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.

中文翻译：

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Haar null 和 Haar 微薄集的基数不变量

一个子集X一个波兰团体G是哈尔空如果存在一个 Borel 概率测度 μ 和一个 Borel 集乙包含X使得 μ(gBh) = 0 对于每个G,H∈G. 一套X是哈尔微薄如果存在紧度量空间ķ, 一个连续函数F：ķ→G和一个 Borel 集乙包含X这样F-1(gBh) 是微薄的ķ对于每个G,H∈G. 我们计算（在ZFC) 这两个 σ-理想的四个基本不变量（add, cov, non, cof）对于最简单的非局部紧波兰群，即在这种情况下$G = \mathbb {Z}^\omega$. 事实上，大多数结果也适用于可分离的 Banach 空间，并且许多结果适用于承认双边不变度量的波兰团体。这回答了第一作者和 Vidnyánszky 的问题。