).Suppose lX ( ν )( ϕ ) < υ ( ϕ ). Choose any a ∈ ( lX ( ν )( ϕ ) , υ ( ϕ )). Then for each t ∈ (0 ,
1] there exists χ t ∈ Υ ϕ t such that υ ( tχ t ) < a . Choose ε > ε ) a < υ ( ϕ ). Put δ = ε . Choose n ∈ N such that δ n < υ ( ϕ ). Put ψ n +1 = δ nX and ψ i = δ i − χ δ i for i ∈ { , . . . , n } . We have υ ( ψ i ) < υ ( ϕ ) for each i ∈ { , . . . , n + 1 } .Put ψ = ⊕ n +1 i =1 ψ i . Then υ ( ψ ) = ⊕ n +1 i =1 υ ( ψ i ) < υ ( ϕ ). On the other hand ϕ ≤ ψ andwe obtain a contradiction. (cid:3) TARAS RADUL
Hence we obtain, in fact, that the monad M •∪ is isomorphic to a submonad of V I with functorial part acting on compactum X as SX . Let us remark that thismonad is one of monads generated by t-norms considered by Zarichnyi [20]. Thusthe following question seems to be natural: can we generalize the results of thispaper to any continuous t-norms? References [1] G. Choquet
Theory of Capacity,
An.l’Instiute Fourie (1953-1954), 13–295.[2] J.Eichberger, D.Kelsey, Non-additive beliefs and strategic equilibria , Games Econ Behav (2000) 183–215.[3] Eilenberg S., Moore J., Adjoint functors and triples , Ill.J.Math., (1965), 381–389.[4] I.Gilboa, Expected utility with purely subjective non-additive probabilities , J. of MathematicalEconomics (1987) 65–88.[5] I.D. Hlushak, O.R.Nykyforchyn, Submonads of the capacity monad , Carpathian J. of Math. (2008) 56–67.[6] V.P. Maslov, S.N. Samborskii,
Idempotent Analysis , Adv. Soviet Math., vol. 13, Amer. Math.Soc., Providence, 1992.[7] O.R.Nykyforchyn,
The Sugeno integral and functional representation of the monad of lattice-valued capacities , Topology (2009) 137–148.[8] O.R.Nykyforchyn, L-Convexity and Lattice-Valued Capacities , Journal of Convex Analysis (2014) 29–52.[9] O.R.Nykyforchyn, M.M.Zarichnyi, Capacity functor in the category of compacta , Mat.Sb. (2008) 3–26.[10] T.Radul,
A functional representation of capacity monad , Topology (2009) 100–104.[11] T.Radul, On functional representations of Lawson monads , Applied Categorical Structures, (2001), 457–463.[12] T.Radul, On strongly Lawson and I-Lawson monads , Boletin de Matematicas, (1999),69–75.[13] T.Radul, A functional representation of the hyperspace monad , Comm.Math.Univ. of Car-olinae, (1997), 165–168.[14] T.Radul, Hyperspace as intersection of inclusion hyperspaces and idempotent measures , Mat.Stud., (2009), 207–210.[15] T.N.Radul, M.M.Zarichnyi, Monads in the category of compacta , Uspekhi Mat.Nauk. (1995) 83-108.[16] D.Schmeidler, Subjective probability and expected utility without additivity , Econometrica (1989) 571–587.[17] Teleiko A., Zarichnyi M., Categorical Topology of Compact Hausdorff Spaces, VNTL Pub-lishers. Lviv, 1999.[18] M.Sugeno, Fuzzy measures and fuzzy integrals , A survey. In Fuzzy Automata and DecisionProcesses. North-Holland, Amsterdam: M. M. Gupta, G. N. Saridis et B. R. Gaines editeurs.89–102. 1977[19] Zhenyuan Wang, George J.Klir
Generalized measure theory , Springer, New York, 2009.[20] M.M.Zarichnyi,