Taras Banakh
Cardinal characteristics of the ideal of Haar null sets

Comment.Math.Univ.Carolinae 45,1 (2004) 119-137.

Abstract:We calculate the cardinal characteristics of the $\sigma $-ideal $\Cal H\Cal N(G)$ of Haar null subsets of a Polish non-locally compact group $G$ with invariant metric and show that $cov(\Cal H\Cal N(G)) \leq \frak b\leq \mathop {\fam \z@ max}\nlimits@ \{\frak d,non(\Cal N)\}\leq non(\Cal H\Cal N(G))\leq cof(\Cal H\Cal N(G)) \kern -0.86pt > \kern -0.86pt \mathop {\fam \z@ min}\nlimits@ \{\frak d,non(\Cal N)\}$. If $G=\DOTSB \prod@ \slimits@ _{n\geq 0}G_n$ is the product of abelian locally compact groups $G_n$, then $add(\Cal H\Cal N(G)) \penalty -\@M = add(\Cal N)$, $cov(\Cal H\Cal N(G))=\mathop {\fam \z@ min}\nlimits@ \{\frak b, cov(\Cal N)\}$, $non(\Cal H\Cal N(G))= \mathop {\fam \z@ max}\nlimits@ \{\frak d,non(\Cal N)\}$ and $cof(\Cal H\Cal N(G))\geq cof(\Cal N)$, where $\Cal N$ is the ideal of Lebesgue null subsets on the real line. Martin Axiom implies that $cof(\Cal H\Cal N(G))>2^{\aleph _0}$ and hence $G$ contains a Haar null subset that cannot be enlarged to a Borel or projective Haar null subset of $G$. This gives a negative (consistent) answer to a question of S. Solecki. To obtain these estimates we show that for a Polish non-locally compact group $G$ with invariant metric the ideal $\Cal H\Cal N(G)$ contains all $o$-bounded subsets (equivalently, subsets with the small ball property) of $G$.

Keywords: Polish group, Haar null set, Martin Axion, cardinal characteristics of an ideal, $o$-bounded set, the small ball property
AMS Subject Classification: 03E04, 03E15, 03E17, 03E35, 03E50, 03E75, 22A10, 28C10, 54A25, 54H11