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Universal measure zero, large Hausdorff dimension, and nearly Lipschitz maps

Ondřej Zindulka (2012)

Fundamenta Mathematicae

We prove that each analytic set in ℝⁿ contains a universally null set of the same Hausdorff dimension and that each metric space contains a universally null set of Hausdorff dimension no less than the topological dimension of the space. Similar results also hold for universally meager sets. An essential part of the construction involves an analysis of Lipschitz-like mappings of separable metric spaces onto Cantor cubes and self-similar sets.

Universally measurable sets in generic extensions

Paul Larson, Itay Neeman, Saharon Shelah (2010)

Fundamenta Mathematicae

A subset of a topological space is said to be universally measurable if it is measured by the completion of each countably additive σ-finite Borel measure on the space, and universally null if it has measure zero for each such atomless measure. In 1908, Hausdorff proved that there exist universally null sets of real numbers of cardinality ℵ₁, and thus that there exist at least 2 ℵ ₁ such sets. Laver showed in the 1970’s that consistently there are just continuum many universally null sets of reals....

Upper envelopes of inner premeasures

Heinz König (2000)

Annales de l'institut Fourier

The paper resumes one of the themes initiated in the final sections of the celebrated “Theory of Capacities” of Choquet 1953-54. It aims at comprehensive versions in the spirit of the author’s recent work in measure and integration.

Upper Estimate of Concentration and Thin Dimensions of Measures

H. Gacki, A. Lasota, J. Myjak (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

We show upper estimates of the concentration and thin dimensions of measures invariant with respect to families of transformations. These estimates are proved under the assumption that the transformations have a squeezing property which is more general than the Lipschitz condition. These results are in the spirit of a paper by A. Lasota and J. Traple [Chaos Solitons Fractals 28 (2006)] and generalize the classical Moran formula.

Upper estimates on self-perimeters of unit circles for gauges

Horst Martini, Anatoliy Shcherba (2016)

Colloquium Mathematicae

Let M² denote a Minkowski plane, i.e., an affine plane whose metric is a gauge induced by a compact convex figure B which, as a unit circle of M², is not necessarily centered at the origin. Hence the self-perimeter of B has two values depending on the orientation of measuring it. We prove that this self-perimeter of B is bounded from above by the four-fold self-diameter of B. In addition, we derive a related non-trivial result on Minkowski planes whose unit circles are quadrangles.

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