A class of sets whose distance set fills an interval
A classification of points on the Sierpinski gasket.
A combinatorial theorem on the existence of a separating element and its applications to sequences and -derivations of measures
A common generalization of Kelley's theorem (concerning measures on Boolean algebra) and on von Neumann's minimax theorem
A complete characterization of R-sets in the theory of differentiation of integrals
Let be the family of open rectangles in the plane ℝ² with a side of angle s to the x-axis. We say that a set S of directions is an R-set if there exists a function f ∈ L¹(ℝ²) such that the basis differentiates the integral of f if s ∉ S, and almost everywhere if s ∈ S. If the condition holds on a set of positive measure (instead of a.e.) we say that S is a WR-set. It is proved that S is an R-set (resp. a WR-set) if and only if it is a (resp. a ).
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 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 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 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 Dieudonné theorem for lattice group-valued measures
A version of Dieudonné theorem is proved for lattice group-valued modular measures on lattice ordered effect algebras. In this way we generalize some results proved in the real-valued case.
A discrete version of an open problem and several answers.
A dominated convergence theorem for real-valued summable set functions