Topology and Measure

Topsøe, Flemming

doi:10.1007/bfb0069481

Cited by 162 publications

(92 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…The following lemma summarizes the properties about the weak-star semi-continuity topology that we have mentioned and provides some additional useful characterizations (see [72,Theorem 8.1]). …”

Section: 4mentioning

confidence: 99%

See 1 more Smart Citation

Abstract framework for the theory of statistical solutions

Bronzi

Mondaini

Rosa

2016

Journal of Differential Equations

View full text Add to dashboard Cite

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.

show abstract

Section: 4mentioning

confidence: 99%

“…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

View full text Add to dashboard Cite

show abstract

“…The following final application (see [16] in this connection) gives a very general theorem and shows in a sense how far some of the previous theorems can be generalized in finite situations.…”

Section: Corollarymentioning

confidence: 84%

A general measure extension procedure

Sultan¹

1978

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. We give a very general and flexible way of producing measure extensions. We obtain as corollaries many well-known and important measure extension and integral representation theorems as well as the main theorems of several recent papers.Introduction. The question of when one can extend a measure (finitely, countably, or arbitrarily additive) is a very old one. Many different procedures over the years have been applied to obtain different types of significant extension theorems, mostly in topological spaces. Many of these have found numerous important applications in diverse branches of mathematics. Some of the more recent extension theorems have found applications in solving topological questions and are giving a great deal of insight into the interplay between measure and topology (see, in particular, [4], [5], [6]). In this paper we give a single, unifying procedure, which allows one to get simultaneously many of the well-known measure extension theorems. The method presented here is particularly useful for a variety of reasons: (1) It lends itself immediately to applications in the finite or totally finite cases. (2) It gives a great deal of flexibility in the actual construction of the measure extension. (3) In most of the important cases where the extension of the measure is not unique, this procedure gives all possible ways of extending the given measure. (4) The method ties in very naturally with many well-known representation theorems, and one obtains many of these as corollaries. (5) The actual construction of the measure extension is simple.In the very simplest cases we obtain many nontrivial extension theorems for locally compact T2 spaces. When our results are applied appropriately in many of the other cases, we obtain the main theorems of Definitions. By a paving (X, E) we mean a set X together with a lattice £ of subsets of X. We will suppress X and just say that £ is a paving. If 0 E £

show abstract

“…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

View full text Add to dashboard Cite

Topology and Measure

Cited by 162 publications

References 0 publications

Abstract framework for the theory of statistical solutions

Abstract framework for the theory of statistical solutions

A general measure extension procedure

Measure and Integration: An attempt at unified systematization

Contact Info

Product

Resources

About