  We give an analytical approach to the definition of additive and
multiplicative free convolutions which is based on the theory of Nevanlinna and
of Schur functions. We consider the set of probability distributions as a
semigroup $\bold M$ equipped with the operation of free convolution and prove a
Khintchine type theorem for the factorization of elements of this semigroup. An
element of $\bold M$ contains either indecomposable ("prime") factors or it
belongs to a class, say $I_0$, of distributions without indecomposable factors.
In contrast to the classical convolution semigroup in the free additive and
multiplicative convolution semigroups the class $I_0$ consists of units (i.e.
Dirac measures) only. Furthermore we show that the set of indecomposable
elements is dense in $\bold M$.
