Related papers: Infinite measures on Cantor spaces

We study algorithmic randomness properties for probability measures on Cantor space. We say that a measure $\mu$ on the space of infinite bit sequences is ML absolutely continuous if the non-ML-random bit sequences form a null set with…

Let $M$ be a compact $n$-dimensional Riemanian manifold, End($M$) the set of the endomorphisms of $M$ with the usual $\mathcal{C}^0$ topology and $\phi: M\to\mathbb{R}$ continuous. We prove that there exists a dense subset of $\mathcal{A}$…

Let $(X, d)$ be a compact metric space and let $\mathcal{M}(X)$ denote the space of all finite signed Borel measures on $X$. Define $I \colon \mathcal{M}(X) \to \R$ by \[ I(\mu) = \int_X \int_X d(x,y) d\mu(x) d\mu(y), \] and set $M(X) =…

Let $\mathcal{M}(X,\mathcal{A})$ be the ring of all real valued measurable functions defined over the measurable space $(X,\mathcal{A})$. Given an ideal $I$ in $\mathcal{M}(X,\mathcal{A})$ and a measure $\mu:\mathcal{A}\to[0,\infty]$, we…

Let $P$ be a Borel probability measure on $\mathbb R^2$ supported by the Cantor dusts generated by a set of $4^u,\ u\geq 1$, contractive similarity mappings satisfying the strong separation condition. For this probability measure, we…

For a large class of Cantor sets on the real-line, we find sufficient and necessary conditions implying that a set has positive (resp. null) measure for all doubling measures of the real-line. We also discuss same type of questions for…

Given a surface $M$ and a Borel probability measure $\nu$ on the group of $C^2$-diffeomorphisms of $M$, we study $\nu$-stationary probability measures on $M$. Assuming the positivity of a certain entropy, the following dichotomy is proved:…

In this paper we define a new class of metric spaces, called multi-model Cantor sets. We compute the Hausdorff dimension and show that the Hausdorff measure of a multi-model Cantor set is finite and non-zero. We then show that a bilipschitz…

Let $X$ be a locally compact Polish space. Let $\mathbb K(X)$ denote the space of discrete Radon measures on $X$. Let $\mu$ be a completely random discrete measure on $X$, i.e., $\mu$ is (the distribution of) a completely random measure on…

The structure of the set of local dimensions of a self-similar measure has been studied by numerous mathematicians, initially for measures that satisfy the open set condition and, more recently, for measures on $\mathbb{R}$ that are of…

Let mu(r) be the Bernoulli measure on the Cantor space given as the infinite product of two-point measures with weights r and 1-r. It is a long-standing open problem to characterize those r and s such that mu(r) and mu(s) are topologically…

In this paper we consider nonmeasurablity with respect to sigma-ideals defined be trees. First classical example of such ideal is Marczewski ideal s_0. We will consider also ideal l_0 defined by Laver trees and m_0 defined by Miller trees.…

A general theory of resource-bounded measurability and measure is developed. Starting from any feasible probability measure $\nu$ on the Cantor space $\C$ and any suitable complexity class $C \subseteq \C$, the theory identifies the subsets…

We obtain characterizations of positive Borel measures $\mu$ on $\B^n$ so that some weighted Hardy-Sobolev are imbedded in $L^p(d\mu)$, where $w$ is an $A_p$ weight in the unit sphere of $\C^n$.

Let $\{f_i\}_{i=1}^N$ be a set of equi-contractive similitudes on $\mathbb{R}^1$ satisfying the finite-type condition. We study the asymptotic quantization error for self-similar measures $\mu$ associated with $\{f_i\}_{i=1}^N$ and a…

A measure $\mu$ on the unit circle $\mathbb{T}$ belongs to Steklov class $\mathcal{S}$ if its density $w$ with respect to the Lebesgue measure on $\mathbb{T}$ is strictly positive: $\inf_{\mathbb{T}} w > 0$. Let $\mu$, $\mu_{-1}$ be…

While finite non-commutative operator systems lie at the foundation of quantum measurement, they are also tools for understanding geometric iterations as used in the theory of iterated function systems (IFSs) and in wavelet analysis. Key is…

We investigate the traceability of positive integral operators on $L^2(X,\mu)$ when $X$ is a Hausdorff locally compact second countable space and $\mu$ is a non-degenerate, $\sigma$-finite and locally finite Borel measure. This setting…

For a measurable space ($X,\mathcal{A}$), let $\mathcal{M}(X,\mathcal{A})$ be the corresponding ring of all real valued measurable functions and let $\mu$ be a measure on ($X,\mathcal{A}$). In this paper, we generalize the so-called…

In this paper, we study the spectrality of a class of Moran measures $\mu_{\mathcal{P},\mathcal{D}}$ on $\mathbb{R}$ generated by $\{(p_n,\mathcal{D}_n)\}_{n=1}^{\infty}$, where $\mathcal{P}=\{p_n\}_{n=1}^{\infty}$ is a sequence of positive…