A conjecture of Ulam on the invariance of measures in Gilbert's cube
A continous version of Orlicz-Pettis theorem via vector-valued Henstock-Kurzweil integrals
We show that a Pettis integrable function from a closed interval to a Banach space is Henstock-Kurzweil integrable. This result can be considered as a continuous version of the celebrated Orlicz-Pettis theorem concerning series in Banach spaces.
A continuity property of the dimension of the harmonic measure of Cantor sets under perturbations
A continuous version of Liapunov's convexity theorem
A convergence theorem concerning real-valued interval functions.
A Convergence Theorem for Measures in Regular Hausdorff Spaces.
A converse of the Arsenin–Kunugui theorem on Borel sets with σ-compact sections
Let f be a Borel measurable mapping of a Luzin (i.e. absolute Borel metric) space L onto a metric space M such that f(F) is a Borel subset of M if F is closed in L. We show that then is a set for all except countably many y ∈ M, that M is also Luzin, and that the Borel classes of the sets f(F), F closed in L, are bounded by a fixed countable ordinal. This gives a converse of the classical theorem of Arsenin and Kunugui. As a particular case we get Taĭmanov’s theorem saying that the image of...
A convolution inequality concerning Cantor-Lebesgue measures.
A covering property of some classes of sets in
A Criterion for a Process to Be Prime.
A criterion for pure unrectifiability of sets (via universal vector bundle)
Let m,n be positive integers such that m < n and let G(n,m) be the Grassmann manifold of all m-dimensional subspaces of ℝⁿ. For V ∈ G(n,m) let denote the orthogonal projection from ℝⁿ onto V. The following characterization of purely unrectifiable sets holds. Let A be an -measurable subset of ℝⁿ with . Then A is purely m-unrectifiable if and only if there exists a null subset Z of the universal bundle such that, for all P ∈ A, one has . One can replace “for all P ∈ A” by “for -a.e. P ∈...
A criterion for Toeplitz flows to be topologically isomorphic and applications
A dynamical system is said to be coalescent if its only endomorphisms are automorphisms. The question whether there exist coalescent ergodic dynamical systems with positive entropy has not been solved so far and it seems to be difficult. The analogous problem in topological dynamics has been solved by Walters ([W]). His example, however, is not minimal. In [B-K2], a class of strictly ergodic (hence minimal) Toeplitz flows is presented, which have positive entropy and trivial topological centralizers...
A cut salad of cocycles
We study the centraliser of locally compact group extensions of ergodic probability preserving transformations. New methods establishing ergodicity of group extensions are introduced, and new examples of squashable and non-coalescent group extensions are constructed.
A cylinder flow arising from irregularity of distribution
A generalization of the Davenport-Erdős construction of normal numbers
We extend the Davenport and Erdős construction of normal numbers to the case.
A Daniell integral approach to nonstandard Kurzweil-Henstock integral
A workable nonstandard definition of the Kurzweil-Henstock integral is given via a Daniell integral approach. This allows us to study the HL class of functions from . The theory is recovered together with a few new results.
A Decomposition of a Functional as a Difference of Two Positive Functionals
A Decomposition Theorem for Additive Set-Functions, with Applications to Pettis Integrals and Ergodic Means.
A derivative to match the v-integral.
A description of stochastic systems using chaotic maps.