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Cardinal invariants of Haar null and Haar meager sets
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 151 / Issue 5 / October 2021
- Published online by Cambridge University Press:
- 27 October 2020, pp. 1568-1594
- Print publication:
- October 2021
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- Article
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A subset X of a Polish group G is Haar null if there exists a Borel probability measure μ and a Borel set B containing X such that μ(gBh) = 0 for every g, h ∈ G. A set X is Haar meager if there exists a compact metric space K, a continuous function f : K → G and a Borel set B containing X such that f−1(gBh) is meager in K for every g, h ∈ G. We calculate (in ZFC) the four cardinal invariants (add, cov, non, cof) of these two σ-ideals for the simplest non-locally compact Polish group, namely in the case
$G = \mathbb {Z}^\omega$. In fact, most results work for separable Banach spaces as well, and many results work for Polish groups admitting a two-sided invariant metric. This answers a question of the first named author and Vidnyánszky.
