Note on sets of positive measure
THEOREM. Let Ai, A2, • , Av be p sets of positive measure lying in euclidean n space. Then there exist p n-dimensional spheres Si, 52, • • • , Sp such that for every set of p points sv, (*> = 1,2, • • • , p), belonging respectively to these spheres, there exists a set of p points av, (v = 1, 2, • • • , p), lying respectively in A\, A2, • • • , Ap, such that the sets [av] and {sv} are congruent. Moreover, there exists a set of p congruent spheres Sv satisfying the condition just stated and a positive number 8 such that for every selected {sv}, with sv belonging to Svy the associated {av} may be so chosen that a\ ranges over a set of measure >h.