  On the torus group, on the group of p-adic integers and on the p-adic
solenoid, we give a construction of an arbitrary weakly infinitely divisible
probability measure using a random element with values in a product of
(possibly infinitely many) subgroups of the real numbers. As a special case of
our results, we have a new construction of the Haar measure on the p-adic
solenoid.
