The Heine-Borel theorem states that any open cover of the closed interval Œa; b contains a nite subcover. We give a new, constructive proof inspired by the concept of a greedy algorithm. In the case when the initial cover consists only of intervals, our proof is optimal in that it constructs a nite subcover with the fewest possible number of elements. Our work leaves open a number of questions...

Let X ⊆ 2 . Consider the class of all Borel F ⊆ X × 2 with null vertical sections Fx, x ∈ X. We show that if for all such F and all null Z ⊆ X, ⋃ x∈Z Fx is null, then for all such F , ⋃ x∈X Fx 6= 2 . The theorem generalizes the fact that every Sierpiński set is strongly meager and was announced in [P]. A Sierpiński set is an uncountable subset of 2 which meets every null (i.e., measure zero) se...

In recent years, much work in descriptive set theory has been focused on the Borel complexity of naturally occurring classification problems, in particular, the study of countable Borel equivalence relations and their structure under the quasi-order of Borel reducibility. Following the approach of Louveau and Rosendal for the study of analytic equivalence relations, we study countable Borel qua...

We present a theorem which generalizes some known theorems on the existence of nonmeasurable (in various senses) sets of the form X+Y . Some additional related questions concerning measure, category and the algebra of Borel sets are also studied. Sierpiński showed in [14] that there exist two sets X, Y ⊆ R of Lebesgue measure zero such that their algebraic sum, i.e. the set X + Y = {x + y : x ∈...

Consider a standard probability space (X,μ), i.e., a space isomorphic to the unit interval with Lebesgue measure. We denote by Aut(X,μ) the automorphism group of (X,μ), i.e., the group of all Borel automorphisms of X which preserve μ (where two such automorphisms are identified if they are equal μ-a.e.). A Borel equivalence relation E ⊆ X is called countable if every E-class [x]E is countable a...

We show that there is a complete, consistent Borel theory which has no “Borel model” in the following strong sense: There is no structure satisfying the theory for which the elements of the structure are equivalence classes under some Borel equivalence relation and the interpretations of the relations and function symbols are Borel in the natural sense. We also investigate Borel isomorphisms be...

We study the surjectivity of, and existence of right inverses for, asymptotic Borel map in Carleman-Roumieu ultraholomorphic classes defined by regular sequences sense E. M. Dyn'kin. extend previous results J. Schmets Valdivia, V. Thilliez, authors, show prominent role played an index associated with sequence introduced Thilliez. The techniques involve variation, integral transforms characteriz...

We analyze the technique used by Adams and Kechris (2000) to obtain their results about Borel reducibility of countable Borel equivalence relations. Using this technique, we show that every Σ1 equivalence relation is Borel reducible to the Borel bi-reducibility of countable Borel equivalence relations. We also apply the technique to two other classes of essentially uncountable Borel equivalence...

In this supplement, we provide the axiomatic characterization of two representations used in the main paper (henceforth denoted AS). In Section S1, we show that with only slight modification of the axioms, the representation theorem from Dekel, Lipman, and Rustichini (2001) can be adapted to our setting of preferences over finite menus of lotteries. In Section S2, we show that the representatio...

It is well known due to Hahn and Mazurkiewicz that every Peano continuum a continuous image of the unit interval. We prove an assignment, which takes as input produces output mapping whose range continuum, can be realized in Borel measurable way. Similarly, we find assignment any nonempty compact metric space assigns from Cantor set onto space. To this end use Burgess selection theorem. Finally...