In the study ([5]) of the geometrically regular weighted shifts (GRWS), signed representing measures, which we call Berger-type charges, played an important role. Motivated by their utility in that context, we establish a general theory for Berger-type charges. We give the first result of which we are aware showing that k–hyponormality alone, as opposed to subnormality, yields measure/charge-related information. More precisely, for signed countably atomic measures with a decreasing sequence of atoms, we prove that k-hyponormality of the associated shift forces positivity of the densities of the largest \(k+1\) atoms. Further, for certain completely hyperexpansive weighed shifts, we exhibit a Berger-type charge representation, in contrast but related to the classical Lévy-Khinchin representation. We use Berger-type charges to investigate when a non-subnormal GRWS weighted shift may be scaled to become conditionally positive definite, and close with an example indicating a distinction between the study of moment sequences and the study of weighted shifts.
