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Вестник Самарского университета. Естественнонаучная серия. Том 24 № 2 2018
УДК 515.12 DOI: 10.18287/2541-7525-2018-24-2-28-32
T.F. Zhuraev, A.Kh. Rakhmatullaev, Z.O. Tursunova1
SOME VALUES SUBFUNCTORS OF FUNCTOR PROBALITIES MEASURES IN THE CATEGORIES COMP
This article is dedicated to the preservation by subfunctors of the functor P of spaces of probability measures countable dimension and extensor properties of spaces of probability measures subspaces.
Key words: probability measures, dimension, the Z-set, homotopy dense, strong discrete approximation properties.
Citation. Zhuraev T.F., Rakhmatullaev A.Kh., Tursunova Z.O. Some values subfunctors of functor probalities measures in the categories Comp. Vestnik Samarskogo universiteta. Estestvennonauchnaia seriia [Vestnik of Samara University. Natural Science Series], 2018, no. 24, no. 2, pp. 28-32. DOI: http://doi.org/10.18287/2541-7525-2018-24-2-28-32 [in Russian].
MSC: 54B15, 54B30, 54B35, 54C05, 54C15, 54C60, 54D30.
1. Introduction
Let X be a topological space. By C(X) is denoted the ring of all continuous real valued functions on the space X with the compact-open topology. The diagonal product of all mappings at C(X) is defined by the embedding of X intoRC(X).
If X is compact, then closed span of its images is a convex compact space which is denoted by P(X) [6]. On the other hand the probability measure functor P is covariant functor acting in the category of compact spaces and their continuous maps. P(X) is a convex subspace of a linear space M(X) conjugate to the space C(X) of continuous functions on X with the weak topology, consisting of all non-negative functional y (i.e.y (p) > 0) for every non-negative p € C (X) with unit norm [2,7]. For a continuous map f : X ^ Y the mapping
P (f): P (X) ^ P (Y) is defined as follows (P (f) (y)) p = y (f ◦ f).
The space P(X) is naturally embedded The base of neighborhoods of a measure y € P (X)
consists of all sets of the form O (y1, p1 ,p2,..., fk,e) = {y' € P (X) : |n (pi) — y! (fi)\ < £,i = l,k, } where e > 0, fi,p2,..,pk € C (X) are arbitrary functions.
2. About a topology on a subspace of the space of probability measures
Let F be a subfunctor of P with a finite support. Then the base of neighborhoods of a measure =
__s + 1
= m° • ... + m° • S(xs) € f (X) consists of sets of the form O < y0,U1, ...,US >= {y € F (X) : y = ^ },
i=i
where yi € M +(X) is the set of all non-negative functional and ||yi+1y < e, suppyi c U^HlyH — m°\ < e for i = 1,...,S, where U1,...,US — are neighborhoods of points x1,...,xS with disjoint closures.
!© Zhuraev T.F., Rakhmatullaev A.Kh., Tursunova Z.O., 2018
Zhuraev Tursunboy Faizievich ([email protected]), Department of General Mathematics, Tashkent State Pedagogical University named after Nizami, 27, Bunyodkor Street, Tashkent, 100070, Republik of Uzbekistan.
Rakhmatullaev Alimby Khasanovich ([email protected]), Department of Higher Mathematics, Tashkent Institute of Irrigation and Agricultural Mechanization Engineers, 39, Kari Niyazov Street, Tashkent, 100000, Republic of Uzbekistan.
Tursunova Zulayho Omonullaevna ([email protected]), Department of General Mathematics, Tashkent State Pedagogical University named after Nizami, 27, Bunyodkor Street, Tashkent, 100070, Republik of Uzbekistan.
In fact, first we show that the set 0 < jo,Ui, ...,US,e > contains a neighborhood of the measures jo in the weak topology. For each i = 1, ...,S we take the function fi : X ^ I, satisfying the conditions: fi([Ui]) = = 1,fj,(\} [Uj ]) = 0. Furthermore, we take the function fs+\_ : X ^ I so that fs+i (X\U |J ...UUS ) = 1,
j=i
and fs+1 ({x1,..., xs}) = 0. Now let us check the inclusion
O (j,fi,..,fs,fs+i,e/2) C O < (po,Ui,...Ua,e). (2.1)
We present a measure j G O (j0, f i,..., fs,fs+i, e/2) in the form j = ji + ...+js +js+i, where suppji C Ui for i = 1, ...,S,suppji C X\ (Ui U ... U Us). Then § > \j (fs+i) — j (fs+i) = \j (fs+i)|. But js+i < j, so Js+i (fs+i) < § at the same time, by definition of the function fs+\_ we have js+i (fs+i) = Js+i (1®) = = U^l So, Us+^l < § < e. To prove the inclusion (1) it remains to show that |||j|| — to°| < e. We have § > \jo (fi) — J (fi)\ > \jo (fi)\ — \j (fi)\ = to0 — \(ji + ... + j + Js+i) (fi)\ = fi /by definition of the function / = to° — J + js+i) (fi) = to0 — ji (f) — js+i (fi) = to0 — HjiH — js+i (fi). Consequently,
to
— Hull < § + js+i (fi) < § + Js+i (1®) = f + lljs+ill < f + § = e.
On the other hand, f > Hi (tyi) + Hs+i (tyi) —m0 = ||Hi|| — mi + Hs+i (tyi) thus ||hI| — m0 < f. The Inequality — m0\ < £ and the inclusion (1) are proved.
We now show that in every neighborhood of the base O (ho,tyi,tyf, ...,tyk,£) there is a neighborhood of the form O < h0,U1, ...,US,3 >. It is enough to consider the neighborhood of the form O (^0,ty,£),since the family of neighborhoods of the measure^ in the form O < h0,U1,..,Us,3 > is directed down by inclusion / intersection of a finite number of neighborhoods of this type contains a neighborhood of the same form /. This follows from the validity of the inclusion
1 2
The main part of checking is the following:
O < jo, Ui n U§ n ... n Ui n U§,- min {¿i, ¿§} >C O < Uo, U\,..., Ui,Si > nO < jo, U§,.., U§,02 > (2.2)
j(Uj) = j(Ui n u§) + j(Uj\Ui n u§) < j(Ui n u§) + j(X\ u (Ui n u§)) <
Ti n U§) + ,,(Uj \Ui n U§ ) ^ ,,(Ui n U§) + ,,(Y^ I (Ui I
e= i
< J (U/ D U§) + § min {¿i, ¿§} < jU D U§) + § ¿j. Therefore, for the measure j from the left side of proved inclusion (3.1) we have
jo(Uj) — j(Uj) < jo(Uj) — j(Ui n U§) = too — j(Ui n U§) < § min ^i, ¿2} < ¿j on the other hand
j(Uj ) — jo(Uj ) < j(Ui n U§) + § ¿j — too < § min {¿ij§} + § ¿j < ¿j. It remains to find a neighborhood of the form O < jo,Ui, ...,US^ > in the neighborhood O(jo,f,e). Since O(jo,Xf,Xe) = O(jo,f,e), for A > 0, we can assume that ||f|| < 1. Moreover, one can also assume that f > 0. For ¿ > 0 we take disjoint neighborhoods Ui of the points xi so that ocsillations of the function f on Ui was less than ¿.
Then \jo(f) — j(f)\ ^ \TOof(xi) — J fdj\ + ... + \TOoof(xs) — f fdj\ + \ J fdj\. Further
«1 us x\u1u...uus
\TOof(xi) — J fdj\ = \TOof(xi) — J f(xi)dj +f f(xi)dj —f fdj\ < TOof(xi) — f f(xH)dj +
Ui Ui Ui Ui Ui
+ H [f(xi) — f]dj\ < f(xi)\mo — Hjill\ + f \f(xi) — f\dj < f(xi)¿ + ¿H jiH < 2¿. Therefore, for ¿ < (§Sf+ i) the
Ui Ui
inclusion O < jo,Ui, ...,US^ >C O(jo,f,e) holds.
3. Basic notions and conventions
It is known that for an infinite compact space X, the space P(X) is homeomorphic to the Hilbert cube Q
[5], where Q = n —1,1] , —1,1] is the segment in the real line R. For a natural number n e N by Pn (X)
i= 1 i
we denote the set of all probability measures with support consisting of at most n points, i.e. Pn (X) = = {h e P (X) : \suppH\ < n}.The compact Pn (X) is convex combinations of
n
Dirac measures of the form: h = m 15xi + m25X2 +... + mn3Xn= 1,mi ^ 0,xi e X, 3Xi — is the Dirac
i=
oo
measure at the point xi. By 3 (X) we denote the set of all Dirac measures and Pu (X) = |J Pn (X). Recall
n=
that the space Pf (X) c P (X) consists of all probability measures in the form h = m 13Xl + mf3X2 + ... + + mk3Xk of finite supports, for each of which mi > k+i for some i [2,7]. For a natural n put Pf,n = Pf n Pn for the compact x. For compact X Pf,n (X) = {h ^ Pf (X) : \supp h\ ^ n} and hold. For the compact X by Pc(X) we denote the set of all measures h e P(X), support of each of which is contained to one of the components of the compact X [7].
We say that a functor F1 is a subfunctor (respectively ontofunctor) of a functor F2, if there is a natural transformation h : F1 ^ F1 such that for every object X the mapping h (X) : F1(X) ^ F2(X) is a monomorphism (epimorphism). By exp we denote the well known hyperspace functor of closed subsets. For example, the identity functor Id is a subfunctor of the functor expn, where expnX = {F G expX : |F| ^ n} and n— of n-degree is a ontofunctor of expn and SPg. A normal subfunctor F of the functor Pn is uniquely determined by its value F(n) on n where {n} denotes n-point set {0,1, ...,n — 1}. Note that Pn(n) is the (n —
— 1)-dimensional simplex an-1. Any subset of (n — 1) - dimensional simplex an-1 defines a normal subfunctor of the functor Pn, if it is invariant with respect to simplicial mappings to itself.
Definition [7]. A normal subfunctor Fof the functor Pn is locally convex if the set F(n) is locally convex. An example which is not a normal subfunctor of the functor Pn is the functor Pn of probability measures, whose supports contains in one of components of a space. One of the examples of locally convex subfunctors of the functor Pn is a functor SPn = SP^, where Sn is a group of homeomorphisms (permutation group) of n-point set.
w
Definition [1,8]. We say that a space X is countable dimension (shortly X G c■ d), if X = |J Xn, where
n=1
dimXn < ж for each n. In particular, X is a countable union of zero-dimensional spaces, i.e. dim Xi = 0 for every Xi.
Theorem 1. If X G c ■ d, then Pfn(X) G c ■ d for each n G N.
Proof. Let X G c ■ d. Then X is a countable union of finite-dimensional spaces dim Xi < ж in the sense
w w
of dim. In this case, Pf,n(X) is a countable union of Pf,n(Xi), i.e. Pf,n(X) = Pf,n(U Xi) = |J Pf,n(Xi).
i=1 i=1
By [9] for each i G N the compact Pf,n (Xi) is finite-dimensional in the sense of dim, i.e. dimPf,n (Xi) < ж, more accurately, dim Pfn (Xi) ^ n dimXi + dimPf n (n) = n dimXi + n — 1. In this case dim Pfn (n) = n —
— 1, since Pf,n (n) is a part of the (n — 1)-dimensional simplex Sn-1 spanned by the points {1, 2,...,n — 1}, i.e. for each i G N the space Pf,n (Xi) is finite-dimensional. Hence, Pf,n (X) is a countable union of finite-dimensional spaces. So Pf,n (X) G c■ d. If X is a countable union of zero-dimensional spaces dimXi = 0, then dimPf,n (Xi) = n — 1 for each i G N. In this case, Pf,n (X) is also a countable union of finite-dimensional
spaces, i.e. Pf,n (X) G c ■ d.Theorem is proved.
w
From the equation Pf (X) = U Pf,n (X), in the particular case we have.
n=1
Corollary 1. If the compact X is a c ■ d space, then Pf (X) G c ■ d.
Let X be a finite-dimensional compact. Then the space Pf,n (X) is also finite-dimensional. More accurately, dim Pf,n (X) ^ n dim X + n — 1 = n (dim X + 1) — 1. On the other hand, there is an open and closed mapping decreasing dimension of spaces. Fibers of the mappings rjn are similar cell, i.e. fibers are contractible to a point.
Theorem 2. Suppose ф : X ^ Y is a continuous surjective open mapping between the infinite compacts X and Y. Then the mapping Pf,n (ф) : Pf,n (X) ^ Pf,n (Y) is also open.
Proof. Let X and Y be infinite compacts and let the mapping ф : X ^ Y be surjective and open. Then by the normality of the functor Pf,n (ф) the mapping Pf,n ^)is surjective. In this case, we have the following commutative diagram
Pf,n (X) Pf,n (Y)
± fn ± fn (3Л)
S (X) [6 (ф)] S (Y)
where S (X) and S (Y) are Dirac measures on compacts X and Y. Let у (x) = m1Sxo + m2SX2 + ... + + mkSxk, r'X n (уо (X)) = 6x0, Pf,n (ф)(уо (x)) = m1SyO + m2Sy2 + ... + mkSyk.
From the fact that the mapping f n, S (ф) is open and the diagram (3) is commutative, it follows that the mapping Pf,n (ф) is open. Commutativity of diagram (3) follows from Lemma 2 of Uspensky's work [3]. Theorem 2 is proved.
Similarly as theorem 2, one can proof the following.
Theorem 3. For infinite compacts X and Y a surjective map is open if and only if the map Pf (ф) : Pf (X) ^ Pf (Y) is open.
Corollary 2. If X G c ■ d, then Pn (X) G c ■ d, Pu (X) G c ■ d and Pu (X) G A (N) R. Let X be a topological space and let A с X. A set A is called homotopy dense in X, if there is a homotopy h : X x [0,1] ^ X such that h (x, 0) = idx and h : (X x (0,1]) с A. A set A is called homotopy void if complement of A is homotopy dense in X. The set A с X is called the Z—set in X [4], if A is closed and for each cover U G cov (X) there is a map f : X ^ X such that (f, idx) ~< U and f (X) П A = 0. Theorem 4. For any infinite compact X and for each n G N the compact Pn (X)is the Z—set in Pu (X). Proof. By infinity of metric compact X the space Pu (X) is convex and a locally convex metric space. So, Pu (X) G A (N) R. On the other hand, the space is compact. It is obvious that Pn (X) is a subspace
of Pu (X), since the compact Pf,n (X) is a subset of the compact Pn (X). We fix a measure y0 = kSX1 +
+ k ¿X2 + ... + ~k ¿Xk .
Let [0,1] is the unit interval. We construct a homotopy h (y,t) : Pu (X) x [0,1] ^ Pu (X) getting h(y,t) = = (1 — t)y + tyo.
Obviously, h(y, 0) = y i.e. h(y, 0) = idP^X) and h(Pu(X) x (0,1]) c Pu(X)\Pn(X). This means that n € N for any subspace Pu(X)\Pn(X) homotopically dense in Pu(X). Then the set Pn(X) is homotopically small in Pu(X). Hence, by one of the results in [4], the subspace Pu(X)\Pn(X) € ANR and Pu (X) \Pn (X) are ANR-spaces. In this case, from theorem 1.4.4. [4] it follows that Pu (X) is the Z—set in Pu (X). Theorem 4 is proved.
Lemma 1. For any infinite compact X each compact subset A of Pu (X) is a Z—set, i.e. Pu (X) has the compact Z—property.
Proof. Let X be an infinite compact, A is compact subset, i.e. A c Pu (X). Consider the set AnPn (X) = = An. It's obvious that P1 (X) c P2 (X) c ... C Pn (X) c ... By theorem 4, the set is a Z—set in Pu (X)
for each n € N. Then A = |J An is a — Z—set and is closed in Pu (X). Then by one of the results in [4]
n=1
A is a Z—set in Pu (X). Lemma 1 is proved.
From Theorem 4 and Lemma 1, in particular, the cases arise. Corollary 2. For any infinite compact X the followings hold:
a) The compact Pf,n (X) is a Z—set in Pu (X) for all n € N.
b) The compact Pf (X) is also Z—set in Pu (X). Corollary 3. For an arbitrary infinite compact X we have:
a) For each n € N the subspace Pu (X) \Pf,n (X) is an ANR space y homotopically dense in Pu (X).
b) The subspace Pu (X) \Pf,n (X) is ANR and homotopically dense in Pu (X).
We say that X has strongly discrete approximation property (shortly, SDAP) if for every map f : Q x x N ^ X and for every cover U € cov (X) there exists a mapping f : Q x N ^ X such that f, f) ~< U and the family f (Q x {n})} is discrete in X.
Let {x1,x2, ...,xn+1} be an (n+1)-point subset of the compact X. Fix the measure y0 = n+1 ¿X1 + n+1 ¿X2 + + ... + n+i¿Xn+1. It is clear that y0€Pn (X) and y0 € Pu (X). We construct a homotopy h (y,t) : Pu (X) x x [0,1] ^ Pu (X) getting h (y,t) = (1 — t) y+ty0. It is known that h (y, 0) = idP^(X) and h (y, (0,1])nPn (X) = = 0. By the structure of the space Pu (X) an by the definition of the homotopy this satisfies the condition of problem 10,1.4 of work [4], i.e. the set Pn (X) is a strongly Z-set in.
Therefore, Pu (X) is a strongly set and Pu (X) € ANR, i.e. the following is true.
Theorem 5. For any infinite compact X the space Pu (X) has strongly discrete approximation property, i.e. Pu (X) € SDAP.
References
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Т.Ф. Жураев, A.X. Рахматуллаев, З.О. Турсунова2
СВОЙСТВА ПОДФУНКТОРОВ ФУНКТОРА ВЕРОЯТНОСТНЫХ МЕР
В КАТЕГОРИЯХ COMP
Данная заметка посвящена сохранению подфункторами функтора P вероятностных мер пространств счетной размерности и экстензорным свойствам подпространств пространства вероятностных мер.
Ключевые слова: вероятностные меры, размерность, Z-множество, гомотопически плотно, сильное дискретное аппроксимационное свойство.
Цитирование. Zhuraev T.F., Rakhmatullaev A.Kh., Tursunova Z.O. Some values subfunctors of functor probalities measures in the categories Comp // Вестник Самарского университета. Естественнонаучная серия. 2018. Т. 24. № 2. С. 28-32. DOI: http://doi.org/10.18287/2541-7525-2018-24-2-28-32.
MSC: 54B15, 54B30, 54B35, 54C05, 54C15, 54C60, 54D30.
Статья поступила в редакцию 23/V/2018. The article received 23/V/2018.
This work is licensed under a Creative Commons Attribution 4.0 International License.
2Жураев Турсунбой Файзиевич ([email protected]), кафедра общей математики, Ташкентский государственный педагогический университет имени Низами, 100070, Республика Узбекистан, г. Ташкент, ул. Бунедкор, 27.
Рахматуллаев Алимбай Хасанович ([email protected] ), кафедра высшей математики, Ташкентский институт инженеров ирригации и механизации сельского хозяйства, 100000, Республика Узбекистан, г. Ташкент, ул. Кары-Ниязи, 39.
Турсунова Зулайхо Омонуллаевна ([email protected]), кафедра общей математики, Ташкентский государственный педагогический университет имени Низами, 100070, Республика Узбекистан, г. Ташкент, ул. Бунедкор, 27.
