  Let $H_\omega$ be a self-adjoint Jacobi operator with a potential sequence
$\{\omega(n)\}_n$ of independently distributed random variables with continuous
probability distributions and let $\mu_\phi^\omega$ be the corresponding
spectral measure generated by $H_\omega$ and the vector $\phi$. We consider
sets $A(\omega)$ which depend on $\omega$ in a particular way and prove that
$\mu_\phi^\omega(A(\omega))=0$ for almost every $\omega$. This is applied to
show equivalence relations between spectral measures for random Jacobi matrices
and to study the interplay of the eigenvalues of these matrices and their
submatrices.
