On certain regularity properties of Haar-null sets

Pandelis Dodos¹

¹ Department of Mathematics National Technical University of Athens Zografou Campus, 157 80 Athens, Greece

Fundamenta Mathematicae, Tome 181 (2004) no. 2, pp. 97-109

Résumé

Let $X$ be an abelian Polish group. For every analytic Haar-null set $A\subseteq X$ let $T(A)$ be the set of test measures of $A$. We show that $T(A)$ is always dense and co-analytic in $P(X)$. We prove that if $A$ is compact then $T(A)$ is $G_\delta $ dense, while if $A$ is non-meager then $T(A)$ is meager. We also strengthen a result of Solecki and we show that for every analytic Haar-null set $A$, there exists a Borel Haar-null set $B\supseteq A$ such that $T(A)\setminus T(B)$ is meager. Finally, under Martin's Axiom and the negation of Continuum Hypothesis, some results concerning co-analytic sets are derived.

Détail

DOI : 10.4064/fm181-2-1

Keywords: abelian polish group every analytic haar null set subseteq set test measures always dense co analytic prove compact delta dense while non meager meager strengthen result solecki every analytic haar null set there exists borel haar null set supseteq setminus meager finally under martins axiom negation continuum hypothesis results concerning co analytic sets derived