Compactness in spaces of measures

Topsøe, Flemming

doi:10.4064/sm-36-3-195-212

Cited by 73 publications

(49 citation statements)

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“…In order to pass to the limit, one needs a compactness result for measures. A suitable result of this kind in our abstract framework is the one developed by Topsoe along his work on a generalization of Prohorov's Theorem ( [57]) to spaces which are not necessarily Polish ( [71,72,73]). …”

Section: Introductionmentioning

confidence: 99%

Abstract framework for the theory of statistical solutions

Bronzi

Mondaini

Rosa

2016

Journal of Differential Equations

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Abstract. An abstract framework for the theory of statistical solutions is developed for general evolution equations, extending the theory initially developed for the three-dimensional incompressible Navier-Stokes equations. The motivation for this concept is to model the evolution of uncertainties on the initial conditions for systems which have global solutions that are not known to be unique. Both concepts of statistical solution in trajectory space and in phase space are given, and the corresponding results of existence of statistical solution for the associated initial value problems are proved. The wide applicability of the theory is illustrated with the very incompressible Navier-Stokes equations, a reaction-diffusion equation, and a nonlinear wave equation, all displaying the property of global existence of weak solutions without a known result of global uniqueness.

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Section: Introductionmentioning

confidence: 99%

Abstract framework for the theory of statistical solutions

Bronzi

Mondaini

Rosa

2016

Journal of Differential Equations

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“…A number of new results on the F-space question have been discovered recently: [26], [16], [11] and [36]. Tops0e [36] has an especially interesting characterization of w*-compactness versus tightness in Mt.…”

Section: Definitions and Preliminariesmentioning

confidence: 99%

“…Tops0e [36] has an especially interesting characterization of w*-compactness versus tightness in Mt. The remaining authors have, apparently, independently shown that a hemicompact espace is a F-space, as is any complete metric space [16].…”

Section: Definitions and Preliminariesmentioning

confidence: 99%

Bounded Continuous Functions on a Completely Regular Space

Sentilles¹

1972

Transactions of the American Mathematical Society

138

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Abstract. Three locally convex topologies on C(X) are introduced and developed, and in particular shown to coincide with the strict topology on locally compact A1 and yield dual spaces consisting of tight, -r-additive and x where p. is a bounded regular Borel measure on X. This yields a very satisfactory relationship between the topology on X, the space C(X), a natural class of linear functionals on it, and those measures on Jfthat measure at least the sets determined by the topology on X in the usual way, the Borel sets.This kind of representation was subsequently extended to locally compact spaces : first to functionals on the space C0(X) of continuous functions vanishing at infinity, and then further, to the bounded continuous functions on X. The last result, due to R. C. Buck, demanded the use of a locally convex topology, the strict topology, rather than a norm topology. In both extensions the same satisfactory relationship between measure and topology was obtained.In this paper we begin the development of locally convex topologies for C(X) which extend this kind of representation to its last reasonable setting, completely regular Hausdorff spaces. This setting appears to be ultimate in the sense of Hewitt's example of a regular space upon which the only continuous functions are constants. Definitions and preliminaries.The actual work of integral representation of linear forms has been done by other authors, going back to Aleksandrov [1] and, following his work, by Varadarajan [39], and later Knowles [21] and more recently Kirk [20] and Moran ([24], [25]). Our work relies heavily on theirs and will not extend the representations they have obtained but will relate these works to earlier

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“…A decisive prelude was the short paper of Kisyński [1968], which produced the final class of Borel-Radon measures on Hausdorff topological spaces via inner regularity. In no time then Topsøe [1970aTopsøe [ ][1970b realized that this procedure opens the road to unification. However, these articles and the subsequent Pollard-Topsøe [1975] and Topsøe [1976] [1978] did not yet present a full systematization.…”

Section: Introductionmentioning

confidence: 99%

“…The situation turned around with an innocent step which the present author took in an analysis course [1969/70], thus at the same time with Kisyński [1968] and Topsøe [1970aTopsøe [ ][1970b: He observed that the old proof of the extension theorem carries over verbatim from rings to that particular class of lattices described above (of course with an adequate notion of content), provided that instead of ϕ • one uses the formation ϕ σ : P(X) → [0, ∞], defined for an isotone set function ϕ : S → [0, ∞] with ϕ(∅) = 0 on a set system S with ∅ ∈ S to be ϕ σ (A) = inf{ lim l→∞ ϕ(S l ) : (S l ) l in S with S l ↑ some subset ⊃ A}. Thus ϕ σ is isotone with ϕ σ (∅) = 0 as well.…”

Section: Introductionmentioning

confidence: 99%

Measure and Integration: An attempt at unified systematization

König

2012

Measure and Integration

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Compactness in spaces of measures

Cited by 73 publications

References 0 publications

Abstract framework for the theory of statistical solutions

Abstract framework for the theory of statistical solutions

Bounded Continuous Functions on a Completely Regular Space

Measure and Integration: An attempt at unified systematization

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