(In)homogeneous invariant compact convex sets of probability measures

It is proved that for any iterated function system of contractions on a complete metric space there exists an invariant compact convex sets of probability measures of compact support on this space. A similar result is proved for the inhomogeneous compact convex sets of probability measures of compact support. Анотація. Математичні підвалини теорії фракталів запропонував Дж. Гатчінсон у 80-х роках минулого століття. Зокрема, він означив поняття атрактора (або інваріантного об’єкта) для ітерованої системи стискуючих відображень (скорочено IFS) на повному метричному просторі і довів існування таких атракторів у гіперпросторі (просторові непорожніх компактних підмножин) та просторі ймовірнісних мір з компактними носіями на повному метричному просторі. Доведення Гатчінсона використовують принцип стискуючих відображень і, зокрема, потребують відповідної метризації простору ймовірнісних мір. Незабаром аналогічні результати було отримано і для неоднорідних атракторів (тобто атракторів з приєднаними ущільнюючими множинами), які є природними узагальненнями інваріантних множин та інваріантних мір. У цій статті ми запроваджуємо поняття інваріантного об’єкта для IFS у просторі компактних опуклих множин ймовірнісних мір з компактними носіями у повному метричному просторі. Такі компактні опуклі множини мір мають численні застосування у теорії очікуваної корисності. На відміну від гіперпростору компактних опуклих підмножин повного метричного простору, інваріантні об’єкти в якому виглядають регулярними, атрактори IFS у просторі компактних опуклих множин мір зберігають іррегулярність, притаманну фрактальним множинам. Одним з основних результатів є теорема існування та єдиності інваріантної компактної опуклої множини ймовірнісних мір з компактними The authors are indebted to the referee for his/her valuable comments. 2010 Mathematics Subject Classification: 37E25, 54E35 UDC 515.12


INTRODUCTION
Hutchinson [7] proved the existence of invariant sets and invariant probability measures for the iterated function systems in the complete metric spaces. The structure of these two proofs is similar and it exploits, in particular, the functoriality of the constructions involved (i.e., the hyperspaces and spaces of probability measures) as well as existence of special metrizations. This led to several generalizations of the existence results, in particular, to the cases of inclusion hyperspaces (i.e., two-valued measures) [11] and idempotent measures on ultrametric spaces [9].
Another approach is applied in [10] and it is proved therein that there exists an invariant idempotent measure (see [18] for topological aspects of the theory of idempotent measures) for an iterated function system on a complete metric space.
Recently, a related notion of inhomogeneous invariant set and measure was introduced in [15]. The properties of these sets and measures were studied in various publications (see, e.g., [5,1,13]).
The compact convex sets of probability measures are used in the maxmin expected utility (MEU) theory [6].
2. PRELIMINARIES 2.1. Hyperspaces. Let exp X denote the set of all nonempty compact subsets of a Tychonov space X. A base of the Vietoris topology on exp X consists of the sets of the form where n P N and U 1 , . . . , U n are open sets in X. The obtained space is called the hyperspace of X.
Actually, exp is a functor in the category of Tychonov spaces and continuous maps. Given a map f : X Ñ Y , the map exp f : exp X Ñ exp Y acts as follows: exp f (A) = f (A), A P exp X.
If (X, d) is a metric space, then the Vietoris topology on exp X is induced by the Hausdorff metric d H , where O r (C) stands for the open r-neighborhood of a subset C.
By u X : exp exp X = exp 2 X Ñ exp X we denote the union map. This map is known to be well defined and, in the case of metric space, nonexpanding.
2.2. Kantorovich metric. By P (X) we denote the space of probability measures on a compact Hausdorff space X. We regard the set of probability measures on X also as a set of normed linear functionals on the Banach space C(X) of continuous real-valued functions on X. Given µ P P (X), we let µ(φ) = ş X φdµ, φ P C(X). The set P (X) is endowed with the weak* topology. A base of this topology is comprised by the sets of the form Let (X, d) be a compact metric space. By 1-LIP(X) we denote the set of all nonexpanding functions on X, i.e. functions φ : X Ñ R satisfying for all x, y P X. The Kantorovich metric d K on the space of probability measures P (X) is defined as follows: Every continuous map f : X Ñ Y between compact spaces induces the map P (f ) : P (X) Ñ P (Y ) defined by P (f )(µ)(A) = µ(f´1(A)) for any µ P P (X) and any measurable subset A Ă Y . In terms of functionals, P (f )(µ)(φ) = µ(φf ) for all µ P P (X) and φ P C(Y ).
Actually, P is a functor in the category Comp of compact Hausdorff spaces.
There is a procedure of extensions of functors from the category Comp to the category of Tychonov spaces [4]. In the case of the functor P , this procedure gives the space of probability measures of compact support. Recall that the support of µ P P (X) is the minimal closed set A Ă X such that µ(XzA) = 0. Alternatively, the support of µ is the minimal closed set A Ă X with the property that, for all φ, ψ P C(X), φ| A = ψ| A implies µ(φ) = µ(ψ).

Convex sets of probability measures.
Let X be a compact Hausdorff space. Denote by ccP(X) the hyperspace of closed convex subsets of the space P (X). Given a continuous map f : X Ñ Y between compact spaces, we define the map ccP(f ) : ccP(X) Ñ ccP(Y ) as follows: It is known that ccP is a functor on the category Comp (see, e.g. [16]). Given A P ccP(X), we say that the set Ytsupp(µ) | µ P Au is the support of A (denoted supp(A)). (Hereafter, for any set Y in a topological space, we denote by Y its closure). Again, applying construction from [4] we extend the functor ccP onto the category of Tychonov spaces. We preserve the notation ccP for this extension.
For any metrizable space X, the space ccP(X) is exactly the hyperspace of closed convex subsets A of P (X) such that supp(A) is compact. Now, assume that X is compact and define a map as follows, see [12]. First, for any compact convex subset K of a locally convex space, denote by b K : P (K) Ñ K the barycenter map. Since P (X) is a subset of the dual space C(X) 1 endowed with the weak* topology, the hyperspace ccP(X) can be regarded as a compact convex subset of a locally convex space [14] and therefore one can consider the barycenter map b ccP(X) : P (ccP(X)) Ñ ccP(X).
Finally, define θ X by the formula Note that the continuity of θ X is a consequence of the continuity of the barycenter map [3, Chapt. III, §3, Corollary of Proposition 9] and the union map [17,Proposition 5.2].
In the case when B is a compact convex subset of the convex hull of a set tM 1 , . . . , M n u, where M 1 , . . . , M n P P (ccP(X)), we have Now, let (X, d) be a metric space. We endow ccP(X) with the Hausdorff metric induced by the Kantorovich metric on P (X). By [8, Proposition 3.2], the map θ X : ccP 2 (X) Ñ ccP(X) is nonexpanding.
Let c¨d(x, y) for all x, y P X. As mentioned above, the 1-Lipschitz maps are also called nonexpanding.

RESULTS
Let (X, d) be a complete metric space and tf 1 , f 2 , . . . , f n u be a finite family of contractions on X (that is, an iterated function system, IFS). Let us consider the discrete topology on the set t1, 2, . . . , nu. Then the space P (t1, 2, . . . , nu) can be regarded as the standard (n´1)-dimensional simplex ∆ n´1 in R n , For B P ccP(t1, 2, . . . , nu) define the map Φ B : ccP(X) Ñ ccP(X) as follows. Let A P ccP(X) and g A : t1, 2, . . . , nu Ñ ccP(X) be the map sending i to ccP(f i )(A). Then we set Φ B (A) = θ X (ccP(g A )(B)).
We say that A P ccP(X) is an invariant set of probability measures for tf 1 , f 2 , . . . , f n u and B whenever A = Φ B (A). Proof. We first consider the case of compact space X. Note that the map Φ B is a contraction. This follows from the fact that the functor ccP preserves c-maps and the map θ X is nonexpanding. By the Banach Contraction Principle, there exists a unique A P ccP(X) such that A = Φ B (A).
In the case of noncompact space X, consider the map Ψ : exp X Ñ exp X defined as follows: is compact for any D P exp X. Now, consider an arbitrary C P ccP(X) and let K = supp(C). Then the the above arguments show that there exists an invariant closed convex set of probability measures A 0 P ccP(Y ) Ă ccP(X). □ Suppose that we are given an IFS tf 1 , f 2 , . . . , f n u on X, B is an element of ccP (t0, 1, . . . , nu), and C P ccP(X). For any A P ccP(X) let g 1 A,C : t0, 1, 2, . . . , nu Ñ ccP(X) be defined by the formulas: . Then the set A satisfying A = Φ 1 (A) is called an inhomogeneous invariant convex set of probability measures. Proof. Let B = tµu P ccP(t1, 2, . . . , nu), for some µ P P (t1, 2, . . . , nu), We start with A 0 = tν 0 u P ccP(X). Then clearly and this easily implies that the invariant set of probability measures A 8 in this case is tν 8 u, where ν 8 is the invariant measure in the sense of [7] corresponding to the IFS tf 1 , . . . , f n u and µ = A similar statement can be formulated and proved in the inhomogeneous case. Therefore our considerations are in some sense extensions of known results from [7] and [15] on probability measures.

FUNCTIONAL APPROACH
Let X be a compact Hausdorff space. Every A P ccP(X) determines a functional F A : C(X) Ñ R defined as follows: Proof. Denote by Without loss of generality one may assume that there exists µ P AzB. Since B is compact, there are φ 1 , . . . , φ k P C(X), for some k P N, such that Since p(B) is compact and convex, it follows from the hyperplane separation theorem that there exists a linear functional l : that sup νPB l(p(ν)) ă l(p(µ)).
Then there exists (l 1 , . . . , l k ) P R k such that Let τ˚be the weak* topology on the set F = tF A | A P ccP(X)}, i.e., the topology induced from the product topology on R C(X) . A base of this topology is comprised by the sets of the form where A 0 P ccP(X), φ 1 , . . . , φ n P C(X), ε ą 0.
If maxtµ(φ 0 ) | µ P Au = µ 0 (φ) for some µ 0 P A, then there exists i P t1, . . . , nu such that µ 0 P Oxµ i ; φ 0 ; εy. Then One can similarly prove that Proof. Due to compactness of X, the space ccP(X) is compact, and the assertion follows from the hausdorffness of F and Proposition 4.1. □ Now the mentioned functional representation A Þ Ñ F A of compact convex sets of probability measures allows us to obtain a purely functional proof of the main results of this paper in the spirit of [10, Theorem 1].

REMARKS
In the case when X = R n and the maps f 1 , . . . , f n are similarities, one can find many pictures of invariant and inhomogeneous sets in the literature.
The invariant probability measures can be visualized in a gray scale by using the random iteration algorithm (see [2, Chapt. IX] for details). An open problem is that of visualization of invariant convex sets of probability measures.