  For an $r$-tuple $(\gamma_1,\ldots,\gamma_r)$ of special orthogonal $d\times
d$ matrices, we say that the Euclidean $(d-1)$-dimensional sphere $S^{d-1}$ is
$(\gamma_1,\ldots,\gamma_r)$-divisible if there is a subset $A\subseteq
S^{d-1}$ such that its translations by the rotations $\gamma_1,\ldots,\gamma_r$
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.
