Weak mixing for nonsingular Bernoulli actions of countable amenable groups

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 ${⨂}_{g\in G}(A,{\mu }_g)$ 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 ${\mathbb{Z}}^d$-actions are ergodic. As a byproduct, we prove a weak version of the pointwise ratio ergodic theorem for nonsingular actions of $G$.