  Let $G$ be an amenable discrete countable infinite group,
  $A$ a finite set, and $(\mu_g)_{g\in G}$ a family of probability measures on
$A$ such that $\inf_{g\in G}\min_{a\in A}\mu_g(a)>0$. It is shown (among other
results) that if the Bernoulli shiftwise action of $G$ on the infinite product
space $\bigotimes_{g\in G}(A,\mu_g)$ is nonsingular and conservative then it is
weakly mixing. This answers in positive a question by Z.~Kosloff who proved
recently that the conservative Bernoulli $\Bbb Z^d$-actions are ergodic. As a
byproduct, we prove a weak version of the pointwise ratio ergodic theorem for
nonsingular actions of $G$.
