  We consider Poisson Boolean percolation on $\mathbb R^d$ with power-law
distribution on the radius with a finite $d$-moment for $d\ge 2$. We prove that
subcritical sharpness occurs for all but a countable number of power-law
distributions. This extends the results of Duminil-Copin--Raoufi--Tassion where
subcritical sharpness is proved under the assumption that the radii
distribution has a $5d-3$ finite moment. Our proofs techniques are different
from their paper: we do not use randomized algorithm and rely on specific
independence properties of Boolean percolation, inherited from the underlying
Poisson process.
  We also prove supercritical sharpness for any distribution with a finite
$d$-moment and the continuity of the critical parameter for the truncated
distribution when the truncation goes to infinity.
