Radon-Nikodym type theorem and Lebesgue decomposition for positive operator valued measures

We introduce a Radon-Nikodym-type derivative for positive operator-valued measures (POVMs) with respect to a scalar measure, whose existence is characterized via the theory of closable Hermitian forms. This formulation unifies and extends several existing results on the existence of operator-valued densities in quantum mechanics and functional analysis. In the commutative case, we establish a necessary and sufficient condition for the existence of the Radon-Nikodym derivative and characterize it explicitly. We also present concrete examples of POVMs that admit a derivative under the proposed definition but fail to do so under previously established definitions. A notable feature of the proposed framework is that the operators defining the Radon-Nikodym derivative of a POVM F are self-adjoint and affiliated with the von Neumann algebra generated by F , a structurally relevant property with potential applications in quantum theory. Finally, we establish a Lebesgue-type decomposition for POVMs, extending classical decomposition results to the positive operator-valued setting.