Quasi-measures and dimension theory

Wheeler, Robert F.

doi:10.1016/0166-8641(95)00009-6

Cited by 24 publications

(17 citation statements)

References 6 publications

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“…Define a quasi-state η on C(∆) by η(H) := ζ τ (Φ * H). The WheelerShakhmatov Theorem [26], [44] implies that every quasi-state on a normal topological space (and hence on any metric space) of covering dimension ≤ 1 is linear. Hence η is linear.…”

Section: Symplectic Quasi-states On Surfacesmentioning

confidence: 99%

Quasi-states and symplectic intersections

Entov¹,

Polterovich²

2006

Comment. Math. Helv.

136

257

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We establish a link between symplectic topology and a recently emerged branch of functional analysis called the theory of quasi-states and quasi-measures (also known as topological measures). In the symplectic context quasi-states can be viewed as an algebraic way of packaging certain information contained in Floer theory, and in particular in spectral invariants of Hamiltonian diffeomorphisms introduced recently by Yong-Geun Oh. As a consequence we prove a number of new results on rigidity of intersections in symplectic manifolds. This work is a part of a joint project with Paul Biran.

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Section: Symplectic Quasi-states On Surfacesmentioning

confidence: 99%

Quasi-states and symplectic intersections

Entov¹,

Polterovich²

2006

Comment. Math. Helv.

136

257

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“…Later we will show that there are situations when measures are nowhere dense in the space of all topological measures on X. A positive topological measure has a (necessarily unique) extension to a regular Borel measure on X if and only if for any open sets U and V we have µ(U ∪ V ) ≤ µ(U ) + µ(V ) (see [19]), while for a signed topological measure the condition for extension to a measure is µ(U ) + µ(V ) = µ(U ∪ V ) + µ(U ∩ V ) (see [11]). Remark 1.1.…”

Section: Ii) µ(U ) = Sup{µ(c) : C ⊆ U C ∈ C(x)} For All U ∈ O(x)mentioning

confidence: 99%

“…Work on the theory of topological measures and dimension theory initiated by R. Wheeler, and continued by D. Shakmatov, D. Fremlin, and D. Grubb, describes a very different relationship between measures and topological measures on a space of dimension not greater than 1. See [19], [18], [10] or [12] (a more general result). …”

Section: Proof Of Theorem 33 Let µ Be a Simple Topological Measure mentioning

confidence: 99%

Density in the space of topological measures

Butler

2002

Fund. Math.

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Abstract. Topological measures (formerly "quasi-measures") are set functions that generalize measures and correspond to certain non-linear functionals on the space of continuous functions. The goal of this paper is to consider relationships between various families of topological measures on a given space. In particular, we prove density theorems involving classes of simple, representable, extreme topological measures and measures, hence giving a way of approximating various topological measures by members of different classes.

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“…Then µ f extends uniquely to a Borel measure on R (all topological measures on one-dimensional spaces extend uniquely to regular Borel measures, cf. [14]). Hence we have the definition The integrals (with respect to topological measures), and hence topological measures, were shown in [1] to be uniformly continuous.…”

Section: Topological Measuresmentioning

confidence: 99%

“…The circle is one-dimensional and hence all topological measures extend to regular Borel measures (cf. [14]). Accordingly T * is homeomorphic to the circle, establishing the assertion.…”

Section: If C ∈ C(x) and A ∈ A(c) Let A *mentioning

confidence: 99%

The homology of spaces of simple topological measures

Johansen

Rustad

2003

Fund. Math.

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Abstract. The simple topological measures X* on a q-space X are shown to be a superextension of X. Properties inherited from superextensions to topological measures are presented. The homology groups of various subsets of X * are calculated. For a qspace X, X * is shown to be a q-space. The homology of X * when X is the annulus is calculated. The homology of X * when X is a more general genus one space is investigated. In particular, X * for the torus is shown to have a retract homeomorphic to an infinite product of circles.

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Quasi-measures and dimension theory

Cited by 24 publications

References 6 publications

Quasi-states and symplectic intersections

Quasi-states and symplectic intersections

Density in the space of topological measures

The homology of spaces of simple topological measures

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