A theorem of Cramér and Wold revisited

Abstract:

Let $H = \{ (x,y);x > 0\} \subseteq {{\mathbf {R}}^2}$ and let $E$ be a Borel subset of $H$ of positive Lebesgue measure. We prove that if $\mu$ and $\upsilon$ are two probability measures on ${{\mathbf {R}}^2}$ such that $\mu (\sigma (E)) = \upsilon (\sigma (E))$ for all rigid motions $\sigma$ of ${{\mathbf {R}}^2}$, then $\mu = \upsilon$ This generalizes a well-known theorem of Cramér and Wold.