This paper develops Lyapunov–type criteria for practical and Lipschitz stability of generalized ordinary differential equations (GODEs). After formulating these stability concepts within the GODE framework, we obtain new sufficient conditions for their validity and establish converse Lyapunov theorems that characterize both properties through appropriate Lyapunov functionals. The resulting criteria provide quantitative information on the boundedness of solutions and on their dependence on initial data, while accommodating systems that may display jumps, discontinuities, or memory effects generated by nonabsolute integrators. We further extend the analysis to functional Volterra–Stieltjes integral equations (FV–SIEs), which can be embedded into the GODE formalism, and demonstrate how the Lyapunov methods developed here apply in that setting. Illustrative examples are provided to clarify the implementation and scope of the theory.
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.
