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Haar Null Sets and the Consistent Reflection of Non-meagreness

  • Márton Elekes (a1) (a2) and Juris Steprāns (a3)

A subset X of a Polish group G is called Haar null if there exist a Borel set B ⊃ X and Borel probability measure μ on G such that μ(gBh) = 0 for every g; h ∊ G. We prove that there exist a set X ⊂ R that is not Lebesgue null and a Borel probability measure μ such that μ (X + t) = 0 for every t ∊ R. This answers a question from David Fremlin’s problem list by showing that one cannot simplify the definition of a Haar null set by leaving out the Borel set B. (The answer was already known assuming the Continuum Hypothesis.)

This result motivates the following Baire category analogue. It is consistent with ZFC that there exist an abelian Polish group G and a Cantor set C ⊂ G such that for every non-meagre set X ⊂ G there exists a t ∊ G such that C ∩ (X + t) is relatively non-meagre in C. This essentially generalizes results of Bartoszyński and Burke–Miller.

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[1] Bartoszyński, T., On perfectly meager sets. Proc. Amer. Math. Soc. 130(2002), no. 4, 11891195.
[2] Bartoszyński, T. and Judah, H., Set theory. On the structure of the real line. A. K. Peters,Wellesley, MA, 1995.
[3] Burke, M. R. and Miller, A.W., Models in which every nonmeager set is nonmeager in a nowhere dense Cantor set. Canad. J. Math. 57(2005), no. 6, 11391154.
[4] Christensen, J. P. R., On sets of Haar measure zero in abelian Polish groups. Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces (Jerusalem, 1972). Israel J. Math. 13(1972), 255260.
[5] Christensen, J. P. R., Measure theoretic zero sets in infinite dimensional spaces and applications to differentiability of Lipschitz mappings. Actes du Deuxième Colloque d’Analyse Fonctionnelle de Bordeaux (Univ.Bordeaux, 1973). Publ. Dép. Math. (Lyon) 10(1973), no. 2, 2939.
[6] Ciesielski, K. and Shelah, S., Category analogue of sup-measurability problem. J. Appl. Anal. 6(2000), no. 2, 159172.
[7] Darji, U. B. and Keleti, T., Covering R with translates of a compact set. Proc. Amer. Math. Soc. 131(2003), no. 8, 25932596.
[8] Dougherty, R. and Mycielski, J., The prevalence of permutations with infinite cycles. Fund. Math. 144(1994), no. 1, 8994.
[9] Erdőos, P. and Kakutani, S., On a perfect set. Colloquium Math. 4(1957), 195196.
[10] Falconer, K. J., The geometry of fractal sets. Cambridge Tracts in Mathematics, 85, Cambridge University Press, Cambridge, 1986.
[11] Hunt, B. R., The prevalence of continuous nowhere differentiable functions. Proc. Amer. Math. Soc. 122(1994), no. 3, 711717.
[12] Hunt, B. R., Sauer, T., and Yorke, J. A., Prevalence: a translation-invariant “almost every” on infinite-dimensional spaces. Bull. Amer. Math. Soc. (N.S.) 27(1992), no. 2, 217238.
[13] A. S. Kechris, , Classical descriptive set theory. Graduate Texts in Mathematics, 156, Springer-Verlag, 1995.
[14] Mattila, P., Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability. Cambridge Studies in Advanced Mathematics, 44, Cambridge University Press, Cambridge, 1995.
[15] A. Rosłanowski, and Shelah, S., Measured creatures. Israel J. Math. 151(2006), 61110.
[16] Zajíček, L., On differentiability properties of typical continuous functions and Haar null sets. Proc. Amer. Math. Soc. 134(2006), no. 4, 11431151.
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Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
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