Divisibility of spheres with measurable pieces

For an r -tuple (γ_1,…,γ_r) of special orthogonal d× d matrices, we say that the Euclidean (d-1) -dimensional sphere Sᵈ⁻¹ is (γ_1,…,γ_r) -divisible if there is a subset A⊆ Sᵈ⁻¹ such that its translations by the rotations γ_1,…,γ_r partition the sphere. Motivated by some old open questions of Mycielski and Wagon, we investigate the version of this notion where the set A has to be measurable with respect to the spherical measure. Our main result shows that measurable divisibility is impossible for a "generic" (in various meanings) r -tuple of rotations. This is in stark contrast to the recent result of Conley, Marks and Unger which implies that, for every "generic" r -tuple, divisibility is possible with parts that have the property of Baire.