  Convergence of non-uniform spherical averages is obtained for
measure-preserving actions of free groups. This result generalizes theorems of
Grigorichuk, Nevo and Stein [in particular, a simpler proof of the Nevo-Stein
theorem about uniform spherical averages is obtained.]
  The proof uses the Markov operator approach, first proposed by R.I.
Grigorchuk. To a measure-preserving action of a free group and a matrix of
weights, a Markov operator is assigned in such a way that convergence of
spherical averages with corresponding weights is equivalent to convergence of
powers of the Markov operator. That last is obtained using Rota's
"Alternierende Verfahren"; the 0-2 law for Markov operators in the form of
Kaimanovich; and a suitable maximal inequality.
