ON UNIVERSAL MAPS AND SPACES OF PROBABILITY MEASURES WITH FINITE SUPPORTS. M.M. Zarichnyĭ

Transcription

1 Œ â â ç i áâã iù 1993, ãáª 2 Š ON UNIVERSAL MAPS AND SPACES OF PROBABILITY MEASURES WITH FINITE SUPPORTS M.M. Zarichnyĭ Abstract. M. Zarichny, On universal maps and spaces of probability measures with finite support, Math. Stud. 2 (1993) 78{82. Recently, the author has constructed the map φ : R Q, R = lim R n, Q = lim Q n which can be characterized by some universality conditions. It is shown that the map P φ is homeomorphic to φ (by P we denote the functor of probability measures with nite supports acting in the category of k ω -spaces). In the last decade, the considerable progress has been reached in the theory of R -manifolds and Q -manifolds (see [1{4]). Recall that R = lim R n, Q = lim Qn, Q is the Hilbert cube [5]. Recently, it has been found by the author [6] that the spaces R and Q are tightly connected, namely, that there exists a map φ : R Q which can be characterized by some natural conditions and possesses many remarkable properties. We begin with necessary denitions. Denote by K(K ω ) the class of (nitedimensional) metrizable compacta. For any class C of topological spaces denote by C the class of spaces that can be represented as direct limits of sequences of elements of the class C and closed embeddings. For any classes A, B of topological spaces let M(A, B) denote the class of maps f : X Y, where X A and Y B (all the maps are assumed to be continuous). Denote by Map the category whose objects are maps of topological spaces. A morphism i = (i, i ) of a map f : X X to a map g : Y Y consists of maps i : X Y and i : X Y making the diagram f X i Y g X i Y commutative. The morphism i = (i, i ) is called embedding if i and i are embeddings. If the maps i and i are inclusions, we say that f is a submap of g (briey, f g) Mathematics Subject Classification. 54C55, 57N Typeset by AMS-TEX

2 ON UNIVERSAL MAPS AND SPACES OF PROBABILITY MEASURES 79 A map f M(K ω, K ) is called strongly (ω, )-universal if for any pair of maps (g, h), g, h M(K ω, K), g h, and embedding i : h f there exists an embedding j : g f extending i. Note that this denition of (ω, )-universality is equivalent to that of [6]. Theorem 1. There exists (unique up to isomorphism in Map) strongly (ω, )- universal map φ : R Q. Proof. See [6]. In this paper we realize the strongly (ω, )-universal map φ as an ane map of convex subsets of linear topological spaces. Recall the construction of the space of probability measures with nite supports. For each compact Hausdor space X we denote by P n X the space of probability measures on X whose support consists of n points (see e.g. [7]). Recall that the support of a measure µ = α i δ xi P n X is the set supp(µ) = {x i α i > 0} (here δ x denotes the Dirac measure supported on x X). Let X be k ω -space i.e. X = lim X i where X 1 X 2... is a sequence of compact subsets of X. Let P X = lim P n X n. It is easy to see that this denition of P X does not depend on the representation X = lim X n. For any continuous map f : X Y of k ω -spaces the map P f : P X P Y is dened as follows P f( α i δ xi ) = α i δ f (xi ). Obviously, we obtain the functor P acting in the category of k ω -spaces. Note that the space P X is naturally embeddable as a convex subset of the free topological linear space of the space X and the map P f can be considered as a restriction of a linear map of free linear topological spaces. By exp X we denote the hyperspace of topological space X and by P we denote the probability measure functor acting in the category of compact Hausdor spaces (see [7]). We need the following Lemma. Let X, Y be metrizable compacta and n N. The map q : P n X P n Y exp P (X Y ), q(µ 1, µ 2 ) = {ν P (X Y ) P pri (ν) = µ i, i = 1, 2}, is continuous. Proof. Let (µ 0, ν 0 ) P n X P n Y and (µ i ), (ν i) be sequences in P nx and P n Y converging to µ 0 and ν 0, respectively. It is sucient to prove that for every λ 0 P (X Y ) with P pr 1 (λ 0 ) = µ 0, P pr 2 (λ 0 ) = ν 0 there exists a sequence (λ i ) in P (X Y ) such that lim i λ i = λ 0 and P pr 1 (λ i ) = µ i, P pr 2 (λ i ) = ν i. Let supp(µ 0 ) = {x 1,..., x k }, supp(ν 0 ) = {y 1,..., y l } and U 1,..., U k (V 1,..., V l ) be disjoint neighbourhoods of the points x 1,..., x k (y 1,..., y l ). Let λ 0 = k p=1 r=1 l γ pr δ (xp,y r ) and µ i = k α ip µ ip + α iµ i, ν i = p=1 l β ir ν ir + β iν i, r=1

3 80 M.M. ZARICHNY I where µ ip, µ i, ν ir, ν i are probability measures such that supp(µ ip) U p, supp(µ i ) X \ (U 1 U k ), supp(ν ir ) V r, supp(ν r) Y \ (V 1 V l ). Let λ i = k l γ pr µ ip ν ir and ε i = sup{ε 0 εp pr1 (λ i ) µ i, εp pr2 (λ i ) ν i }. p=1 r=1 Dene the elements λ i by the following formula: λ i = ε i λ i + ((µ i ε i P pr 1 (λ i)) (γ i ε i P pr 2 (λ i)))/ µ i ε i P pr 1 (λ i). Obviously, P pr 1 (λ i ) = µ i, P pr 2 (λ i ) = γ i. Since lim i λ i = λ 0, we have lim i ε i = 1 and consequently lim i λ i = λ. Lemma is proved. Theorem 2. The map P φ is homeomorphic to φ. Proof. We have to prove (ω, )-universality of the map P φ. Given maps f : A A, g : B B, f, g M(K ω, K), f g, and embedding j = (j, j ) : g P φ, nd n N such that j (B ) P n I n, j (B ) P n Q n. Here we assume that R = lim I n, Q = lim Q n, where Q 1 Q 2... is a sequence of copies Q i of the Hilbert cube and Z-embeddings, and φ(i n ) Q n. Since P n+1 Q n+1 contains P n Q n as Z-set and P i Q = Q (see [7]), there exists an embedding i : A P n+1 Q n+1 extending the embedding j : B P n Q n P n+1 Q n+1 (see [5]) for the properties of Z-sets in Q. Let S = {(a, x) A Q n+1 x supp(i f(a))}. For each a A let i a : supp(i f(a)) S be a map dened by the formula: i a (y) = (a, y), y supp(i f(a)). Dene the map ζ : A P n+1 S by the formula: ζ(a) = P i a (i f(a)). Note that the map ζ is continuous, by the arguments of [8]. As A.N.Dramishnikov remarked [9], dim S = dim A <. Let α = pr 2 S : S Q n+1, where pr 2 : A Q n+1 Q n+1 is the projection. Consider the pullback Z π 2 I n+1 π 1 φ I n+1 =φ n+1 S α Q n+1. There exists a nite-dimensional separable metrizable AR-space T that contains S as a closed subset (see e.g. [10]). Without loss of generality, assume that Z T I k, for some k N, and the map π 1 is the restriction of the projection pr 1 : T I k T. Since the functor P n preserves the class of absolute retracts (see [7]), the space P n+1 (T I k ) is an AR(M)-space. Denote by p i the projection map of the space T I k onto the i-th factor, i = 1, 2. Strong (ω, )-universality of the map φ implies that there exist embeddings κ : T I k I l, γ : T Q l for some l n + 1 making the diagram

4 ON UNIVERSAL MAPS AND SPACES OF PROBABILITY MEASURES 81 T I k κ I l Z π 2 I n+1 p 1 π 1 φ n+1 φ l T S α Q n+1 γ Q l commutative. Since P n+1 (T I k ) is AR(M)-space [11], there exists an extension h : A P n+1 (κ(t I k )) of the map j. Let a A and q(a) = {µ P (κ(t I k )) P (p 2 κ 1 )(µ) = P (p 2 κ 1 )(h(a)), P (p 1 κ 1 (µ) = ζ(a)}. Obviously, q(a) is a compact convex subset of P r (κ(t I k )) for some r N and it follows from Lemma that the map q : A exp P r (κ(t I k )) is continuous. Assuming that P r (κ(t I k )) P I l is anely embedded into a Hilbert space l denote by ξ : conv(l) L L the nearest point map (here conv(l) denotes the hyperspace of nonempty compact convex subsets of L, see [12]). Dene the map i 1 : A I l by the formula: i 1(a) = ξ(q(a), h(a)), a A. If a B, then h(a) = j (a) and P (p 1 κ 1 )(j (a)) = P (p 1 κ 1 φ 1 n+1 φ n+1 )(i (a)) = P (γ 1 φ n+1 )(j (a)) = P γ 1 ((j g)(a)) = (P γ 1 i f)(a) = ζ(a). Consequently, h(a) q(a) and i 1(a) = j (a). Note that for each a A and µ q(a) we have P φ(µ) = (P (φ κ) P κ 1 )(µ) = (P (γ p 1 ) P κ 1 )(µ) = (P γ ζ)(a) = P (γ i a )((i f)(a)) = P (α i a )((i f)(a)) = (i f)(a), and consequently, P φ i 1 = i f. Finally, modify the map i 1 to obtain an embedding i : A P (R ). By (ω, )-universality, we can assume that the space I l lies as I l {0} in the copy I l I m R, m = 2 dim A +1 and (φ I l ) pr 1 = φ (I l I m ). For each y I m let s y : I l I l I m denotes the map dened by the formula: s y (x) = (x, y), x I l. Let ψ : A I m be a map such that ψ(a) = 0 for every a B and ψ (A \B ) embeds A \B into I m \{0}. Dene the map i : A P l (I l I m ) P (R ) by the formula: i (a) = P s ψ(a) (i 1(a)), a A. It is easy to see that i is an embedding such that P φ i = i f and i B = j. Thus, the map P φ is strongly (ω, )-universal and, by Theorem 3 of [6], P φ = φ. Theorem is proved.

5 82 M.M. ZARICHNY I R E F E R E N C E S 1. Sakai K. On R -manifolds and Q -manifolds // Topol. Appl V.18, N1. P Sakai K. On R -manifolds and Q -manifolds, II: Innite deciency // Tsukuba J. Math V.8, N1. P Vo Thang Liem. On innite deciency in R -manifolds // Trans. Amer. Math. Soc V. 288, N1. P å.ž. à âà ç áª à ã«ìâ âë «ï ª â àëå ª«áá áª ç à ëå - à // ªà. â. ãà , N Chapman T. Lectures on Hilbert cube manifolds.-cbms Reg. Conf. Ser. Math., N28. Providence, à ç ë Œ. ã ªâ àë, à ë ã àá «ì ë â à ï ê ªâ ëå à «á«â «ì áâ ª ªâ Œ à // Œ â â ª. ãç ë âàã ë , àçãª.. Š à â ë äã ªâ àë ª â à ª ªâ, á «îâ ë à âà ªâë Q- à ï // á å â. ãª , ë àçãª.. Œï ª â à ï, ç ë à âà ªæ äã ªâ àë // á å â. ãª , ë à è ª.. Š à â ë äã ªâ àë á «îâ ë íªáâ àë à à áâ n // á å â. ãª , ë àiç Œ.Œ. à ªâ à æiï ª ªâ å AE(n)- à áâ ài // ªà. â. ãà , N àçãª.. à ïâ áâ ë àë â «// á å â. ãª , ë ãà ª. «ç áª ªâ à ë à áâà áâ.{œ.:ˆ, Department of Mechanics and Mathematics, Lviv University, Universytetska 1, Lviv, , Ukraine Received