  This work studies the interplay between Green functions, the index of
determinacy of spectral measures and interior finite rank perturbations of
Jacobi operators. The index of determinacy quantifies the stability of
uniqueness of solutions of the moment problem. We give results on the constancy
of this index in terms of perturbations of the corresponding Jacobi operators.
The permanence of the $N$-extremality of a measure is also studied. A measure
$\mu$ is $N$-extremal when the polynomials are dense in $L_2(\mathbb{R},\mu)$.
As a by-product, we give a characterization of the index in terms of cyclic
vectors. We consider a new inverse problem for Jacobi operators in which
information on the place where the interior perturbation occurs is obtained
from the index of determinacy.
