Topological measure theory for completely regular spaces and their projective covers

Wheeler, Robert F.

doi:10.2140/pjm.1979.82.565

Cited by 6 publications

(3 citation statements)

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“…The developments in topological measure theory are propelled by Alexandrov and Varadarajan, considering that the topological spaces are always completely regular as well as Hausdorff [6,7]. The fundamental question in measure theory and its topological variants is the extensibility of σ−algebras [8].…”

Section: Motivation and Contributionsmentioning

confidence: 99%

“…Furthermore, the algebra-based topological separation of subspaces also depends on W(X). In the case of a completely regular topological space, an extremely disconnected space (i.e., closure of open set is open) exists, where the corresponding Baire sets become reduced and the zero-sets are easy to identify [7]. In other words, the topological determination of measure compactness becomes simpler in this setting.…”

Section: Motivation and Contributionsmentioning

confidence: 99%

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Topological Sigma-Semiring Separation and Ordered Measures in Noetherian Hyperconvexes

Bagchi

2022

Symmetry

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The interplay between topological hyperconvex spaces and sigma-finite measures in such spaces gives rise to a set of analytical observations. This paper introduces the Noetherian class of k-finite k-hyperconvex topological subspaces (NHCs) admitting countable finite covers. A sigma-finite measure is constructed in a sigma-semiring in a NHC under a topological ordering of NHCs. The topological ordering relation maintains the irreflexive and anti-symmetric algebraic properties while retaining the homeomorphism of NHCs. The monotonic measure sequence in a NHC determines the convexity and compactness of topological subspaces. Interestingly, the topological ordering in NHCs in two isomorphic topological spaces induces the corresponding ordering of measures in sigma-semirings. Moreover, the uniform topological measure spaces of NHCs need not always preserve the pushforward measures, and a NHC semiring is functionally separable by a set of inner-measurable functions.

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Section: Motivation and Contributionsmentioning

confidence: 99%

Section: Motivation and Contributionsmentioning

confidence: 99%

Topological Sigma-Semiring Separation and Ordered Measures in Noetherian Hyperconvexes

Bagchi

2022

Symmetry

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“…The topological concepts we use are explained in Willard [27] and Bourbaki [2]. For an interesting paper on topological measure theory see, for example, Wheeler [26].…”

Section: Introductionmentioning

confidence: 99%

Strictly Positive Radon Measures

Casteren

1994

Journal of the London Mathematical Society

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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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Topological measure theory for completely regular spaces and their projective covers

Cited by 6 publications

References 40 publications

Topological Sigma-Semiring Separation and Ordered Measures in Noetherian Hyperconvexes

Topological Sigma-Semiring Separation and Ordered Measures in Noetherian Hyperconvexes

Strictly Positive Radon Measures

Measures on topological spaces

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