For the sphere, yes. But for the plane, oo + oo = oo, so it's not immediately obvious why measurability must be ruled out. —Dan Allan Wechsler wrote: ----- Maybe I'm not understanding, but I think we can prove right away that the measurability requirement is not attainable. Two congruent measurable sets, after all, must have the same measure. And I think that if A is congruent to the union of the disjoint sets B and C, its measure must be at least |B| + |C|. But maybe I am forgetting some horrible pathological case. -----