Miroslav Zelený

Descriptive set Theory of Families of Small Sets

This is a survey paper on the descriptive set theory of hereditary families of closed sets in Polish spaces. Most of the paper is devoted to ideals and σ-ideals of closed or compact sets.

The Dynkin system generated by balls in ℝ^{} contains all Borel sets

We show that for every d ∈ N each Borel subset of the space Rd with the Euclidean metric can be generated from closed balls by complements and countable disjoint unions. Let X be a nonempty set and S ⊂ 2 . Following [B, p. 8] we say that S is a Dynkin system if (D1) X ∈ S, (D2) A ∈ S ⇒ X \A ∈ S, (D3) if An ∈ S are pairwise disjoint, then ⋃∞ n=1 An ∈ S. Some authors use the name σ-class instead of Dynkin system. The smallest Dynkin system containing a system T ⊂ 2 is denoted by D(T ). Let P be a metric space. The system of all closed balls in P (of all Borel subsets of P , respectively) will be denoted by Balls(P ) (Borel(P ), respectively). We will deal with the problem of whether D(Balls(P )) = Borel(P ). (?) One motivation for such a problem comes from measure theory. Let μ and ν be finite Radon measures on a metric space P having the same values on each ball. Is it true that μ = ν? If D(Balls(P )) = Borel(P ), then obviously μ = ν. If P is a Banach space, then μ = ν again (Preiss, Tǐser [PT]). But Preiss and Keleti ([PK]) showed recently that (?) is false in infinite-dimensional Hilbert spaces. We prove the following result. Theorem 1. Let d ∈ N, and let R be equipped with the Euclidean metric. Then D(Balls(R)) = Borel(R). This theorem was partially proved by Olejček ([O]), who proved it for d = 2, 3 (the case d = 1 is easy and well-known). Several of Olejček’s ideas will also be used in our proof of the general statement. In fact, we prove a slightly more general result: Each Borel subset of the space R (with the Euclidean metric) can be generated from closed balls by countable monotone unions, countable monotone intersections and countable disjoint unions. Received by the editors February 11, 1998. 1991 Mathematics Subject Classification. Primary 28A05, 04A15. This research was supported by Research Grant GAUK 190/1996 and GAČR 201/97/1161. c ©1999 American Mathematical Society 433 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

On separable determination of σ-P-porous sets in Banach spaces

Abstract We use a method involving elementary submodels and a partial converse of Foran lemma to prove separable reduction theorems concerning Souslin σ - P -porous sets where P can be from a rather wide class of porosity-like relations in complete metric spaces. In particular, we separably reduce the notion of Souslin cone small set in Asplund spaces. As an application we prove that a continuous approximately convex function on an Asplund space is Frechet differentiable up to a cone small set.

Inscribing closed non-σ-lower porous sets into Suslinnon-σ-lower porous sets

The main aim of this paper is to prove that every non-σ-lower porous Suslin set in a topologically complete metric space contains a closed non-σ-lower porous subset. In fact, we prove a general result of this type on “abstract porosities.” This general theorem is also applied to ball small sets in Hilbert spaces and to σ-cone-supported sets in separable Banach spaces.

On proper Shapley values for monotone TU-games

The Shapley value of a cooperative transferable utility game distributes the dividend of each coalition in the game equally among its members. Given exogenous weights for all players, the corresponding weighted Shapley value distributes the dividends proportionally to their weights. A proper Shapley value, introduced in Vorob’ev and Liapounov (Game Theory and Applications, vol IV. Nova Science, New York, pp 155–159, 1998), assigns weights to players such that the corresponding weighted Shapley value of each player is equal to her weight. In this contribution we investigate these proper Shapley values in the context of monotone games. We prove their existence for all monotone transferable utility games and discuss other properties of this solution.

Additive Families of Low Borel Classes and Borel Measurable Selectors

An important conjecture in the theory of Borel sets in non-separable metric spaces is whether any point-countable Borel-additive family in a complete metric space has a σ-discrete refinement. We confirm the conjecture for point-countable Π 0 3 -additive families, thus generalizing results of R. W. Hansell and the first author. We apply this result to the existence of Borel measurable selectors for multivalued mappings of low Borel complexity, thus answering in the affirmative a particular version of a question of J. Kaniewski and R. Pol. Faculty of Mathematics and Physics, Charles University, Sokolovska 83, Czech Republic e-mail: spurny@karlin.mff.cuni.cz zeleny@karlin.mff.cuni.cz Received by the editors January 24, 2008. Published electronically August 3, 2010. The work is a part of the research project MSM 0021620839 financed by MSMT and partly supported by the grants GA CR 201/06/0198, GA CR 201/06/0018, and GA CR 201/07/0388. AMS subject classification: 54H05, 54E35.

On singular boundary points of complex functions

Let f be a complex valued function from the open upper halfplane E of the complex plane. We study the set of all z∈∂E such that there exist two Stoltz angles V 1 , V 2 in E with vertices in z ( i.e. , V i is a closed angle with vertex at z and V i \{ z } ⊂ E , i = 1, 2) such that the function f has different cluster sets with respect to these angles at z . E. P. Dolzhenko showed that this set of singular points is G ∂σ and σ-porous for every f . He posed the question of whether each G ∂σ σ-porous set is a set of such singular points for some f . We answer this question negatively. Namely, we construct a G ∂ porous set, which is a set of such singular points for no function f .

Universal and complete sets in martingale theory

The Doob convergence theorem implies that the set of divergence of any martingale has measure zero. We prove that, conversely, any

A remark on the Debs–Saint-Raymond theorem

G\_{\delta\sigma}

A note on intersections of non-Haar null sets

subset of the Cantor space with Lebesgue-measure zero can be represented as the set of divergence of some martingale. In fact, this is effective and uniform. A consequence of this is that the set of everywhere converging martingales is