Let X be an abelian Polish group. For every analytic Haar-null set A ⊆ 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 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 ⊇ A such that T(A)∖ T(B) is meager. Finally, under Martin’s Axiom and the negation of Continuum Hypothesis, some results concerning co-analytic sets are derived.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm181-2-1,
author = {Pandelis Dodos},
title = {On certain regularity properties of Haar-null sets},
journal = {Fundamenta Mathematicae},
volume = {184},
year = {2004},
pages = {97-109},
zbl = {1043.28013},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm181-2-1}
}
Pandelis Dodos. On certain regularity properties of Haar-null sets. Fundamenta Mathematicae, Tome 184 (2004) pp. 97-109. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm181-2-1/