In this paper the authors prove that continuously indexed frames in separable Hilbert spaces with no redundancy (or excess) are equivalent to discretely indexed sets. The result applies to general not necessarily separated Hilbert spaces, by restricting the analysis to closed countably generated subspaces. More specific, let be a Hilbert space and a measure space. Then a generalized frame in indexed by is a family ; such that: (a) , is measurable; (b) there are such that , . Recall also a measurable subset of is called an atom if and contains no measurable subset such that . The main result reads (the reviewer takes the liberty to fix a typo):
Theorem 2.2: Let be a generalized frame in indexed by and assume . Then for every countable subset of , there exists a countable collection of disjoint measurable sets such that for all in the closed linear span of , where is a set of complex numbers depending on , and denotes the characteristic function of . In particular, if is an infinite dimensional separable space, then is isometrically isomorphic to the weighted space consisting of all sequences with , where for a fixed collection of disjoint -atoms .