In this article we determine that an operator-valued measure (OVM) for Banach spaces is actually a weak∗ measure, and then we show that an OVM can be represented as an operator-valued function if and only if it has -finite variation. By the means of direct integrals of Hilbert spaces, we introduce and investigate continuous generalized frames (continuous operator-valued frames, or simply CG frames) for general Hilbert spaces. It is shown that there exists an intrinsic connection between CG frames and positive OVMs. As a byproduct, we show that a Riesz-type CG frame does not exist unless the associated measure space is purely atomic. Also, a dilation theorem for dual pairs of CG frames is given.
"Radon–Nikodym theorems for operator-valued measures and continuous generalized frames." Banach J. Math. Anal. 11 (2) 363 - 381, April 2017. https://doi.org/10.1215/17358787-0000008X