  The paper considers the convergence to equilibrium for measure solutions of
the spatially homogeneous Boltzmann equation for hard potentials with angular
cutoff. We prove the exponential sharp rate of strong convergence to
equilibrium for conservative measure solutions having finite mass and energy.
The proof is based on the regularizing property of the iterated collision
operators, exponential moment production estimates, and some previous results
on the exponential rate of strong convergence to equilibrium for square
integrable initial data. We also obtain a lower bound of the convergence rate
and deduce that no eternal solutions exist apart from the trivial stationary
solutions given by the Maxwellian equilibrium. We finally use these convergence
rates in order to deduce global-in-time strong stability of measure solutions.
