Let $(Z,T_Z)$ be a minimal non-periodic flow which is either symbolic or strictly ergodic. Any topological extension of $(Z,T_Z)$ is Borel isomorphic to an almost 1-1 extension of $(Z,T_Z)$. Moreover, this isomorphism preserves the affine-topological structure of the invariant measures. The above extends a theorem of Furstenberg-Weiss (1989). As an application we prove that any measure-preserving transformation which admits infinitely many rational eigenvalues is measure-theoretically isomorphic to a strictly ergodic toeplitz...

Let (X, μ) be a σ-finite measure space and let τ be an ergodic invertible measure preserving transformation. We study the a.e. convergence of the Cesàro-α ergodic averages associated with τ and the boundedness of the corresponding maximal operator in the setting of Lp,q(wdμ) spaces.

It is well-known that a probability measure $\mu$ on the circle $𝕋$ satisfies $\parallel {\mu }^n{*f-\int f\mathrm{d}m\parallel }_p\to 0$ for every $f\in L_p$, every (some) $p\in [1,\infty )$, if and only if $|\widehat{\mu }\left(n\right)|lt;1$ for every non-zero $n\in \mathbb{Z}$ ($\mu$ is strictly aperiodic). In this paper we study the a.e. convergence of ${\mu }^n*f$ for every $f\in L_p$ whenever $pgt;1$. We prove a necessary and sufficient condition, in terms of the Fourier–Stieltjes coefficients of $\mu$, for the strong sweeping out property (existence of a Borel set $B$ with $lim\; sup{\mu }^n*1_B=1$ a.e. and $lim\; inf{\mu }^n*1_B=0$ a.e.). The results are extended to general compact Abelian groups $G$ with Haar...

Bellow and Calderón proved that the sequence of convolution powers ${\mu }_{n}f\left(x\right)={\sum }_{k\in \mathbb{Z}}{\mu }^n\left(k\right)f\left(T^{k}x\right)$ converges a.e, when $\mu$ is a strictly aperiodic probability measure on $\mathbb{Z}$ such that the expectation is zero, $E\left(\mu \right)=0$, and the second moment is finite, $m_2\left(\mu \right)\<\infty$. In this paper we extend this result to cases where $m_2\left(\mu \right)=\infty$.

Let T be a d×d matrix with integer entries and with eigenvalues >1 in modulus. Let f be a lipschitzian function of positive order. We prove that the series ${\sum }_{n=1}^{\infty }{c}_{n}f\left(T^{n}x\right)$ converges almost everywhere with respect to Lebesgue measure provided that ${\sum }_{n=1}^{\infty }{\left|c_n\right|}^{2}lo{g}^{2}n<\infty$.

We present a new characterization of Lebesgue measurable functions; namely, a function f:[0,1]→ ℝ is measurable if and only if it is first-return recoverable almost everywhere. This result is established by demonstrating a connection between almost everywhere first-return recovery and a first-return process for yielding the integral of a measurable function.

We prove an extension of a result by Peres and Solomyak on almost sure absolute continuity in a class of symmetric Bernoulli convolutions.

We show that whenever the $q$-dimensional Minkowski content of a subset $A\subset {ℝ}^d$ exists and is finite and positive, then the “S-content” defined analogously as the Minkowski content, but with volume replaced by surface area, exists as well and equals the Minkowski content. As a corollary, we obtain the almost sure asymptotic behaviour of the surface area of the Wiener sausage in ${ℝ}^d$, $d\ge 3$.

The aim of this paper is to survey and discuss, very briefly, some ways how to introduce, within the framework of possibilistic measures, a notion analogous to that of conditional probability measure in probability theory. The adjective “analogous” in the last sentence is to mean that the conditional possibilistic measures should play the role of a mathematical tool to actualize one’s degrees of beliefs expressed by an a priori possibilistic measure, having obtained some further information concerning...