On sets of Haar measure zero in abelian polish groups

Christensen, Jens Peter

doi:10.1007/bf02762799

Cited by 221 publications

(221 citation statements)

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“…Christensen [1,Theorem 2] proved that 0 ∈ int (A − A) for each universally measurable set A, which is not Haar null in an abelian Polish group X. In fact this theorem generalizes Steinhaus' theorem.…”

Section: Introductionmentioning

confidence: 95%

Some analogies between Haar meager sets and Haar null sets in abelian Polish groups

Jabłońska

2015

Journal of Mathematical Analysis and Applications

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In the paper we would like to pay attention to some analogies between Haar meager sets and Haar null sets. Among others, we will show that 0 ∈ int (A− A) for each Borel set A, which is not Haar meager in an abelian Polish group. Moreover, we will give an example of a Borel non-Haar meager set A ⊂ c 0 such that int (A + A) = ∅. Finally, we will define D-measurability as a topological analog of Christensen measurability, and apply our generalization of Piccard's theorem to prove that each D-measurable homomorphism is continuous. Our results refer to the papers [1], [2] and [4].

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Section: Introductionmentioning

confidence: 95%

Some analogies between Haar meager sets and Haar null sets in abelian Polish groups

Jabłońska

2015

Journal of Mathematical Analysis and Applications

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“…'After [5] went to press, it was discovered that Christensen [3] gave an analogous definition of a "Haar zero set" on abelian Polish groups; see [6].…”

Section: Introductionmentioning

confidence: 99%

The prevalence of continuous nowhere differentiable functions

Hunt¹

1994

Proc. Amer. Math. Soc.

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Abstract. In the space of continuous functions of a real variable, the set of nowhere dilferentiable functions has long been known to be topologically "generic". In this paper it is shown further that in a measure theoretic sense (which is different from Wiener measure), "almost every" continuous function is nowhere dilferentiable. Similar results concerning other types of regularity, such as Holder continuity, are discussed.

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“…Moreover, in a recent result by Smith, Hirsch, and the author, it is shown that under the same hypotheses the generic bounded solution of (3) converges towards a homogeneous equilibrium [4]. In particular, the set of nonhomogeneous equilibria must also be sparse (in the sense of prevalence; see [8,3]). This indicates that item (3) in Theorem 2 cannot be strengthened to a substantially larger set of nonhomogeneous equilibria.…”

Section: Theorem 2 There Exists a Reaction Diffusion System (3) Undementioning

confidence: 95%

On a Smale theorem and nonhomogeneous equilibria in cooperative systems

Enciso¹

2008

Proc. Amer. Math. Soc.

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Abstract. A standard result by Smale states that n dimensional strongly cooperative dynamical systems can have arbitrary dynamics when restricted to unordered invariant hyperspaces. In this paper this result is extended to the case when all solutions of the strongly cooperative system are bounded and converge towards one of only two equilibria outside of the hyperplane.An application is given in the context of strongly cooperative systems of reaction diffusion equations. It is shown that such a system can have a continuum of spatially inhomogeneous steady states, even when all solutions of the underlying reaction system converge to one of only three equilibria.

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On sets of Haar measure zero in abelian polish groups

Cited by 221 publications

References 1 publication

Some analogies between Haar meager sets and Haar null sets in abelian Polish groups

Some analogies between Haar meager sets and Haar null sets in abelian Polish groups

The prevalence of continuous nowhere differentiable functions

On a Smale theorem and nonhomogeneous equilibria in cooperative systems

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