  For a general radially symmetric, non-increasing, non-negative kernel $h\in L
^ 1 _{loc} ( R ^ d)$, we study the rigidity of measurable sets in $R ^ d$ with
constant nonlocal $h$-mean curvature. Under a suitable "improved integrability"
assumption on $h$, we prove that these sets are finite unions of equal balls,
as soon as they satisfy a natural nondegeneracy condition. Both the radius of
the balls and their mutual distance can be controlled from below in terms of
suitable parameters depending explicitly on the measure of the level sets of
$h$. In the simplest, common case, in which $h$ is positive, bounded and
decreasing, our result implies that any bounded open set or any bounded
measurable set with finite perimeter which has constant nonlocal $h$-mean
curvature has to be a ball.
