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Product liftings and densities with lifting invariant and density invariant sections

Kazimierz Musiał, W. Strauss, N. Macheras (2000)

Fundamenta Mathematicae

Given two measure spaces equipped with liftings or densities (complete if liftings are considered) the existence of product liftings and densities with lifting invariant or density invariant sections is investigated. It is proved that if one of the marginal liftings is admissibly generated (a subclass of consistent liftings), then one can always find a product lifting which has the property that all sections determined by one of the marginal spaces are lifting invariant (Theorem 2.13). For a large...

Product Pre-Measure

Noboru Endou (2016)

Formalized Mathematics

In this article we formalize in Mizar [5] product pre-measure on product sets of measurable sets. Although there are some approaches to construct product measure [22], [6], [9], [21], [25], we start it from σ-measure because existence of σ-measure on any semialgebras has been proved in [15]. In this approach, we use some theorems for integrals.

Products of non- σ -lower porous sets

Martin Rmoutil (2013)

Czechoslovak Mathematical Journal

In the present article we provide an example of two closed non- σ -lower porous sets A , B ⊆ ℝ such that the product A × B is lower porous. On the other hand, we prove the following: Let X and Y be topologically complete metric spaces, let A ⊆ X be a non- σ -lower porous Suslin set and let B ⊆ Y be a non- σ -porous Suslin set. Then the product A × B is non- σ -lower porous. We also provide a brief summary of some basic properties of lower porosity, including a simple characterization of Suslin non- σ -lower porous sets in topologically...

Propriétés arithmétiques et dynamiques du fractal de Rauzy

Ali Messaoudi (1998)

Journal de théorie des nombres de Bordeaux

Dans ce travail, nous construisons explicitement deux isomorphismes métriques partout continus. L’un entre le système dynamique symbolique associé à la substitution σ : 0 ↦ 01 , 1 ↦ 02 , 2 ↦ 0 et une rotation sur le tore 𝕋 2 ; l’autre, entre le système adique stationnaire [33] associé à la matrice de la substitution et la même rotation. Pour cela, nous étudions les propriétés arithmétiques de la frontière d’un ensemble compact de ℂ appelé “fractal de Rauzy”. Les constructions se généralisent aux substitutions de la forme σ k : 0 ↦ 01 , 1 ↦ 02 , ⋯ k - 1 ↦ 0 k , k ↦ 0 ...

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