A generalized topological measure theory

Kirk, R. B.; Crenshaw, J. A.

doi:10.1090/s0002-9947-1975-0369648-7

Cited by 12 publications

(8 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For additional information concerning basic regularity properties of measures, see [4,13,14,31,184,185,193,239,258,265,267,275,336,382,389,416,509,515,516,517,5601 …”

Section: Proolmentioning

confidence: 99%

Measures on topological spaces

Богачев¹

1998

J Math Sci

View full text Add to dashboard Cite

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...

show abstract

“…For additional information concerning basic regularity properties of measures, see [4,13,14,31,184,185,193,239,258,265,267,275,336,382,389,416,509,515,516,517,5601 …”

Section: Proolmentioning

confidence: 99%

Measures on topological spaces

Богачев¹

1998

J Math Sci

View full text Add to dashboard Cite

show abstract

“…A more general case is considered by Kirk and Crenshaw. The following theorem is proved in [4]. THEOREM…”

Section: S Hegdementioning

confidence: 99%

“…There have been attempts to generalize the AlexandrofFs theorem to a wider class of functions. Among them are the works of Kirk [3], Kirk and Crenshaw [4]. Given a set X and a uniformly closed algebra A of bounded real valued functions on X which contains the constants and separates the points of X, there exists, as a consequence of Stone-Weierstrass theorem, a compact Hausdorff space X A such that X can be embedded into X A as a dense subspace of X A and A is isomorphic as a Banach lattice to C(X A ) [2, p. 276].…”

Section: S Hegdementioning

confidence: 99%

Some Smoothness Properties of Measures on Topological Spaces

Hegde

1978

Can. math. bull.

View full text Add to dashboard Cite

AbstrctV. S. Varadarajan has classified the bounded linear functional on the algebra C(X) of bounded continuous functions on a topological space X, according to the properties of their smoothness and related this classification to the corresponding natural classification of finitely additive regular measures on the zero sets of X. In this paper, some of these results are extended to the linear functionals on an arbitrary uniformly closed algebra A of bounded functions on a set X.

show abstract

“…LEMMA 2.1 (Kirk and Crenshaw [10,Proposition 1.2]). Let F be a subset of X, then F belongs to R[_ C W] if and only if there are sets W τ , V τ in °U (z=l, 2, •••, n) such that the following conditions hold:…”

Section: )mentioning

confidence: 99%