Topological spaces whose Baire measure admits a regular Borel extension

Ohta, Haruto; Tamano, Kenichi

doi:10.1090/s0002-9947-1990-0946425-5

Cited by 6 publications

(14 citation statements)

References 27 publications

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“…[] Examples of this kind were considered in [536,537,370]. In order to construct such an example, it suffices to have a Baire measure # on X possessing a full measure discrete Baire set T of cardinality c and vanishing on all singletons.…”

Section: Extensions Of Measuresmentioning

confidence: 99%

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Measures on topological spaces

Богачев¹

1998

J Math Sci

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IntroductionIntegration on topological spaces is a field of mathematics which could be defined as the intersection of functional analysis, general topology, and probability theory. However, at different epochs the roles of these three ingredients were different, and, moreover, very often none of the three exerted a dominating'influence. For example, the theory of topological groups and analysis on manifolds gave rise to questions concerning Haar measures, Riemannian volumes, and other measures on locally compact spaces, and their influence was so strong that until recently many fundamental books on integration dealt exclusively with locally compact spaces. On the other hand, quantum fields and statistical physics provide problems of a totally different type, and this circumstance results in another trend in the theory of integration. At present, measure theory is especially strongly influenced by the intensive developmdnt of infinite-dimensional analysis in a broad sense, including stochastic analysis, dynamic systems, and the theory of representations of groups. This development involves measures on complicated infinite-dimensional manifolds and functional spaces. Recent investigations in population genetics have given rise to measure-valued diffusions, which, in turn, lead to such objects as measures on spaces of measures.The main aim of this survey is to present a systematic exposition of the integration theory on topological spaces, having in mind the indicated tendencies. Therefore, the target readership includes topologists, functional analysts, probabilists, and mathematical physicists. However, the main accent is put on analytical and probabilistic aspects rather than on purely topological or set-theoretic concepts. For this reason, no special topological knowledge is assumed.There are several books and recent surveys presenting modern measure theory on topological spaces. Schwartz' book [457] remains a standard reference book for basic questions. The results obtained after the publication of this book and many more special issues with a strong emphasis on general topology are discussed in excellent survey papers [183, 185] and [540]. A good introduction to the whole direction is given in Chapter 1 of the very informative monograph [527]. In addition, there are a number of books on gene:al measure theory, functional analysis, and probability theory, which include material on integration on topological spaces (the corresponding references are given in the text). However, there is no systematic exposition of modern theory oriented foward nontopologists. In addition, not all aspects of tile theory which are important for applications (in particular, those mentioned above) have been discussed in the literature. The central topics of this survey are:(i) regularity properties of measures on general topological spaces and specific spaces that arise in probability theory and functional analysis, including extension theorems for measures, (ii) transformations of measures and related problems such as conditional me...

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Section: Extensions Of Measuresmentioning

confidence: 99%

“…We shall present the main results in this direction following [542] and [370]. By Theorem 3.3.14, every normal countably paracompact space is MaHk.…”

Section: Extensions Of Measuresmentioning

confidence: 99%

Measures on topological spaces

Богачев¹

1998

J Math Sci

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“…Note that a topological space X is Baire-dominated if X is a cb-space [1] or X is cozero-dominated [10]. In particular, X is Baire-dominated if X is countably compact or X is normal and countably paracompact.…”

Section: Applicationsmentioning

confidence: 99%

On Extremal Extensions of Regular Contents and Measures

Adamski¹

1994

Proceedings of the American Mathematical Society

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“…Mařik's theorem says, then, that every normal non-Mařik spaces is Dowker. Ohta and Tamano [15] call a normal space X quasi-Mařik if every Baire measure on X admits some Borel extension. In this terminology, a counter-example to the measure extension problem is a non quasi-Mařik Dowker space.…”

Section: Introductionmentioning

confidence: 99%

“…Wheeler [26] asks if there are Dowker spaces that are Mařik. Ohta and Tamano [15] ask if quasi-Mařik non Mařik Dowker spaces exist. And Fremlin [12] asks for a non-Mařik Dowker space (namely, for a counter-example for the measure extension problem).…”

Section: Introductionmentioning

confidence: 99%