A theorem of Cramér and Wold revisited

Abstract

Let H={( x,y):x>0)⊆R² and let E be a Borel subset of H of positive Lebesgue measure We prove that if μ and ν are two probability measures on R² such that μ(σ(E))=ν{σ(E)) for all rigid motions σ of R². then μ=ν This generalizes a well-known theorem of Cramér and Wold.