  The goal is to extend Gleason's notion of a frame function, which is
essential in his fundamental theorem in quantum measurement, to a more general
function acting on 1-tight, so-called, Parseval frames. We refer to these
functions as Gleason functions for Parseval frames. The reason for our
generalization is that positive operator valued measures (POVMs) are
essentially equivalent to Parseval frames, and that POVMs arise naturally in
quantum measurement theory. We prove that under the proper assumptions, Gleason
functions for Parseval frames are quadratic forms, as well as other results
analogous to Gleason's original theorem. Further, we solve an intrinsic problem
relating Gleason functions for Parseval frames of different lengths. We use
this solution to weaken the hypotheses in the finite dimensional version of
Busch's theorem, that itself is an analog of Gleason's mathematical
characterization of quantum states.
