CC BY
62
Онлайн-доступ к журналу: http: / / mathizv.isu.ru
Серия «Математика»
2018. Т. 25. С. 33-45
УДК 517.987 MSG 28С15
DOI https://doi.org/10.26516/1997-7670.2018.25.33
Ways of obtaining topological measures on locally compact spaces
S. V. Butler
University of California Santa Barbara, Santa Barbara, USA
Abstract. Topological measures and quasi-linear functionals generalize measures and linear functionals. Deficient topological measures, in turn, generalize topological measures. In this paper we continue the study of topological measures on locally compact spaces. For a compact space the existing ways of obtaining topological measures are (a) a method using super-measures, (b) composition of a q-function with a topological measure, and (c) a method using deficient topological measures and single points. These techniques are applicable when a compact space is connected, locally connected, and has a certain topological characteristic, called "genus", equal to 0 (intuitively, such spaces have no holes). We generalize known techniques to the situation where the space is locally compact, connected, and locally connected, and whose Alexandroff one-point compactification has genus 0. We define super-measures and q-functions on locally compact spaces. We then obtain methods for generating new topological measures by using super-measures and also by composing q-functions with deficient topological measures. We also generalize an existing method and provide a new method that utilizes a point and a deficient topological measure on a locally compact space. The methods presented allow one to obtain a large variety of finite and infinite topological measures on spaces such as R" , half-spaces in R" , open balls in R" , and punctured closed balls in R" with the relative topology (where n > 2).
Keywords: topological measure, deficient topological measure, solid-set function, super-measure, q-function.
This paper belongs to a series of papers devoted to study of topological measures, deficient topological measures, and their corresponding nonlinear functionals on locally compact spaces. The main focus of this paper is techniques for generating new topological measures on a locally compact,
1. Introduction
locally connected and connected space whose one-point compactification has genus 0. Such spaces include Rra, half-planes in Rra, open balls in Rra, and punctured closed balls in Rra with the relative topology (n > 2).
The study of topological measures (initially called quasi-measures) and corresponding quasi-linear functionals began with papers by J. F. Aarnes [1-3]. Deficient topological measures were first defined and used by A. Rustad and 0. Johansen in [10], and later independently rediscovered by M. Svistula, see [14] and [15]. Application of topological measures and quasi-linear functionals to symplectic topology has been studied in numerous papers (beginning with [12]) and a monograph [13]. All this work is done for compact spaces.
Topological measures, deficient topological measures and some ways to obtain them when X is locally compact are studied by the author in [7] and [8]. In this paper we develop analogs on locally compact spaces of techniques that exist for compact spaces with genus 0. These are methods for generating new topological measures from super-measures, via q-functions, and by utilizing a deficient topological measure and a point. When X is compact, the method of super-measures was first developed in [4]; the method of q-functions first appeared in [5], and was discussed in [10] and [6]. One method that utilizes a deficient topological measure and a point first appeared in [10].
In this paper X is a locally compact, connected, and locally connected space. By a component of a set we always mean a connected component. We denote by E the closure of a set E. A set А С X is called bounded if A is compact. A set А С X is called solid if A is connected and X \ A has only unbounded components. We denote by |J a union of disjoint sets.
Several collections of sets are used often. These include: OiX\ the collection of open subsets of X; C€(X\ the collection of closed subsets of X; JT(X), the collection of compact subsets of X; and st{X) = ^(X)Uff(X). By ,Xq(X) we denote the collection of finite unions of disjoint compact connected sets. ¿P{X) is the power set of X. We use subscripts s or с to indicate (open, compact) sets that are, respectively, solid or connected. For example, ЖС{Х) is the collection of compact connected subsets of X. Given any collection <g С ¿^(X), we denote by <S* the subcollection of all bounded sets belonging to S. For example, ¿//(X) = иХД) is
the collection of bounded open solid and compact solid sets.
Definition 1. Let X be a topological space and ц be a set function on £, a family of subsets of X. We say that /л is finite г/ sup{|/x(A)| : A ^ £} < M < oo; ¡л is compact-finite if\/j,(K)\ < oo for any К € Ж(Х); ¡i is simple if it assumes only values 0 and 1.
We consider set functions that are not identically oo.
2. Preliminaries
We will need the following two results (see, for example, section 2 in [7]).
Lemma 1. Let K c U, K € U € 0{X) in a locally compact,
locally connected space X. If either K or U is connected there exist a bounded open connected set V and a compact connected set C such that K<ZV<ZC<ZU. One may take C = V.
Lemma 2. LetX be a locally compact and locally connected space. Suppose K <ZU, Kg Jf(X), U € 0{X). Then there exists C € J^o(X) such that K c c CU.
The next two lemmas can be found in section 3 of [7].
Lemma 3. If K c U, K € Jf(X), U € <ff*s(X) then there exists C € Jf8(X) such that K <ZC ZU.
Lemma 4. Let K C V, K € Xg{X), V € 0(X). Then there exists W € ff*s(X) such that K<ZW<ZW<ZV.
Definition 2. A topological measure on X is a set function ¡j, : 'if(X) u G(X) —)• [0, oo] satisfying the following conditions:
(tm1) if A, B, A u B <e jt(x) u 0{X) then fx(A u B) = a) + ¡¿(B)] (TM2) fx(U) = sup{fx(K) : K € K C U} for U € 0{X);
(TM3) fx(F) = inf{/x(£7) : U € &(X), F QU} for F € ^(X).
Definition 3. A deficient topological measure on a locally compact space X is a set function v on 'if(X) U G(X) —> [0, oo] which is finitely additive on compact sets, inner compact regular, and outer regular, i.e. :
(DTM1) ifCHK = 0, C, K <e jt(X) then v(C U K) = v{C) + v(K); (DTM2) u{U) = sup{v(C) : C C U, C € Jf(X)} for U € 0{X); (DTM3) v{F) = mi{u{U) : F <Z U, U € &(X)} for F €
For a closed set F, v(F) = oo iff v(lJ) = oo for every open set IJ containing F.
Remark 1. For more information about topological measures and deficient topological measures on locally compact spaces, their properties, and examples see [7] and [8]. We point out that a deficient topological measure v is monotone, countably additive on open sets, z/(0) = 0, and v is superadditive, i.e. if \_\teTAt <Z A, where At,A € G(X) and at
most one of the closed sets is not compact, then v(A) > ^t&Tv(At).
Remark 2. Let v be a deficient topological measure on X. If X is locally compact and locally connected then by Lemma 2 for each open set U
v(U) = sup {v(K) : K ZU, K € Jto(X)}.
If X is locally compact, connected, and locally connected, then from Lemma 1
v(X) = sup{v{K) : К € ЖС{Х)},
and considering for a compact connected set С С X its solid hull С € Jfs(X),C С С (see section 3 in [7] for detail), we also obtain
v(X) = sup{v(K) : К € Xs(X)}.
We denote by TM(X) and DTM(X), respectively, the collections of all topological measures on X, and all deficient topological measures on X. By M(X) we denote the collection of all Borel measures on X that are inner regular on open sets and outer regular (restricted to 0(X) U^(X)).
Remark 3. Let X be locally compact. We have:
M{X) с TM{X) с DTM(X). (2.1)
For proper inclusions in (2.1) and criteria for a deficient topological measure to be a topological measure or a measure in M(X) see sections 4 and 6 in [8], and section 9 in [7].
Definition 4. A function A : s^*(X) —>• [0, oo) is a solid set function on X if
n n
(si) whenever |J С С, С,С\ € Xs{X), we have £ Л (С,) < Л(С);
г=1 г=1
(s2) Л(С7) = sup{A(К) : К С [J, К £ Ж8(Х)} for U € (s3) \{К) = inf{A(f/) : К Q U, U € 0*(Х)} for К € Xs(X);
п
(s4) if А = ULi Ai, А, е then Х(А) = £ Л(Аг).
г=1
Theorem 1, Theorem 2, and Lemma 5 below are proved in [7], section 8.
Theorem 1. Let X be locally compact, connected, locally connected. A solid set function on X extends uniquely to a compact-finite topological measure on X. If a solid set function Л is extended to a topological measure ¡л then the following holds: if X is simple, then so is ¡л; if sup{X(K) : К € Jfs(X)} = M < oo then ¡л is finite and ц{Х) = M.
Theorem 2. The restriction X of a compact-finite topological measure /л to s^*(X) is a solid set function, and /л is uniquely determined by X.
Remark 4. We will summarize the extension procedure for obtaining a topological measure ¡л from the a solid set function Л. First, for a compact connected set С we have: /л(С) = \{C)—where С = СU|_|ie/ Bi
is a solid hull of C, and {Bi : i € 1} is the family of bounded components of X \ C. The set C is compact solid, and all Bi are bounded open solid sets.
For C € Jto(X), that is, for a compact set C which is the union of finitely many disjoint compact connected sets C1,..., Cn, we have: /x(C) =
T^iKCi).
For an open set IJ we have: ¡jl(IJ) = sup{¡~i(K) : K C [7, K € J^o(X)}, and for a closed set F let ¡i{F) = inf{/x([7) : F c U, U € &(X)}.
Remark 5. When X is compact, a set is called solid if it and its complement are both connected. For a compact space X we define a certain topological characteristic, genus. See [3] for more information about genus g of the space. We are particularly interested in spaces with genus 0. A compact space has genus 0 iff any finite union of disjoint closed solid sets has a connected complement. Another way to describe the "g = 0" condition is the following: if the union of two open solid sets in X is the whole space, their intersection must be connected. (See [9].) Intuitively, X does not have holes or loops. In the case where X is locally path connected, g = 0 if the fundamental group ni(X) is finite (in particular, if X is simply connected). Knudsen [11] was able to show that if Hl(X) = 0 then g{X) = 0, and in the case of CW-complexes the converse also holds.
Lemma 5. Let X be a locally compact space whose one-point compactifi-cation X has genus 0. If A £ stf* (X) then any solid partition of A is the set A itself.
Remark 6. From Lemma 5 it follows that for any locally compact space whose one-point compactification has genus 0 the last condition of Definition 4 holds trivially. This is true, for example, for Rra, (Rra)+, an open ball in Rra, or for a punctured closed ball in Rra with the relative topology (n > 2).
Example 1. Let X = [r2, I be a straight line and p a point of X not on the line I. For A € 8f*{X) define ¡¿(A) = 1 if A n I / 0 and p € A; otherwise, let ¡jl(A) = 0. It is easy to verify the first three conditions of Definition 4. From Remark 6 it follows that /x is a solid set function on X. By Theorem 1 ¡i extends uniquely to a topological measure on X, which we also call /J,. Note that /x is simple. We claim that /x is not a measure. Let F be the closed half-plane determined by I which does not contain p. Then using Remark 4 we have ¡jl(F) = /x(X \ F) = 0, and /x(X) = 1. Failure of subadditivity shows that /x is not a measure.
Example 2. Let X be a locally compact space whose one-point compactification has genus 0. Let n be a natural number. Let P be the set of distinct 2n + 1 points. For each A € =s//(X) let v(A) = i/n if jjA = 2i or 2i + l, where j\A is the number of points in An P. We claim that v is a solid
set function. By Remark 6 we only need to check the first three conditions of Definition 4. The first one is easy to see. Using Lemma 3 and Lemma 4 it is easy to verify the next two conditions. The solid set function v extends to a unique topological measure on X. This topological measure assumes values 0,1/n,..., 1.
3. Super-measures on a locally compact space
If X is compact, one way to obtain a large collection of topological measures on X is to use super-measures (see [4], for example). In this section we shall generalize this technique to locally compact spaces. First, we adapt the definition of a super-measure.
Definition 5. A super-measure on a countable set E is a function v : 2?{E) [0, oo] such that v{A U B) > v{A) + v{B) and v{A) < oo for any finite subsets A and B of E.
Note that a super-measure is a monotone set function.
Theorem 3. LetX be a locally compact, connected, locally connected space whose one-point compactification has genus 0. Let E be a countable subset of X such that each bounded subset of X contains finitely many points from E, and let v be a super-measure on E. Define function ¡j, on bounded solid subsets of X by
¡i(A) = v(A n E).
Then [i is a solid set function on X which extends uniquely to a compact-finite topological measure on X.
Proof. By Remark 6 we only need to check the first three conditions of Definition 4. Condition (si) in Definition 4 is satisfied because v is a super-measure. Lemma 3 and Lemma 4 help to verify conditions (s2) and (s3). It is easy to see that /x is compact-finite. □
Example 3. Let X be Rra or [R™ , and E be the set of points with integer
Here is the cardinality of the
coordinates. Let v(A) =
— \AC\E\
set An E, and [x] denotes the whole part of a real number. Then v is a super-measure on E, and by Theorem 3 we obtain the topological measure ¡i on X. Note that /л is not a measure as it is not subadditive: it is easy to see that a compact solid set with positive z/-value can be covered by finitely many solid sets each of which has zero z/-value.
4. Q-functions on a locally compact space
In this section we shall generalize the techniques of q-functions for obtaining topological measures on X to the situation where X is locally compact.
We begin by adapting the definition of a q-function. Definition 6. A function f : [0, oo) —> [0, oo) is called a q-function if
(i) / is right-continuous
(ii) f(x) + f(y)<f(x + y)
Remark 7. From Definition 6 it follows that
a) /(0) = 0
b) / is non-decreasing
c)
Definition 7. The split spectrum of a deficient topological measure v is the set sp(y) = {a € (0, oo) : there exist CgJs(I), U £ ff*s(X), C <ZU with v(C) = u{U) = a}.
Remark 8. The definition of a split spectrum of a topological measure on a compact space was first given in [6]. It is easy to see that when X is compact and v is a topological measure Definition 7 is equivalent to Definition 3.4 in [6].
Theorem 4. LetX be a locally compact, connected, locally connected space whose one-point compactification has genus 0, v be a compact-finite deficient topological measure on X and f be a q-function. Define function ¡j, on bounded solid subsets of X by letting ¡jl(C) = f(v(C)) for C € Jfs(X) and fx(U) = sup{/MC)) : CCU, C£ JT,(X)} = f(u(U)~) for U € Then
(I) ¡j, = f o v defined as above is a solid set function on X and, hence, extends uniquely to a topological measure on X, which we also call ¡j,.
(II) / is continuous on the split spectrum of v.
Proof. First note that the second equality in the definition of ¡jl(IJ) holds because of regularity of v.
(I) By Remark 6 we only need to check the first three conditions of Definition 4. Suppose that C\,... Cn, C € Xs(X) and C\ U ... U Cn C C. v is a deficient topological measure, so by superadditivity of v (see Remark 1) and the monotonicity of a q-function
n n n
= 52mci)) < f(E"(Ci)) < /mc)) = KC).
i=i i=i i=i
By the the definition of /j.(U), we are only left to show that for every C €
li(C) = inf{/x(£7) : C CU, U € &t(X).
Let C € Jfs(X), so v(C) < oo. Since / is right-continuous and v is outer regular, given e > 0 there exist 5 > 0 and (by Lemma 4) IJ € ff*s{X), ecu such that v(U)-v(C) < 5 and f(y(U))-f(y(C)) < e. Since /(i/(C)) < f{v{U)-) < f{v{U)), we have:
(i(U) - (i(C) = f{y{U)~) - f{y{C)) < f{y{U)) - f{y{C)) < e,
which shows property (s3) of Definition 4 for /x. Thus, ¡j, is a solid set function.
(II) Let a € sp(v) and C € Xs(X), U € 0*(X) be such that C C U, v(C) = v(U) = a. Since ¡j, is a topological measure, we have:
/(a") = f(u(U)~) = KU) = sup MK) : K € Xa(X\ K QU} > niC) = f{y{C)) = f(a).
Thus, f(a~) = f(a) and / is continuous at a.
□
Remark 9. (a) By Theorem 1 we may take v to be a solid set function, (b) In the proof of the first part above we only need / to be right-continuous at v(U), U € 0*8(X).
Example 4. Here are some examples of ^-functions and compositions of ^-functions with topological measures. X is a locally compact, connected, locally connected space whose one-point compactification has genus 0.
(i) The easiest one is the function f(x) = x. Then / o v = v for any topological measure u, where topological measure fov is as in Theorem 4.
(ii) Let e > 0. Define / : [0, oo) ->■ [0, oo) by f(x) = 0 for x € [0, e) and f(x) = x for x > e. Then / is a ^-function. Let m be the Lebesque measure on IR",n > 2 and /j, = / o m. Then ¡i(A) = 0 for any compact solid set A with m(A) < e, and for any open solid bounded set with m(A) < e. Otherwise ¡jl(A) = m(A). A closed ball is a compact solid set. A closed ball B of radius greater than e has ¡jl(B) = m(B) > 0 and can be covered by finitely many closed balls Bi of radius less than e with ¡j,(Bi) = 0. Thus, ¡j, is not subadditive and, hence, can not be a measure.
(iii) Consider / : [0, oo) —> [0, oo) defined by f(x) = i for x € [i, i + 1), where tGN. Then / is a g-function. Let m be the Lebesque measure on X and ¡j, = f o m. Then ¡i is a topological measure that assumes nonnegative integer values. Note that ¡j, is not finite. As in part (ii) it is easy to show that ¡j, is not subadditive and, hence, is not a measure.
Remark 10. The topological measure v in Example 2 can be also obtained by a q-function. Let g = —/, where / is the q-function from part
1 n
(iii)in Example 4. Let m = --+ - • • ¿2n+i), where Si are point masses
2/71 1
at 2n + 1 points which comprise the set P in Example 2. Then v = g o m.
5. DTM and point methods
In this section we will study topological measures obtained by utilizing a deficient topological measure and a point. We call such methods DTM and point methods, and they are presented in Theorem 5 and Theorem 6.
Theorem 5. Let X be a locally compact, connected, locally connected space whose one-point compactification has genus 0. Let v be a deficient topological measure on X such that v(X) < oo and letp € X be an arbitrary point. Define a set function up : s^*(X) —>• [0, oo) by
u(A] _ / ifP^A
VpKA) ~ \ u(X) -u(X\A), ifp € A
Then Up is a solid set function and, hence, extends to a topological measure on X.
Proof. To show that vp is a solid set function by Remark 6 we only need to check the first three conditions of Definition 4. Suppose C1UC2U.. .UCn C C. If p £ C, the first condition is just the superadditivity of v (see Remark 1). Now assume that p is in one of the sets C1,..., Cn, say, p € C\. Since (X \ C) |J C2 U ... U Cn C X \ C\, by superadditivity of v we have: v(X \ C) + u{C2) + ... + v{Cn) < v(X \ Ci). Then
vv{Ci)+up(C2) + ... + up(Cn) = p{X) - p{X \ Ci) + is(C2) + • • • + v(Cn)
< v{X) - v{X \C) = vp(C)
The case when p € C but p £ Ci for i = 1,... ,n can be proved similarly by noticing that (X \ C) |J C\ |J ... |J Cn C X and applying the superadditivuty of v.
Now we shall show inner and outer regularity conditions (s2) and (s3) of Definition 4 for up. Inner and outer regularity is easy to see when a
solid set does not contain p. So assume that p € C, where C € Jf^(X). For an open set W = X \ C and e > 0 find compact K C W for which v(W) - v(K) < e. With U = X\K, we see that C C U and by Lemma 4 there exists V € <^*(X) such that C C V C U. Then
vp(V) - vp(C) = v(X \C)~ v(X \V) = v(W) - v(X \ V)
< v(W) - v(X \U) = v(W) - v(K) < e,
which shows the outer regularity condition (s3) of Definition 4 for up.
Now we will assume p € U, where IJ € and we shall show the
inner regularity For a closed set F = X \ IJ and e > 0 find an open set W such that F C W and v(W) - v(F) < e. Since compact X \ W C U, by Lemma 3 there exists K e Jfs(X) such that X \ W C K c IJ and p € K. Then
vp(U) - Vp(K) = u{X \K) — v(X \U) = v(X \K) — v{F)
< v(W) - v(F) < e,
which shows inner regularity (s2) of Definition 4 for up. □
Example 5. Let v be the topological measure on X = [R2 from Example 1, and let vv be given by Theorem 5 using p from Example 1. Then for A e ,<*(X)
Let C € Jfc(X). From Remark 4,
pp{C) = pp{C)~YJMBi)- (5-2)
m
where Bi are bounded open solid sets and C is a compact solid set.
If p € C then peCbutp^Bi for all i € L Then by (5.1) vp(C) = 1. If p £ C then p may or may not belong to C. If p £ C, then p £ Bi for each i, and vp(C) = 0. If p £ C, but p € C, then p is in some component Bj, and vp(C) = 0. We see that vp(A) = 0 if p £ A and vp(A) = 1 if p € A for A being compact connected, then a finite disjoint union of compact connected sets, then open, and then closed, by Remark 4. Thus, vp on stf(X) is the point mass 5P.
Example 6. Let X = [R2, and let v be a topological measure on X as in Example 2 for P = {pi,P2,P3,P4,P5}, where pi = (4i — 1, 0). For i = 1,..., 5 let Ui be an open disk of radius 3, and let W = IJ\ U U2 U ... U £/5, U = IJ\ U U2 U [/3. Then Ui,... ,U5,U, W are all open bounded solid sets, and W = U U U4 U [/5. Taking p to be any point not in W, consider the
topological measure vp given by Theorem 5. Both v and vv assume values 0,1/2,1. We see that z/p(f/4) = vp(U5) = 0,vp(U) = 1/2, and vp(W) = 1. Thus, Up is not subadditive, hence, it is a topological measure which is not a measure. This is in contrast to Example 5.
Theorem 6. LetX be a locally compact, locally connected, connected space whose one-point compactification has genus 0. Let X be a compact-finite deficient topological measure on X, and let p € X be an arbitrary point. Define a set function Xp : s^*(X) —>• [0, oo) by
Then Xp is a solid set function and, hence, extends to a topological measure on X. If X is compact-finite but not finite, then so is Xp.
Proof. By Remark 6 we only need to check the first three conditions of Definition 4. The first one is easy to see. We shall show the inner and outer regularity conditions of Definition 4 for Xp. Let IJ € 0*(X). The inner regularity is trivial when p U. Now let p € U. Since A{IJ) < oo, for e > 0 choose C such that p € C C U, X(U) — X(C) < e. By Lemma 3 we may assume that C € JJS(X). Then
Example 7. Let X = Rra, n > 2. The Lebesque measure A is a compact-finite deficient topological measure on X, so let Xp be a topological measure on X according to Theorem 6. We claim that Xp is not a measure. Since Ap(X) = sup{AP(K) : K e Jfs(X)}, taking balls of arbitrarily large radius we see that XP(X) = oo. Now let X be covered by countably many open
balls of the same positive radius: X = IJ Bi. Only finitely many of Bi
contain p, and thus have a positive Xp measure. Thus, £ Ap{Bi) < oo, so
i=i
Xp is not subadditive and, hence, can not be a measure.
Acknowledgments: This work was conducted at the Department of Mathematics at the University of California Santa Barbara. The author would like to thank the department for its hospitality and supportive environment.
AP(U) - Xp(C') = X(U) - X(C) < e. The proof of outer regularity uses Lemma 4 and is similar.
□
oo
i= 1
oo
References
1. Aarnes J.F. Quasi-states and quasi-measures. Adv. Math., 1991, vol. 86, no. 1, pp. 41-67. https://doi.org/10.1016/0001-8708 (91) 90035-6
2. Aarnes J.F. Pure quasi-states and extremal quasi-measures. Math. Ann., 1993, vol. 295, pp. 575-588. https://doi.org/10.1007/BF01444904
3. Aarnes J.F. Construction of non-subadditive measures and discretization of Borel measures. Fundamenta Mathematícae, 1995, vol. 147, pp. 213-237. https://doi.org/10.4064/fm-147-3-213-237
4. Aarnes J.F., Butler S.V. Super-measures and finitely defined topological measures. Acta Math. Hungar., 2003, vol. 99 (1-2), pp. 33-42. https://doi.Org/10.1023/A:1024549126938
5. Aarnes J.F., Rustad A.B. Probability and quasi-measures - a new interpretation. Math. Scand., 1999, vol. 85, no. 2, pp. 278-284. https://doi.org/10.7146/math.scand.a-18277
6. Butler S.V. Q-functions and extreme topological measures, J. Math. Anal. Appl, 2005, vol. 307, pp. 465-479. https://doi.Org/10.1016/j.jmaa.2005.01.013
7. Butler S.V. Solid-set functions and topological measures on locally compact spaces, submitted.
8. Butler S.V. Deficient topological measures on locally compact spaces, submitted.
9. Grubb D.J. Irreducible Partitions and the Construction of Quasi-measures. Trans. Amer. Math. Soc., 2001, vol. 353, no. 5, pp. 2059-2072. https://doi.org/10.1090/S0002-9947-01-02764-7
10. Johansen 0., Rustad A. Construction and Properties of quasi-linear func-tionals. Trans. Amer. Math. Soc., 2006, vol. 358, no. 6, pp. 2735-2758. https://doi.org/10.1090/S0002-9947-06-03843-8
11. Knudsen F.F. Topology and the construction of Extreme Quasi-measures. Adv. Math., 1996, vol. 120, no. 2, pp. 302-321. https://doi.org/10.1006/aima.1996.0041
12. Entov M., Polterovich L. Quasi-states and symplectic intersections. ArXiv, 2004.
13. Polterovich L., Rosen D. Function theory on symplectic manifolds. CRM Monograph series, vol. 34, American Mathematical Society, Providence, Rhode Island, 2014. https://doi.org/10.1090/crmm/034
14. SvistulaM.G. A Signed quasi-measure decomposition. Vestnik SamGU. Estestven-nonauchnaia seriia, 2008, vol. 62, no. 3, pp. 192-207. (In Russian).
15. Svistula M.G. Deficient topological measures and functionals generated by them. Sbornik: Ma,them,a,tics, 2013, vol. 204, no. 5, pp. 726-761. https://doi.org/10.1070/SM2013v204n05ABEH004318
Svetlana Butler, Ph. D., Department of Mathematics 552 University Road, Isla Vista CA 93117, University of California Santa Barbara, Santa Barbara, United States of America, tel.: (805)8932955 (e-mail: [email protected])
Received 10.08.18
Способы получения топологических мер на локально компактных пространствах
С. В. Батлер
Калифорнийский университет в Санта-Барбаре, Соединенные Штаты Америки
Аннотация. Топологические меры и квазилинейные функционалы являются обобщением мер и линейных функционалов. Дефектные топологические меры, в
свою очередь, являются обобщением топологических мер. В этой статье мы продолжаем исследование топологических мер на локально компактных пространствах. На компактном пространстве существующие способы получения топологических мер — это (а) метод, использующий супер-меры, (б) композиция q-функции с топологической мерой и (в) метод с использованием дефектных топологических мер и единичных точек. Эти способы применимы, когда компактное пространство является связным, локально связным, а также имеет определённую топологическую характеристику, которая называется «род», равную 0 (интуитивно, у таких пространств нет дыр). Мы обобщаем известные способы на случай, когда пространство локально компактное, связное, локально связное, и его компактификадия Александрова имеет род 0. Мы даём определение супер-мер и q-функций на локально компактном пространстве. Затем мы получаем методы построения новых топологических мер, используя супер-меры, а также композиции q-функций с дефектными топологическими мерами. Мы также обобщаем существующий метод и приводим новый метод с использованием точки и дефектной топологической меры на локально компактном пространстве. Представленные способы позволяют получить большое количество разнообразных конечных и бесконечных топологических мер на таких пространствах, как R™, полупространства в R™, открытые шары в R™, и проколотые замкнутые шары вЕ° с индуцированной топологией (где п > 2).
Ключевые слова: топологические меры, солид-функции, супермеры, q-функ-ции.
Светлана Батлер, Ph. D., математический факультет, Калифорнийский университет в Санта-Барбаре, Соединенные Штаты Америки, Санта-Барбара, 552 University Road, Isla Vista CA 93117, tel.: (805)8932955 (e-mail: [email protected])
Поступила в редакцию 10.08.18
