On the uniqueness of Lebesgue and Borel measures.
On topologies generated by some operators
The paper concerns topologies introduced in a topological space (X, τ) by operators which are much weaker than the lower density operators. Some properties of the family of sets having the Baire property and the family of meager sets with respect to such topologies are investigated.
On transformations of sets in
On translation invariant families of sets
On Ulam's problem on families of measures
On uncountable unions and intersections of measurable sets.
On universal null and universally measurable sets, II
On vector sums of measure zero sets.
On Vitali's theorem for groups of motions of Euclidean space.
On Weakly Measurable Functions
We show that if T is an uncountable Polish space, 𝓧 is a metrizable space and f:T→ 𝓧 is a weakly Baire measurable function, then we can find a meagre set M ⊆ T such that f[T∖M] is a separable space. We also give an example showing that "metrizable" cannot be replaced by "normal".
On -discrete Borel mappings via quasi-metrics
On -porous sets in abstract spaces.
One-to one Carathéodory representation theorem for multifunctions with uncountable values
Opérations de Hausdorff itérées et réunions croissantes de compacts
In this paper, motivated by questions in Harmonic Analysis, we study the operation of (countable) increasing union, and show it is not idempotent: iterations are needed in general to obtain the closure of a class under this operation. Increasing union is a particular Hausdorff operation, and we present the combinatorial tools which allow to study the power of various Hausdorff operations, and of their iterates. Besides countable increasing union, we study in detail a related Hausdorff operation,...
Orbits of linear operators and Banach space geometry
Let T be a bounded linear operator on a (real or complex) Banach space X. If (aₙ) is a sequence of non-negative numbers tending to 0, then the set of x ∈ X such that ||Tⁿx|| ≥ aₙ||Tⁿ|| for infinitely many n’s has a complement which is both σ-porous and Haar-null. We also compute (for some classical Banach space) optimal exponents q > 0 such that for every non-nilpotent operator T, there exists x ∈ X such that , using techniques which involve the modulus of asymptotic uniform smoothness of X.
"Paradoxe" Zerlegung Euklidischer Räume.
Partitions of pairs of reals
Pointwise convergence and the Wadge hierarchy
We show that if is a separable metrizable space which is not -compact then , the space of bounded real-valued continuous functions on with the topology of pointwise convergence, is Borel--complete. Assuming projective determinacy we show that if is projective not -compact and is least such that is then , the space of real-valued continuous functions on with the topology of pointwise convergence, is Borel--complete. We also prove a simultaneous improvement of theorems of Christensen...
Possibly there is no uniformly completely Ramsey null set of size
We show that under the axiom there is no uniformly completely Ramsey null set of size . In particular, this holds in the iterated perfect set model. This answers a question of U. Darji.
Poznámka o lineární míře Vituškinových množin