The generalized Riemann integral of Pfeffer (1991) is defined on all bounded $\mathrm{BV}$ subsets of ${ℝ}^n$, but it is additive only with respect to pairs of disjoint sets whose closures intersect in a set of $\sigma$-finite Hausdorff measure of codimension one. Imposing a stronger regularity condition on partitions of $\mathrm{BV}$ sets, we define a Riemann-type integral which satisfies the usual additivity condition and extends the integral of Pfeffer. The new integral is lipeomorphism-invariant and closed with respect to the formation...

Let X be a complete metric space and write (X) for the family of all Borel probability measures on X. The local dimension $di{m}_{loc}(\mu ;x)$ of a measure μ ∈ (X) at a point x ∈ X is defined by $di{m}_{loc}(\mu ;x)=li{m}_{r\searrow 0}\left(log\mu \left(B(x,r)\right)\right)/\left(logr\right)$ whenever the limit exists, and plays a fundamental role in multifractal analysis. It is known that if a measure μ ∈ (X) satisfies a few general conditions, then the local dimension of μ exists and is equal to a constant for μ-a.a. x ∈ X. In view of this, it is natural to expect that for a fixed x ∈ X, the local dimension...

We prove that every Sobolev function defined on a metric space coincides with a Hölder continuous function outside a set of small Hausdorff content or capacity. Moreover, the Hölder continuous function can be chosen so that it approximates the given function in the Sobolev norm. This is a generalization of a result of Malý [Ma1] to the Sobolev spaces on metric spaces [H1].

We classify all homeomorphisms of the double cover of the Sierpiński gasket in n dimensions. We show that there is a unique homeomorphism mapping any cell to any other cell with prescribed mapping of boundary points, and any homeomorphism is either a permutation of a finite number of topological cells or a mapping of infinite order with one or two fixed points. In contrast we show that any compact fractafold based on the level-3 Sierpiński gasket is topologically rigid.

A class of subsets of ℝⁿ is constructed that have certain homogeneity and non-coincidence properties with respect to Hausdorff and box dimensions. For each triple (r,s,t) of numbers in the interval (0,n] with r < s < t, a compact set K is constructed so that for any non-empty subset U relatively open in K, we have $(di{m}_H\left(U\right),{\underline{dim}}_B\left(U\right),{\overline{dim}}_B\left(U\right))=(r,s,t)$. Moreover, $2^{-n}\le H^r\left(K\right)\le 2n^{r/2}$.

We show that almost every function (in the sense of prevalence) in a Sobolev space is multifractal: Its regularity changes from point to point; the sets of points with a given Hölder regularity are fractal sets, and we determine their Hausdorff dimension.

Subadditivity of capacities is defined initially on the compact sets and need not extend to all sets. This paper explores to what extent subadditivity holds. It presents some incidental results that are valid for all subadditive capacities. The main result states that for all hull-additive capacities (a class that contains the strongly subadditive capacities) there is countable subadditivity on a class at least as large as the universally measurable sets (so larger than the analytic sets).

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...