Seminarium Zakładu Funkcji Rzeczywistych
Seminar of the Division of Real Functions
Time and place: usually on Tuesdays at 10:15 in room 122
- 2019-10-15: Eliza Jabłońska (Pedagogical University of Cracow), Generically Haar-”small” sets in abelian Polish groups
- 2019-10-22: Adam Ostaszewski (London School of Economics, UK), Multivariate regular variation and Popa homomorphisms
Abstract: A subset A of an abelian Polish group X is called Haar-null if there are a Borel set B ⊃ A and a Borel probability measure μ on X such that μ(x + B) = 0 for all x ∈ X. In 2004 Dodos proved that for every Borel Haar-null set A ⊂ X the set of all test measures
is dense, coanalytic, and either meager or comeager in the space of all probabilistic Borel measures on X with Lévy-Prokhorov metric.
We prove a topological counterpart to Dodos’ result using Darji’s notion of a Haar-meager set. More precisely, we prove that for every Borel Haar-meager set A ⊂ X (i.e a set for which there is a continuous function f : 2ω → X such that f−1(A + x) is meager in 2ω for x ∈ X) the set of all witness functions
is dense, coanalytic, and either meager or comeager in in the sapce C(2ω, X) of all continuous functions f : 2ω → X.