On the Existence of Pathological Submeasures and the Construction of Exotic Topological Groups.
On the Extension of a Measure on Lattices
On the extension of measures.
We give necessary and sufficient conditions for a totally ordered by extension family (Ω, Σx, μx)x ∈ X of spaces of probability to have a measure μ which is an extension of all the measures μx. As an application we study when a probability measure on Ω has an extension defined on all the subsets of Ω.
On the Extension of Measures in Relatively Complemented Lattices
On the Extremality of measure extensions.
On the extremality of regular extensions of contents and measures
Let be an algebra and a lattice of subsets of a set . We show that every content on that can be approximated by in the sense of Marczewski has an extremal extension to a -regular content on the algebra generated by and . Under an additional assumption, we can also prove the existence of extremal regular measure extensions.
On the Functional Equation f(x)g(y) = ...hi(aix + biy).
On the geometric structure of the limit set of conformal iterated function systems.
On the Hausdorff dimension of a family of self-similar sets with complicated overlaps
We investigate the properties of the Hausdorff dimension of the attractor of the iterated function system (IFS) {γx,λx,λx+1}. Since two maps have the same fixed point, there are very complicated overlaps, and it is not possible to directly apply known techniques. We give a formula for the Hausdorff dimension of the attractor for Lebesgue almost all parameters (γ,λ), γ < λ. This result only holds for almost all parameters: we find a dense set of parameters (γ,λ) for which the Hausdorff dimension...
On the Hausdorff Dimension of CAT(κ) Surfaces
We prove that a closed surface with a CAT(κ) metric has Hausdorff dimension = 2, and that there are uniform upper and lower bounds on the two-dimensional Hausdorff measure of small metric balls. We also discuss a connection between this uniformity condition and some results on the dynamics of the geodesic flow for such surfaces. Finally,we give a short proof of topological entropy rigidity for geodesic flow on certain CAT(−1) manifolds.
On the Hausdorff dimension of certain self-affine sets
A subset E of ℝⁿ is called self-affine with respect to a collection ϕ₁,...,ϕₜ of affinities if E is the union of the sets ϕ₁(E),...,ϕₜ(E). For S ⊂ ℝⁿ let . If Φ(S) ⊂ S let denote . For given Φ consisting of contracting “pseudo-dilations” (affinities which preserve the directions of the coordinate axes) and subject to further mild technical restrictions we show that there exist self-affine sets of each Hausdorff dimension between zero and a positive number depending on Φ. We also investigate...
On the Hausdorff Dimension of Rademacher Riesz Products.
On the Hausdorff dimension of some fractal sets
On the ideal (v 0)
The σ-ideal (v 0) is associated with the Silver forcing, see [5]. Also, it constitutes the family of all completely doughnut null sets, see [9]. We introduce segment topologies to state some resemblances of (v 0) to the family of Ramsey null sets. To describe add(v 0) we adopt a proof of Base Matrix Lemma. Consistent results are stated, too. Halbeisen’s conjecture cov(v 0) = add(v 0) is confirmed under the hypothesis t = min{cf(c), r}. The hypothesis cov(v 0) = ω 1 implies that (v 0) has the ideal...
On the infimum convolution inequality
We study the infimum convolution inequalities. Such an inequality was first introduced by B. Maurey to give the optimal concentration of measure behaviour for the product exponential measure. We show how IC inequalities are tied to concentration and study the optimal cost functions for an arbitrary probability measure μ. In particular, we prove an optimal IC inequality for product log-concave measures and for uniform measures on the balls. Such an optimal inequality implies, for a given measure,...
On the inner Daniell-Stone and Riesz representation theorems.
On the instability of Hausdorff content.
On the Integration of Radon Measures and Set Functions.
On the integration of set-valued mappings in a Banach space
On the integration on -classes