Extensions of tight set functions with applications in topological measure theory

Adamski, Wolfgang

doi:10.1090/s0002-9947-1984-0735428-9

Cited by 10 publications

(5 citation statements)

References 36 publications

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“…(2) to consider (3) to consider (4) to consider compact families of tight nonnegative Baire measures (which is the case in the definition compact families of not necessarily tight nonnegative Baire measures, weakly convergent sequences of tight nonnegative Baire measures with tight limits, countable compact families of type (1) or (2), (5) to consider in (1)-(4) completely bounded (i.e., precompact) families instead of compact ones, (6) to deal with .M, instead of Adt +. Certainly, there is a lot of other reasonable options.…”

Section: Theorem 8315 If X Is Of Countable Type Then Every Weak*-mentioning

confidence: 99%

“…Since every weakly fundamental sequence of Baire measures has a limit, which is a Baire measure, the modification of (3) indicated in (5) consists of considering weakly fundamental sequences of tight Baire measures. For example, the only difference between (3) and its analog given by (5) lies in the possible nonsequential completeness of .M+(X) with the weak topology (it is easy to see that .A4+(X) is weakly sequentially complete, provided every weak Cauchy sequence is uniformly tight). Obviously, assertions based on (5) and implying uniform tightness are stronger than those based on (1)- (4).…”

Section: Theorem 8315 If X Is Of Countable Type Then Every Weak*-mentioning

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: Theorem 8315 If X Is Of Countable Type Then Every Weak*-mentioning

confidence: 99%

Section: Theorem 8315 If X Is Of Countable Type Then Every Weak*-mentioning

confidence: 99%

Measures on topological spaces

Богачев¹

1998

J Math Sci

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“…the weak topology due to the general Portmanteau lemma. Therefore I is real-valued by statement (1). Let (X n ) n be an antitone sequence in L with X n X for some X ∈ L. Then the general Dini lemma (cf.…”

Section: Proof Of (1) ⇒ (2)mentioning

confidence: 98%

“…Let φ be a bounded isotone, modular set function φ on a lattice S which is stable under countable intersections, and which contains ∅ with φ(∅) = 0. Then we have: (1) φ is an inner premeasure if it satisfies the following conditions:…”

Section: Notations and Preliminariesmentioning

confidence: 99%

Compactness in spaces of inner regular measures and a general Portmanteau lemma

Krätschmer¹

2009

Journal of Mathematical Analysis and Applications

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This paper may be understood as a continuation of Topsoe's seminal paper [F. Topsoe, Compactness in spaces of measures, Studia Math. 36 (1970) 195-212] to characterize, within an abstract setting, compact subsets of finite inner regular measures w.r.t. the weak topology. The new aspect is that neither assumptions on compactness of the inner approximating lattices nor nonsequential continuity properties for the measures will be imposed. As a providing step also a generalization of the classical Portmanteau lemma will be established. The obtained characterizations of compact subsets w.r.t. the weak topology encompass several known ones from literature. The investigations rely basically on the inner extension theory for measures which has been systemized recently by König [H. König, Measure and Integration, Springer, Berlin, 1997; H. König, On the inner DaniellStone and Riesz representation theorems, Doc. Mat. 5 (2000) 301-315; H. König, Measure and integration: An attempt at unified systematization, Rend. Istit. Mat. Univ. Trieste 34 (2002) 155-214]. IntroductionThe most influential result concerning compactness of spaces of measures has been presented by Prokhorov [17] for probability Borel-measures on Polish spaces. His equivalent characterization of relative compactness by uniform tightness has turned out to be an important tool to check convergence in law for many stochastic processes. Therefore nowadays this criterion may be found in most of standard textbooks of probability theory (see e.g. [3,4,7,16]). Another characterization of compact sets of Borel probability measures on Polish spaces has been shown by Huber and Strassen [8] in terms of a continuity property that the upper envelopes of these sets satisfy. Prokhorov as well as Huber and Strassen used the so called topology of weak convergence which is derived from the weak * topology on the algebraic dual of the space of bounded continuous mappings. Of course this topology may be extended to spaces of finite Baire-measures on general topological spaces. This has been done by Varadarajan [22] who has also found an equivalent characterization for compact sets. However the topology of weak convergence relies on hidden regularity properties. Finite Baire-measures are inner regular w.r.t. the functionally closed sets, and in the special context of metrizable spaces they coincide with the finite Borel-measures, being inner regular w.r.t. the closed subsets. But in general finite Borel-measures are not inner regular w.r.t. the closed subsets. So for spaces of such measures the topology of weak convergence is not a reasonable concept because the measures are not uniquely determined by the restrictions of the integrals to bounded continuous mappings. Furthermore it seems to be necessary to impose regularity for the measures to find tractable extensions of the topology of weak convergence. ✩

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“…The notion of Baire sets in classical topology was introduced and studied by Andrzej Szymanski [2].The concept of fuzzy Baire sets in fuzzy topological spaces was introduced and studied by G.Thangaraj and R.Palani [14] in terms of fuzzy open sets and fuzzy residual sets and further studied in [18]. In classical topology, W. Adamski [1] introduced the concept of Baire-separation in topological spaces. Motivated on these lines, the notion of fuzzy Baire -separation in fuzzy topological spaces is introduced in tems of fuzzy Baire sets.…”

Section: Introductionmentioning

confidence: 99%

On Fuzzy Baire-Separated Spaces and Related Concepts

Thangaraj,

Raji

2024

PAMJ

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In this paper, a new class of fuzzy topological spaces, namely fuzzy Baire-separated spaces is introduced in terms of fuzzy Baire sets. Several characterizations of fuzzy Baire-separated spaces are established. It is shown that fuzzy Baire sets lie between disjoint fuzzy P-sets and fuzzy F<sub>σ</sub>- sets in a fuzzy Baire-separated space. Conditions under which fuzzy topological spaces become fuzzy Baire-separated spaces are established. Fuzzy nowhere dense sets are fuzzy closed sets in fuzzy nodec spaces and subsequently a question will arise. Which fuzzy topological spaces [other than fuzzy hyperconnected spaces, fuzzy globally disconnected spaces] have fuzzy closed sets with fuzzy nowhere denseness? For this, fuzzy topological spaces having fuzzy closed sets with fuzzy nowhere denseness are identified in this paper. It is verified that fuzzy ultraconnected spaces are non fuzzy Baire -separated spaces. The means, by which fuzzy weakly Baire space become fuzzy Baire -separated spaces and in turn fuzzy Baire - separated spaces become fuzzy seminormal spaces, are obtained. There are scope in this paper for exploring the inter-relations between fuzzy Baire spaces and Baire -separated spaces.

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Extensions of tight set functions with applications in topological measure theory

Cited by 10 publications

References 36 publications

Measures on topological spaces

Measures on topological spaces

Compactness in spaces of inner regular measures and a general Portmanteau lemma

On Fuzzy Baire-Separated Spaces and Related Concepts

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