ON SOME SUBSETS OF SPACES EQUIPPED WITH TRANSFORMATION GROUPS
Kharazishvili, Kharazishvili · Real Analysis Exchange · 1996
For a given space E equipped with a transformation group G, the notions of a G-thick set and of a G-thin set are introduced and discussed.Some relationships between these notions and the theory of G-invariant (more generally, G-quasiinvariant) measures are considered.Let E be a nonempty basic set and let G be a subgroup of the group Sym(E) of all bijective mappings acting from E onto E. In such a case the pair (E, G) is usually called a space equipped with a transformation group.We recall that a space (E, G) is homogeneous if the group G acts transitively in E, i.e. for any two points x and y of E, there exists a transformation g from G such that g(x) = y.It is well known that homogeneous spaces play an important role in various domains of modern mathematics.We recall also that the group G acts freely in E if, for any two distinct transformations g ∈ G and h ∈ G and for each point x ∈ E, we have g(x) = h(x).More generally, suppose that I is a G-invariant σ-ideal of subsets of E. We say that G acts I-freely in E if, for any two distinct transformations g and h from G, the set {x ∈ E : g(x) = h(x)} belongs to I.Let X be a subset of the basic set E.Definition 1 We say that X is (G, I)-thick (in E) if there exists a family {g k : k < ω} of transformations from G satisfying the relation E \ ∪{g k (X) : k < ω} ∈ I,