Topological Spaces Whose Baire Measure Admits a Regular Borel Extension
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Measure Theory (Graduate Texts in Mathematics)
PaulR. Halmos Measure Theory Springer-VerlagNewYork-Heidelberg-Berlin Managing Editors P. R. Halmos C. C. Moore Indiana University University of California Department of Mathematics at Berkeley Swain Hall East Department of Mathematics Bloomington, Indiana 47401 Berkeley, California 94720 AMS Subject Classifications (1970) Primary: 28 - 02, 28A10, 28A15, 28A20, 28A25, 28A30, 28A35, 28A40, 28A60, 28A65, 28A70 Secondary: 60A05, 60Bxx Library of Congress Cataloging in Publication Data Halmos, Paul Richard, 1914- Measure theory. (Graduate texts in mathematics, 18) Reprint of the ed. published by Van Nostrand, New York, in series: The University series in higher mathematics. Bibliography: p. 1. Measure theory. I. Title. II. Series. [QA312.H26 1974] 515'.42 74-10690 ISBN 0-387-90088-8 All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag. © 1950 by Litton Educational Publishing, Inc. and 1974 by Springer-Verlag New York Inc. Printed in the United States of America. ISBN 0-387-90088-8 Springer-Verlag New York Heidelberg Berlin ISBN 3-540-90088-8 Springer-Verlag Berlin Heidelberg New York PREFACE My main purpose in this book is to present a unified treatment of that part of measure theory which in recent years has shown itself to be most useful for its applications in modern analysis. If I have accomplished my purpose, then the book should be found usable both as a text for students and as a source of refer ence for the more advanced mathematician. I have tried to keep to a minimum the amount of new and unusual terminology and notation. -
Version of 21.8.15 Chapter 43 Topologies and Measures II The
Version of 21.8.15 Chapter 43 Topologies and measures II The first chapter of this volume was ‘general’ theory of topological measure spaces; I attempted to distinguish the most important properties a topological measure can have – inner regularity, τ-additivity – and describe their interactions at an abstract level. I now turn to rather more specialized investigations, looking for features which offer explanations of the behaviour of the most important spaces, radiating outwards from Lebesgue measure. In effect, this chapter consists of three distinguishable parts and two appendices. The first three sections are based on ideas from descriptive set theory, in particular Souslin’s operation (§431); the properties of this operation are the foundation for the theory of two classes of topological space of particular importance in measure theory, the K-analytic spaces (§432) and the analytic spaces (§433). The second part of the chapter, §§434-435, collects miscellaneous results on Borel and Baire measures, looking at the ways in which topological properties of a space determine properties of the measures it carries. In §436 I present the most important theorems on the representation of linear functionals by integrals; if you like, this is the inverse operation to the construction of integrals from measures in §122. The ideas continue into §437, where I discuss spaces of signed measures representing the duals of spaces of continuous functions, and topologies on spaces of measures. The first appendix, §438, looks at a special topic: the way in which the patterns in §§434-435 are affected if we assume that our spaces are not unreasonably complex in a rather special sense defined in terms of measures on discrete spaces. -
1 Sets and Classes
Prerequisites Topological spaces. A set E in a space X is σ-compact if there exists a S1 sequence of compact sets such that E = n=1 Cn. A space X is locally compact if every point of X has a neighborhood whose closure is compact. A subset E of a locally compact space is bounded if there exists a compact set C such that E ⊂ C. Topological groups. The set xE [or Ex] is called a left translation [or right translation.] If Y is a subgroup of X, the sets xY and Y x are called (left and right) cosets of Y . A topological group is a group X which is a Hausdorff space such that the transformation (from X ×X onto X) which sends (x; y) into x−1y is continuous. A class N of open sets containing e in a topological group is a base at e if (a) for every x different form e there exists a set U in N such that x2 = U, (b) for any two sets U and V in N there exists a set W in N such that W ⊂ U \ V , (c) for any set U 2 N there exists a set W 2 N such that V −1V ⊂ U, (d) for any set U 2 N and any element x 2 X, there exists a set V 2 N such that V ⊂ xUx−1, and (e) for any set U 2 N there exists a set V 2 N such that V x ⊂ U. If N is a satisfies the conditions described above, and if the class of all translation of sets of N is taken for a base, then, with respect to the topology so defined, X becomes a topological group. -
Borel Extensions of Baire Measures
FUNDAMENTA MATHEMATICAE 154 (1997) Borel extensions of Baire measures by J. M. A l d a z (Madrid) Abstract. We show that in a countably metacompact space, if a Baire measure admits a Borel extension, then it admits a regular Borel extension. We also prove that under the special axiom ♣ there is a Dowker space which is quasi-Maˇr´ıkbut not Maˇr´ık,answering a question of H. Ohta and K. Tamano, and under P (c), that there is a Maˇr´ıkDowker space, answering a question of W. Adamski. We answer further questions of H. Ohta and K. Tamano by showing that the union of a Maˇr´ıkspace and a compact space is Maˇr´ık,that under “c is real-valued measurable”, a Baire subset of a Maˇr´ıkspace need not be Maˇr´ık, and finally, that the preimage of a Maˇr´ıkspace under an open perfect map is Maˇr´ık. 1. Introduction. The Borel sets are the σ-algebra generated by the open sets of a topological space, and the Baire sets are the smallest σ- algebra making all real-valued continuous functions measurable. The Borel extension problem asks: Given a Baire measure, when can it be extended to a Borel measure? Whenever one deals with Baire measures on a topological space, it is assumed that the space is completely regular and Hausdorff, so there are enough continuous functions to separate points and closed sets. In 1957 (see [Ma]), J. Maˇr´ıkproved that all normal, countably paracompact spaces have the following property: Every Baire measure extends to a regular Borel measure. -
Some Undecidability Results Concerning Radon Measures by R
transactions of the american mathematical society Volume 259, Number 1, May 1980 SOME UNDECIDABILITY RESULTS CONCERNING RADON MEASURES BY R. J. GARDNER AND W. F. PFEFFER Abstract. We show that in metalindelöf spaces certain questions about Radon measures cannot be decided within the Zermelo-Fraenkel set theory, including the axiom of choice. 0. Introduction. All spaces in this paper will be Hausdorff. Let A be a space. By § and Q we shall denote, respectively, the families of all open and compact subsets of X. The Borel o-algebra in X, i.e., the smallest a-algebra in X containing ê, will be denoted by <®. The elements of 9> are called Borel sets. A Borel measure in A' is a measure p on % such that each x G X has a neighborhood U G % with p( U) < + 00. A Borel measure p in X is called (i) Radon if p(B) = sup{ p(C): C G G, C c F} for each F G % ; (ii) regular if p(B) = inî{ p(G): G G @,B c G} for each fieS. A Radon space is a space X in which each finite Borel measure is Radon. The present paper will be divided into two parts, each dealing with one of the following questions. (a) Let A' be a locally compact, hereditarily metalindelöf space with the counta- ble chain condition. Is X a Radon space? (ß) Let A' be a metalindelöf space, and let p be a a-finite Radon measure in X. Is pa. regular measure? We shall show that neither question can be decided within the usual axioms of set theory. -
On Measure-Compactness and Borel Measure-Compactness
CORE Metadata, citation and similar papers at core.ac.uk Provided by Osaka City University Repository Okada, S. and Okazaki, Y. Osaka J. Math. 15 (1978), 183-191 ON MEASURE-COMPACTNESS AND BOREL MEASURE-COMPACTNESS SUSUMU OKADA AND YOSHIAKI OKAZAKI (Received December 6, 1976) 1. Introduction Moran has introduced "measure-compactness' and "strongly measure- compactness' in [7], [8], For measure-compactness, Kirk has independently presented in [4]. For Borel measures, "Borel measure-compactness' and "property B" are introduced by Gardner [1], We study, defining "a weak Radon space", the relation among these proper- ties at the last of §2. Furthermore we investigate the stability of countable unions in §3 and the hereditariness in §4. The authors wish to thank Professor Y. Yamasaki for his suggestions. 2. Preliminaries and fundamental results All spaces considered in this paper are completely regular and Hausdorff. Let Cb(X) be the space of all bounded real-valued continuous functions on a topological space X. By Z(X] we mean the family of zero sets {/" \0) / e Cb(X)} c and U(X) the family of cozero sets {Z \ Z<=Z(X)}. We denote by &a(X) the Baίre field, that is, the smallest σ-algebra containing Z(X) and by 1$(X) the σ-algebra generated by all open subsets of X, which is called the Borel field. A Baire measure [resp. Borel measure} is a totally finite, non-negative coun- tably additive set function defined on Άa(X) [resp. 3)(X)]. For a Baire measure μ, the support of μ is supp μ = {x^Xm, μ(U)>0 for every U^ U(X) containing x} . -
Real Analysis II, Winter 2018
Real Analysis II, Winter 2018 From the Finnish original “Moderni reaalianalyysi”1 by Ilkka Holopainen adapted by Tuomas Hytönen February 22, 2018 1Version dated September 14, 2011 Contents 1 General theory of measure and integration 2 1.1 Measures . 2 1.11 Metric outer measures . 4 1.20 Regularity of measures, Radon measures . 7 1.31 Uniqueness of measures . 9 1.36 Extension of measures . 11 1.45 Product measure . 14 1.52 Fubini’s theorem . 16 2 Hausdorff measures 21 2.1 Basic properties of Hausdorff measures . 21 2.12 Hausdorff dimension . 24 2.17 Hausdorff measures on Rn ...................... 25 3 Compactness and convergence of Radon measures 30 3.1 Riesz representation theorem . 30 3.13 Weak convergence of measures . 35 3.17 Compactness of measures . 36 4 On the Hausdorff dimension of fractals 39 4.1 Mass distribution and Frostman’s lemma . 39 4.16 Self-similar fractals . 43 5 Differentiation of measures 52 5.1 Besicovitch and Vitali covering theorems . 52 1 Chapter 1 General theory of measure and integration 1.1 Measures Let X be a set and P(X) = fA : A ⊂ Xg its power set. Definition 1.2. A collection M ⊂ P (X) is a σ-algebra of X if 1. ? 2 M; 2. A 2 M ) Ac = X n A 2 M; S1 3. Ai 2 M, i 2 N ) i=1 Ai 2 M. Example 1.3. 1. P(X) is the largest σ-algebra of X; 2. f?;Xg is the smallest σ-algebra of X; 3. Leb(Rn) = the Lebesgue measurable subsets of Rn; 4. -
Category-Measure Duality: Convexity, Mid- Point Convexity and Berz Sublinearity
N. H. Bingham, A. J. Ostaszewski Category-measure duality: convexity, mid- point convexity and Berz sublinearity Article (Published version) (Refereed) Original citation: Bingham, N. H. and Ostaszewski, Adam (2017) Category-measure duality: convexity, mid-point convexity and Berz sublinearity. Aequationes Mathematicae. ISSN 0001-9054 DOI: 10.1007/s00010-017-0486-7 Reuse of this item is permitted through licensing under the Creative Commons: © 2017 The Authors CC BY 4.0 This version available at: http://eprints.lse.ac.uk/68940/ Available in LSE Research Online: July 2017 LSE has developed LSE Research Online so that users may access research output of the School. Copyright © and Moral Rights for the papers on this site are retained by the individual authors and/or other copyright owners. You may freely distribute the URL (http://eprints.lse.ac.uk) of the LSE Research Online website. Aequat. Math. c The Author(s) 2017. This article is an open access publication Aequationes Mathematicae DOI 10.1007/s00010-017-0486-7 Category-measure duality: convexity, midpoint convexity and Berz sublinearity N. H. Bingham and A. J. Ostaszewski Abstract. Category-measure duality concerns applications of Baire-category methods that have measure-theoretic analogues. The set-theoretic axiom needed in connection with the Baire category theorem is the Axiom of Dependent Choice, DC rather than the Axiom of Choice, AC. Berz used the Hahn–Banach theorem over Q to prove that the graph of a measurable sublinear function that is Q+-homogeneous consists of two half-lines through the origin. We give a category form of the Berz theorem. -
The Strict Topology in a Completely Regular Setting: Relations to Topological Measure Theory
Can. J. Math., Vol. XXIV, No. 5, 1972, pp. 873-890 THE STRICT TOPOLOGY IN A COMPLETELY REGULAR SETTING: RELATIONS TO TOPOLOGICAL MEASURE THEORY STEVEN E. MOSIMAN AND ROBERT F. WHEELER 1. Introduction. Let X be a locally compact Hausdorff space, and let C*(X) denote the space of real-valued bounded continuous functions on X. An interesting and important property of the strict topology /3 on C*(X) was proved by Buck [2]: the dual space of (C*(X), 13) has a natural representation as the space of bounded regular Borel measures on X. Now suppose that X is completely regular (all topological spaces are assumed to be Hausdorff in this paper). Again it seems natural to seek locally convex topologies on the space C*(X) whose dual spaces are (via the integration pairing) significant classes of measures. Motivated by this idea, Sentilles [24] has considered locally convex topologies /30, 0, and fii on C*(X) which yield as dual spaces the tight, r-additive, and cr-additive Baire measures of Vara- darajan [30]. If X is locally compact, the topologies /30 and fi coincide and are precisely the original strict topology of Buck. The topology /30 on C*(X) has an intuitively appealing description: it is the finest locally convex topology which, when restricted to sets bounded in the supremum norm, coincides with the compact-open topology. However, it is defective with respect to a desirable property of the dual space: a weak*- compact set of tight measures need not be /30-equicontinuous. This bears on a question posed by Buck in the locally compact setting: when is (C*(X),/3) a Mackey space? A partial answer can be found in the work of LeCam [15] and, independently, Conway [4]: if X is cr-compact locally compact, then (C*(X), /3) is a strong Mackey space (i.e., weak*-compact subsets of the dual space are /3-equicon- tinuous). -
5.2 Complex Borel Measures on R
MATH 245A (17F) (L) M. Bonk / (TA) A. Wu Real Analysis Contents 1 Measure Theory 3 1.1 σ-algebras . .3 1.2 Measures . .4 1.3 Construction of non-trivial measures . .5 1.4 Lebesgue measure . 10 1.5 Measurable functions . 14 2 Integration 17 2.1 Integration of simple non-negative functions . 17 2.2 Integration of non-negative functions . 17 2.3 Integration of real and complex valued functions . 19 2.4 Lp-spaces . 20 2.5 Relation to Riemann integration . 22 2.6 Modes of convergence . 23 2.7 Product measures . 25 2.8 Polar coordinates . 28 2.9 The transformation formula . 31 3 Signed and Complex Measures 35 3.1 Signed measures . 35 3.2 The Radon-Nikodym theorem . 37 3.3 Complex measures . 40 4 More on Lp Spaces 43 4.1 Bounded linear maps and dual spaces . 43 4.2 The dual of Lp ....................................... 45 4.3 The Hardy-Littlewood maximal functions . 47 5 Differentiability 51 5.1 Lebesgue points . 51 5.2 Complex Borel measures on R ............................... 54 5.3 The fundamental theorem of calculus . 58 6 Functional Analysis 61 6.1 Locally compact Hausdorff spaces . 61 6.2 Weak topologies . 62 6.3 Some theorems in functional analysis . 65 6.4 Hilbert spaces . 67 1 CONTENTS MATH 245A (17F) 7 Fourier Analysis 73 7.1 Trigonometric series . 73 7.2 Fourier series . 74 7.3 The Dirichlet kernel . 75 7.4 Continuous functions and pointwise convergence properties . 77 7.5 Convolutions . 78 7.6 Convolutions and differentiation . 78 7.7 Translation operators . -
ON MEASURES in FIBRE SPACES by Anthony Karel SEDA
CAHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES ANTHONY KAREL SEDA On measures in fibre spaces Cahiers de topologie et géométrie différentielle catégoriques, tome 21, no 3 (1980), p. 247-276 <http://www.numdam.org/item?id=CTGDC_1980__21_3_247_0> © Andrée C. Ehresmann et les auteurs, 1980, tous droits réservés. L’accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ CAHIERS DE TOPOLOGIE Vol. XXI - 3 (1980) ET GEOMETRIE DIFFERENTIELLE ON MEASURES IN FIBRE SPACES by Anthony Karel SEDA INTRODUCTION. Let S and X be locally compact Hausdorff spaces and let p : S - X be a continuous surjective function, hereinafter referred to as a fibre space with projection p, total space S and base space X . Such spaces are com- monly regarded as broad generalizations of product spaces X X Y fibred over X by the projection on the first factor. However, in practice this level of generality is too great and one places compatibility conditions on the f ibres of S such as : the fibres of S are all to be homeomorphic ; p is to be a fibration or 6tale map; S is to be locally trivial, and so on. In this paper fibre spaces will be viewed as generalized transformation groups, and the specific compatibility requirement will be that S is provided with a categ- ory or groupoid G of operators. -
ABSOLUTE BAIRE SETS <P: X^T
ABSOLUTE BAIRE SETS stelios negrepontis 1. Introduction. Definitions and facts. For background in nota- tion and terminology the reader should consult [2]. We make here the blanket assumption that all spaces to be considered in this note are completely regular Hausdorff topological spaces. A continuous map/: X—>Fis called proper if /is onto, closed, con- tinuous, and if for every compact set K in Y,/"l(K) is compact. The family of Baire sets of a space X is the cr-field generated by the zero-sets of X. The family of Borel sets of a space X is the cr-field generated by the closed sets of X. For a metric space the two families coincide, but in general the family of Borel sets properly contains the family of Baire sets. A subset A of a space X is said to be C*-embedded in X if every real-valued continuous bounded function on A is the restriction of a real-valued continuous (bounded) function on X. Thus, for example, a space X is C*-embedded in its Stone-Cech compactification BX. 2. Absolute Baire spaces. We will now prove our first main result, which characterizes the spaces which are Baire sets in their Stone- Cech compactifications. The characterization, which uses the idea included in the proof of Theorem D, §51, of Halmos's text on mea- sure theory [2], is surprisingly simple. 2.1. Theorem. The /ollowing are equivalent on any space X: (1) X is Baire in BX (i.e. X is an "absolute Baire set"). (2) There is a separable metric space T which is an absolute Borel space (in the class 0/ separable metric spaces) and a proper mapping <p:X^T.