2019/08/15 by Elekes, Márton, Poór, Márk
#FOS: Mathematics #General Topology (math.GN) #Logic (math.LO)
paper · doi:10.48550/arxiv.1908.05776
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 (\rm add, \rm cov, \rm non, \rm cof) of these two σ-ideals for the simplest non-locally compact Polish group, namely in the case G = ℤω. 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 Z. Vidnyánszky.