We prove that for every Borel ideal, the ideal limits of sequences of continuous functions on a Polish space are of Baire class one if and only if the ideal does not contain a copy of Fin × Fin. In particular, this is true for $F_{\sigma \delta }$ ideals. In the proof we use Borel determinacy for a game introduced by C. Laflamme.

Let $X=X\left(t\right),t\in {ℝ}^N$ be a Gaussian random field in ${ℝ}^d$ with stationary increments. For any Borel set $E\subset {ℝ}^N$, we provide sufficient conditions for the image X(E) to be a Salem set or to have interior points by studying the asymptotic properties of the Fourier transform of the occupation measure of X and the continuity of the local times of X on E, respectively. Our results extend and improve the previous theorems of Pitt [24] and Kahane [12,13] for fractional Brownian motion.

Having Polish spaces $𝕏$, $𝕐$ and $\mathbb{Z}$ we shall discuss the existence of an $𝕏\times 𝕐$-valued random vector $(\xi ,\eta )$ such that its conditional distributions $K_x=ℒ(\eta \mid \xi =x)$ satisfy $e(x,K_x)=c\left(x\right)$ or $e(x,K_x)\in C\left(x\right)$ for some maps $e:𝕏\times {ℳ}_1\left(𝕐\right)\to \mathbb{Z}$, $c:𝕏\to \mathbb{Z}$ or multifunction $C:𝕏\to 2^{\mathbb{Z}}$ respectively. The problem is equivalent to the existence of universally measurable Markov kernel $K:𝕏\to {ℳ}_1\left(𝕐\right)$ defined implicitly by $e(x,K_x)=c\left(x\right)$ or $e(x,K_x)\in C\left(x\right)$ respectively. In the paper we shall provide sufficient conditions for the existence of the desired Markov kernel. We shall discuss some special solutions of the $(e,c)$- or $(e,C)$-problem and illustrate...

We prove that a compact family of bounded condensing multifunctions has bounded condensing set-theoretic union. Compactness is understood in the sense of the Chebyshev uniform semimetric induced by the Hausdorff distance and condensity is taken w.r.t. the Hausdorff measure of noncompactness. As a tool, we present an estimate for the measure of an infinite union. Then we apply our result to infinite iterated function systems.

Let α be an irrational and φ: ℕ → ℝ⁺ be a function decreasing to zero. Let $\omega \left(\alpha \right):=sup\theta \ge 1:limin{f}_{n\to \infty }{n}^{\theta }\left|\right|n\alpha |=0$$.Forany\alpha ...$

We investigate the box dimensions of inhomogeneous self-similar sets. Firstly, we extend some results of Olsen and Snigireva by computing the upper box dimensions assuming some mild separation conditions. Secondly, we investigate the more difficult problem of computing the lower box dimension. We give some non-trivial bounds and provide examples to show that lower box dimension behaves much more strangely than upper box dimension, Hausdorff dimension and packing dimension.

The main aim of this paper is to give a simpler proof of the following assertion. Let A be an analytic non-σ-porous subset of a locally compact metric space, E. Then there exists a compact non-σ-porous subset of A. Moreover, we prove the above assertion also for σ-P-porous sets, where P is a porosity-like relation on E satisfying some additional conditions. Our result covers σ-⟨g⟩-porous sets, σ-porous sets, and σ-symmetrically porous sets.

A family of subsets of a set is called a $\sigma$-topology if it is closed under arbitrary countable unions and arbitrary finite intersections. A $\sigma$-topology is perfect if any its member (open set) is a countable union of complements of open sets. In this paper perfect $\sigma$-topologies are characterized in terms of inserting lower and upper measurable functions. This improves upon and extends a similar result concerning perfect topologies. Combining this characterization with a $\sigma$-topological version of Katětov-Tong...