Les propriétés de réduction et de norme pour les classes de Boréliens
Lipschitz-quotients and the Kunen-Martin Theorem
We show that there is a universal control on the Szlenk index of a Lipschitz-quotient of a Banach space with countable Szlenk index. It is in particular the case when two Banach spaces are Lipschitz-homeomorphic. This provides information on the Cantor index of scattered compact sets and such that is a Lipschitz-quotient of (that is the case in particular when these two spaces are Lipschitz-homeomorphic). The proof requires tools of descriptive set theory.
MAD families with strong combinatorial properties
In his paper in Fund. Math. 178 (2003), Miller presented two conjectures regarding MAD families. The first is that CH implies the existence of a MAD family that is also a σ-set. The second is that under CH, there is a MAD family concentrated on a countable subset. These are proved in the present paper.
Marczewski sets, measure and the Baire property
Martin's axiom and medial functions.
Measurability of equivalence classes and MEC-property in metric spaces.
Measurability of some sets of Borel measurable functions on .
Measurable spaces with c.c.c. [Book]
Measure-additive coverings and measurable selectors [Book]
Measure-theoretic unfriendly colorings
We consider the problem of finding a measurable unfriendly partition of the vertex set of a locally finite Borel graph on standard probability space. After isolating a sufficient condition for the existence of such a partition, we show how it settles the dynamical analog of the problem (up to weak equivalence) for graphs induced by free, measure-preserving actions of groups with designated finite generating set. As a corollary, we obtain the existence of translation-invariant random unfriendly colorings...
Mesurabilité des débuts et théorème de section : le lot à la portée de toutes les bourses
Metastability in the Furstenberg-Zimmer tower
According to the Furstenberg-Zimmer structure theorem, every measure-preserving system has a maximal distal factor, and is weak mixing relative to that factor. Furstenberg and Katznelson used this structural analysis of measure-preserving systems to provide a perspicuous proof of Szemerédi’s theorem. Beleznay and Foreman showed that, in general, the transfinite construction of the maximal distal factor of a separable measure-preserving system can extend arbitrarily far into the countable ordinals....
Metric spaces admitting only trivial weak contractions
If (X,d) is a metric space then a map f: X → X is defined to be a weak contraction if d(f(x),f(y)) < d(x,y) for all x,y ∈ X, x ≠ y. We determine the simplest non-closed sets X ⊆ ℝⁿ in the sense of descriptive set-theoretic complexity such that every weak contraction f: X → X is constant. In order to do so, we prove that there exists a non-closed set F ⊆ ℝ such that every weak contraction f: F → F is constant. Similarly, there exists a non-closed set G ⊆ ℝ such that every weak contraction...
Minimality in the -degrees
Mixing actions of groups with the Haagerup approximation property
A countable group Γ has the Haagerup approximation property if and only if the mixing actions are dense in the space of all actions of Γ.
Multiple gaps
We study a higher-dimensional version of the standard notion of a gap formed by a finite sequence of ideals of the quotient algebra 𝓟(ω)/fin. We examine different types of such objects found in 𝓟(ω)/fin both from the combinatorial and the descriptive set-theoretic side.
Non-existence of certain Borel structures
Non-Glimm–Effros equivalence relations at second projective level
A model is presented in which the equivalence relation xCy iff L[x]=L[y] of equiconstructibility of reals does not admit a reasonable form of the Glimm-Effros theorem. The model is a kind of iterated Sacks generic extension of the constructible model, but with an “ill“founded “length” of the iteration. In another model of this type, we get an example of a non-Glimm-Effros equivalence relation on reals. As a more elementary application of the technique of “ill“founded Sacks iterations, we obtain...
Nonmeasurable algebraic sums of sets of reals
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.
Normal numbers and subsets of N with given densities
For X ⊆ [0,1], let denote the collection of subsets of ℕ whose densities lie in X. Given the exact location of X in the Borel or difference hierarchy, we exhibit the exact location of . For α ≥ 3, X is properly iff is properly . We also show that for every nonempty set X ⊆[0,1], is -hard. For each nonempty set X ⊆ [0,1], in particular for X = x, is -complete. For each n ≥ 2, the collection of real numbers that are normal or simply normal to base n is -complete. Moreover, , the...