  This paper introduces the class of selfdecomposable distributions concerning
Boolean convolution. A general regularity property of Boolean selfdecomposable
distributions is established; in particular the number of atoms is at most two
and the singular continuous part is zero. We then analyze how shifting
probability measures changes Boolean selfdecomposability. Several examples are
presented to supplement the above results. Finally, we prove that the standard
normal distribution $N(0,1)$ is Boolean selfdecomposable but the shifted one
$N(m,1)$ is not for sufficiently large $|m|$.
