Vitali’s theorem for invariant measures

Abstract

Csn, lim.,0 XkSn=0, and XkUU,/XkSn_oa. (Xk denotes Lebesgue measure in the space X = Rk. The number a is called a parameter of regularity at x.) The invariance under translation of the set-function Xk suggests the point of view adopted in the present generalization of Vitali's theorem. We consider a group G acting transitively on a Hausdorff space X, the latter endowed with a measure ,u for which u(gB) =,B whenever gCG and B is a b-measurable subset of X. A Vitali covering for a subset A of X is defined in the obvious way. Noticing that if S, denotes the sphere of radius a centered at the origin q in Rk, then Sn +Sn CS2n and XkSa/XkS2a=1/2k, we define regularity of a Vitali cover in terms of a set-theoretic multiplication defined between subsets of X. We replace the spheres Sa by what we call quasispheres. As a con