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Isometries of Spaces of Radon Measures

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Isometries of Spaces of Radon Measures Hindawi Journal of Function Spaces Volume 2017, Article ID 3850817, 4 pages https://doi.org/10.1155/2017/3850817 Research Article Isometries of Spaces of Radon Measures Marek Wójtowicz Instytut Matematyki, Uniwersytet Kazimierza Wielkiego, Pl. Weyssenhoffa 11, 85-072 Bydgoszcz, Poland Correspondence should be addressed to Marek Wojtowicz;´ [email protected] Received 2 May 2017; Accepted 5 June 2017; Published 4 July 2017 Academic Editor: Adrian Petrusel Copyright © 2017 Marek Wojtowicz.´ This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Let Ω and denote a compact metrizable space with card(Ω) ≥ 2 and the unit interval, respectively. We prove Milutin and Cantor- Bernstein type theorems for the spaces M(Ω) of Radon measures on compact Hausdorff spaces Ω. In particular, we obtain the 2ℵ0 following results: (1) for every infinite closed subset of N the spaces M(), M(N),andM(Ω ) are order-isometric; (2) 2m for every discrete space Γ with m fl card(Γ) > ℵ0 the spaces M(Γ) and M( ) are order-isometric, whereas there is no linear 2m homeomorphic injection from () into ( ). 1. Introduction appliedinproofsofourmainresults,giveninSections3and 4. We use the notation from the abstract and Δ denotes the Cantor set endowed with the standard product topology. 2. Preliminaries Throughout this paper, we also assume that every topological space is of cardinality at least 2. For notions and notations Throughout what follows Ω denotes a compact Hausdorff undefined here we refer the reader to the monographs [1–5]. space with card(Ω) ≥ 2, M(Ω) denotes the space of Radon Let us recall that a linear injective operator between two measures on Ω,and(Ω) stands for the space of continuous Banach lattices and is said to be an order isomorphism Ω M(Ω) N −1 functions on ,thatis,thepredualof .By we denote [order-isometry, resp.] if both and are order-preserving the discrete space of positive integers. [with an isometry, resp.]; by [1, Theorem 16.6], in the Let and be two (real or complex) Banach lattices. The + general case, is continuous. lattice is said to be an -latticeifitsnormisadditiveon ; In this paper, we deal with surjective isometries of spaces that is, ‖1 +2‖=‖1‖+‖2‖ for all 1,2 ≥0.Inparticular, of Radon measures defined on compact Hausdorff spaces. In the classical spaces 1() and M(Ω) are typical -spaces the monographs and survey papers devoted to isometries on [9, Chapter 6]. Moreover, every -space is order-isometric function spaces this topic is either completely overlooked [6– to 1() for some measure space (Θ,Σ,)[9, Theorem 3 on 8] or treated marginally [9, Theorem 7 on p. 177, Exercises p. 135]. If is an order-isometry between the real parts of 4–7onp.229];cf.[9,pp.181,226-227].Ouraimisto two given -spaces and , then the extended operator Ω Ω ̃ ̃ show that, in many cases, one can indicate pairs 1, 2 of :→of the form (1 +2)=1 +2 is an isometry, “highly nonhomeomorphic” compact Hausdorff spaces such too [9, p. 139]. This allows us to restrict our considerations to that the spaces M(Ω1) and M(Ω2) are order-isometric. It real M(Ω)-spacesandapplythetheoryofreallinearlattices is a little surprising that here a number of results can be [1–3]. A Banach space is said to be an 1-predual space if ∗ obtained immediately by means of a cardinality argument its dual space is linearly isometric to an -space; see [9, (Propositions A and B below), yet there are cases requiring Chapter 7]. more advanced knowledge. Let bearealBanachlattice.Alinearprojection in In the next section, we list basic notions and results is said to be an order (or a band) projection if 0≤≤for + concerning Riesz spaces, that is, linear lattices, which will be all ∈ ,anditsrange,(), is called aprojectionband.We 2 Journal of Function Spaces write ≅if the Banach lattices and are order-isometric. considered as a dual version of Milutin’s theorem, essentially The symbol Γ denotes the Stone-Cechˇ compactification of a stronger than the classical one. m discrete infinite space Γ,and2 denotes m-copies of the two- Let us notice that condition (1) applies also for spaces element discrete space 2,thatis,the-Cantor cube endowed ℵ of Radon measures built on scattered spaces. The classical with the product topology; thus Δ=2 0 . result [5, Corollary 19.7.7]saysthatif Ω is a scattered compact The following result is an immediate consequence of [10, space (i.e., every nonempty closed subset of Ω has an isolated Theorem 3.4 and Remark 3.5.(ii)]. point), then M(Ω) consists of atomic measures only; that is, M(Ω) is order-isometric to ℓ1(Ω). The examples of scattered Lemma 1. Let and be two real -lattices. If and are spaces are furnished, for example, by order intervals [5, pp. each order-isometric to a projection band of the other space, 151–156], Mrowka´ spaces [14, 15] (cf. [16, Section 3]), and then ≅. Stone spaces of superatomic Boolean algebras [17, Theorem, p. 1146], [18]. Hence we obtain the following complement to B(Ω) The symbol denotes the -algebra of all Borel part (a) of Proposition A. subsets of a compact space Ω,andB0(Ω) denotes the Baire- B(Ω) Ω subalgebra of generated by the class of subsets of . Proposition B. Let Ω1, Ω2 be two infinite compact scattered Ω B(Ω) = B (Ω) In particular, if is metrizable, then 0 .Two spaces. Then condition (1) implies that M(Ω1)≅M(Ω2). topological spaces and are said to be Borel [Baire,resp.] isomorphic if there is a bijection :→such that {() : In this paper, we shall prove and apply the following ∈B()} = B() [resp., {() : ∈ B0()} = B0()]. If theorem which, by the above results, is essential in the class ∈B(Ω),then∩B(Ω) denotes the class {∩ : ∈ B}. of compact Hausdorff spaces nonhomeomorphic either to The next result is proved on page 177 in [9]. products of metrizable spaces or to scattered spaces. Lemma 2. Let denotetheunitcircleintheplaneendowed Theorem 3. Let Ω1, Ω2 be two compact Hausdorff spaces. If with its natural topology. For every infinite discrete space Γ 2m (i) Ω1 and Ω2 are each homeomorphic to a closed subset with m = card(Γ),thespacesM(Γ) and M( ) are order- of the other space, or isometric. (ii) Ω1 and Ω2 are each continuously mapped onto the other space and they are extremally disconnected; 3. The Results then M(Ω1)≅M(Ω2). It is well known that two Polish spaces are Borel isomorphic if and only if they have the same cardinality ([11, p. 451] or [12, Proof. p. 38]). It follows that two compact metrizable spaces Ω1, Ω2 Part (i). Let Ω1 beaclosedsubspaceofΩ2.Weshallshowthat are Baire isomorphic if and only if M(Ω1) is order-isometric to a projection band of M(Ω2).The card (Ω1)=card (Ω2), (1) formula () () fl (∩Ω1), ∈B (Ω2) (3) whence the product of m ≥ℵ0 compact metrizable spaces m m is Baire isomorphic both to the Cantor cube 2 and to . defines an order projection from M(Ω2) onto the projec- Hence,by[13,TheoremH,p.229]and[13,TheoremD,p.239], tion band MΩ (Ω2) of all Radon measures concentrated on we obtain the following. Ω B(Ω1 )=Ω ∩ B(Ω ) M (Ω )∋ 1.Since 1 1 2 ,themapping Ω1 2 ] → ]|Ω is an order-isometry from MΩ (Ω2) onto M(Ω2). Proposition A. Let Ω be a compact metrizable space. 1 1 Similarly, M(Ω2) is order-isometric to a projection band in (a) If Ω is uncountable, then M(Ω) is order-isometric both M(Ω1). By Lemma 1, we obtain that M(Ω1)≅M(Ω2). to M() and to M(Δ). Part (ii).Observethatif maps Ω1 onto Ω2, then there is (b) If (Ω)∈ is a family of compact metrizable spaces with a closed subset of Ω1 such that the restriction | is a card =m ≥ℵ0,then homeomorphism from onto Ω2 (see, e.g., [5, Proposition 7.1.13 and Theorem 24.2.10]). Now we apply part (i). m m M (∏Ω)≅M ( )≅M (2 ). (2) ∈ Corollary 4. Let Ω be a compact space, and let Γ be an infinite m discrete space with m = card Γ.ThenM(Γ) ≅ M(Ω )≅ m m m 2 In particular, each of the spaces M(Ω ), M(Ω × ), M( ). m m m M(Ω × 2 ),andM( ) is order-isometric to M(2 ). Proof. ThisisanimmediateconsequenceofLemma2and It is also known that if Ω is a compact metrizable and part (b) of Proposition A. uncountable space, then, by Milutin’s result [5, Theorem Corollary 5. Ω 21.5.10], the spaces of continuous functions (Ω) and () Let be a metrizable compact space, and let N M() ≅ are linearly homeomorphic yet non-order-isomorphic [5, denote an infinite closed subspace of .Then 2ℵ0 2ℵ0 Theorem 7.8.1], in general. Thus Proposition A may be M(N)≅M(Ω )≅M( ). Journal of Function Spaces 3 ∗ 2ℵ0 Proof. Since contains a homeomorphic copy of N [19, p. Example 9. By Corollary 5, the spaces M(N ) and M( ) ∗ 71], from part (i) of the Theorem we obtain that M() ≅ are order-isometric, yet, by Lemma 7, (N ) does not embed M(N) 2ℵ0 . The remaining order isometries follow from Corol- linearly isomorphically into ( ). lary 4. The corollary below is a generalization of part (b)of Conflicts of Interest Proposition A. The author declares that there are no conflicts of interest Corollary 6. Let Ω be a compact space of weight m ≥ℵ0.If regarding the publication of this paper.
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