  In the study of the geometrically regular weighted shifts (GRWS) -- see [5]
-- signed power 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 L\'{e}vy-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.
