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 $\Phi \left(S\right)={\bigcup }_{1\le j\le t}{\varphi }_j\left(S\right)$. If Φ(S) ⊂ S let $E_{\Phi }\left(S\right)$ denote ${\bigcap }_{k\ge 0}{\Phi }^k\left(S\right)$. 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 $E_{\Phi }\left(S\right)$ of each Hausdorff dimension between zero and a positive number depending on Φ. We also investigate...

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...

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 $\ell {ⁿ}_p$ balls. Such an optimal inequality implies, for a given measure,...

I. Kluvánek suggested to built the Lebesgue integral on a compact interval in the real line by the help of the length of intervals only. In the paper a modification of the Kluvánek construction is presented applicable to abstract spaces, too.

Let Ω be a countable infinite product $\Omega {₁}^{\mathbb{N}}$ of copies of the same probability space Ω₁, and let Ξₙ be the sequence of the coordinate projection functions from Ω to Ω₁. Let Ψ be a possibly nonmeasurable function from Ω₁ to ℝ, and let Xₙ(ω) = Ψ(Ξₙ(ω)). Then we can think of Xₙ as a sequence of independent but possibly nonmeasurable random variables on Ω. Let Sₙ = X₁ + ⋯ + Xₙ. By the ordinary Strong Law of Large Numbers, we almost surely have $E_{*}\left[X₁\right]\le liminfSₙ/n\le limsupSₙ/n\le E*\left[X₁\right]$, where $E_{*}$ and E* are the lower and upper expectations. We ask...

Si dimostra che il funzionale ${\int }_{\mathrm{\Omega }}f(u,Du)dx$ è semicontinuo inferiormente su $W_{loc}^{1,1}(\mathrm{\Omega })$, rispetto alla topologia indotta da $L_{loc}^1(\mathrm{\Omega })$, qualora l’integrando $f(s,p)$ sia una funzione non-negativa, misurabile in $s$, convessa in $p$, limitata nell’intorno dei punti del tipo $(s,0)$, e tale che la funzione $s\mapsto f(s,0)$ sia semicontinua inferiormente su $𝐑$.

We obtain a metrical property on the asymptotic behaviour of the maximal run-length function in the Lüroth expansion. We also determine the Hausdorff dimension of a class of exceptional sets of points whose maximal run-length function has sub-linear growth rate.