Sturm-Liouville operators in the half axis with shifted potentials

We consider Sturm-Liouville operators in the half axis generated by shifts of the potential and prove that Lebesgue measure is equivalent to a measure defined as an average of spectral measures which correspond to these operators. This is then used to obtain results on stability of spectral types under change of parameters such as boundary conditions, local perturbations, and shifts. In particular if for a set of shifts of positive measure the corresponding operators have alpha-singular, singular continuous and ( or) point spectrum in a fixed interval, then this set of shifts has to be unbounded. Moreover, there are large sets of boundary conditions and local perturbations for which the corresponding operators enjoy the same property.