Article

Full entry | PDF   (0.3 MB)
Keywords:
Polish group; Haar null set; Martin Axion; cardinal characteristics of an ideal; $o$-bounded set; the small ball property
Summary:
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 $\operatorname{cov}(\Cal H\Cal N(G)) \leq \frak b\leq \max \{\frak d,\operatorname{non}(\Cal N)\}\leq \operatorname{non}(\Cal H\Cal N(G))\leq \operatorname{cof}(\Cal H\Cal N(G)) \kern -0.86pt > \kern -0.86pt \min \{\frak d,\operatorname{non}(\Cal N)\}$. If $G=\prod_{n\geq 0}G_n$ is the product of abelian locally compact groups $G_n$, then $\operatorname{add}(\Cal H\Cal N(G)) \break = \operatorname{add}(\Cal N)$, $\operatorname{cov}(\Cal H\Cal N(G))=\min\{\frak b, \operatorname{cov}(\Cal N)\}$, $\operatorname{non}(\Cal H\Cal N(G))= \max \{\frak d,\operatorname{non}(\Cal N)\}$ and \linebreak $\operatorname{cof}(\Cal H\Cal N(G))\geq \operatorname{cof}(\Cal N)$, where $\Cal N$ is the ideal of Lebesgue null subsets on the real line. Martin Axiom implies that $\operatorname{cof}(\Cal H\Cal N(G))>2^{\aleph_0}$ 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 $G$.
References:
[Ba$_1$] Banakh T.: Locally minimal groups and their embeddings into products of $o$-bounded groups. Comment. Math. Univ. Carolinae 41.4 (2000), 811-815. MR 1800163
[Ba$_2$] Banakh T.: On index of total boundedness of (strictly) $o$-bounded groups. Topology Appl. 120 (2002), 427-439. MR 1897272 | Zbl 1010.22004
[BNS] Banakh T., Nickolas P., Sanchis M.: Filter games and pathologic subgroups of the countable product of lines. J. Austral. Math. Soc., to appear. MR 2300160
[BP] Banakh T.O., Protasov I.V.: Ball structures and colorings of graphs and groups. VNTL, Lviv, 2003. MR 2392704 | Zbl 1147.05033
[BS] Bartoszyński T., Judah H., Shelah S.: The Cichoń diagram. J. Symb. Log. 58.2 (1993), 401-423. MR 1233917
[BK] Behrends E., Kadets V.: On the small ball property. Studia Math. 148 (2001), 275-287. MR 1880727
[BL] Benyamini Y., Lindenstrauss J.: Geometric Nonlinear Functional Analysis, I. Amer. Math. Soc., 2000. MR 1727673
[C] Christensen J.P.R.: On sets of Haar measure zero in abelian Polish groups. Israel J. Math. 13 (1972), 255-260. MR 0326293
[D] Dougherty R.: Examples of nonshy sets. Fund. Math. 144 (1994), 73-88. MR 1271479
[vD] van Douwen E.K.: The integers and topology. in Handbook of Set-Theoretic Topology , K. Kunen, J.E. Vaughan (Eds.), North-Holland, Amsterdam, 1984, pp.111-167. MR 0776619 | Zbl 0561.54004
[Her] Hernández C.: Topological groups close to being $\sigma$-compact. Topology Appl. 102 (2000), 101-111. MR 1739266
[HRT] Hernández C., Robbie D., Tkachenko M.: Some properties of $o$-bounded and strictly $o$-bounded groups. Appl. General Topology 1 (2000), 29-43. MR 1796930
[He] Heyer H.: Probability Measures on Locally Compact Groups. Springer, 1977. MR 0501241 | Zbl 0528.60010
[Ke] Kechris A.: Classical Descriptive Set Theory. Springer, 1995. MR 1321597 | Zbl 0819.04002
[La] Laver R.: On the consistency of the Borel's conjecture. Acta Math. 137 (1976), 151-169. MR 0422027
[Pa] Paterson A.: Amenability. Math. Surveys and Monographs, vol. 29, Amer. Math. Soc., 1988. MR 0961261 | Zbl 1106.22008
[PZ] Plichko A., Zagorodnyuk A.: Isotropic mappings and automatic continuity of polynomial, analytic, and convex operators. in General Topology in Banach Spaces (T. Banakh, Ed.), Nova Sci. Publ., NY, 2001, pp.1-13. MR 1901530
[Po] Pontryagin L.S.: Continuous Groups. Nauka, Moscow, 1984. MR 0767087 | Zbl 0659.22001
[S$_1$] Solecki S.: Haar null sets. Fund. Math. 149 (1996), 205-210. MR 1383206 | Zbl 0887.28006
[S$_2$] Solecki S.: Haar null and non-dominating sets. Fund. Math. 170 (2001), 197-217. MR 1881376 | Zbl 0994.28006
[Tk$_1$] Tkachenko M.: Introduction to topological groups. Topology Appl. 86 (1998), 179-231. MR 1623960 | Zbl 0955.54013
[Tk$_2$] Tkachenko M.: Topological groups: between compactness and $\aleph_0$-boundedness. in Recent Progress in General Topology, (M. Hušek and J. van Mill, Eds.), North-Holland, 2002. MR 1970010 | Zbl 1029.54045
[Ts] Tsaban B.: $o$-Bounded groups and other topological groups with strong combinatorial properties. submitted, http://arxiv.org/abs/math.GN/0307225 MR 2180906 | Zbl 1090.54034
[THJ] Topsøe F., Hoffmann-Jørgensen J.: Analytic Spaces and their Applications. in Analytic Sets, C. Rogers et al., Academic Press, London, 1980.
[V] Vaughan J.E.: Small uncountable cardinals and topology. in Open Problems in Topology, J. van Mill, G.M. Reed (Eds.), Elsevier Sci. Publ., 1990, pp.197-216. MR 1078647
[Za] Zakrzewski P.: Measures on algebraic-topological structures. in Handbook on Measure Theory, E. Pap (Ed.), North Holland, 2002. MR 1954637 | Zbl 1040.28016

Partner of