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A method for evaluating the fractal dimension in the plane, using coverings with crosses

Claude Tricot (2002)

Fundamenta Mathematicae

Various methods may be used to define the Minkowski-Bouligand dimension of a compact subset E in the plane. The best known is the box method. After introducing the notion of ε-connected set E ε , we consider a new method based upon coverings of E ε with crosses of diameter 2ε. To prove that this cross method gives the fractal dimension for all E, the main argument consists in constructing a special pavement of the complementary set with squares. This method gives rise to a dimension formula using integrals,...

A new proof of Kelley's Theorem

S. Ng (1991)

Fundamenta Mathematicae

Kelley's Theorem is a purely combinatorial characterization of measure algebras. We first apply linear programming to exhibit the duality between measures and this characterization for finite algebras. Then we give a new proof of the Theorem using methods from nonstandard analysis.

A noncommutative version of a Theorem of Marczewski for submeasures

Paolo de Lucia, Pedro Morales (1992)

Studia Mathematica

It is shown that every monocompact submeasure on an orthomodular poset is order continuous. From this generalization of the classical Marczewski Theorem, several results of commutative Measure Theory are derived and unified.

A nonlinear Banach-Steinhaus theorem and some meager sets in Banach spaces

Jacek Jachymski (2005)

Studia Mathematica

We establish a Banach-Steinhaus type theorem for nonlinear functionals of several variables. As an application, we obtain extensions of the recent results of Balcerzak and Wachowicz on some meager subsets of L¹(μ) × L¹(μ) and c₀ × c₀. As another consequence, we get a Banach-Mazurkiewicz type theorem on some residual subset of C[0,1] involving Kharazishvili's notion of Φ-derivative.

A Non-Probabilistic Proof of the Assouad Embedding Theorem with Bounds on the Dimension

Guy David, Marie Snipes (2013)

Analysis and Geometry in Metric Spaces

We give a non-probabilistic proof of a theorem of Naor and Neiman that asserts that if (E, d) is a doubling metric space, there is an integer N > 0, depending only on the metric doubling constant, such that for each exponent α ∈ (1/2; 1), one can find a bilipschitz mapping F = (E; dα ) ⃗ ℝ RN.

A note on almost strong liftings

C. Ionescu-Tulcea, R. Maher (1971)

Annales de l'institut Fourier

Let X be a locally compact space. A lifting ρ of M R ∞ ( X , μ ) where μ is a positive measure on X , is almost strong if for each bounded, continuous function f , ρ ( f ) and f coincide locally almost everywhere. We prove here that the set of all measures μ on X such that there exists an almost strong lifting of M R ∞ ( X , | μ | ) is a band.

A note on almost sure convergence and convergence in measure

P. Kříž, Josef Štěpán (2014)

Commentationes Mathematicae Universitatis Carolinae

The present article studies the conditions under which the almost everywhere convergence and the convergence in measure coincide. An application in the statistical estimation theory is outlined as well.

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