The Lebesgue decomposition theorem and the Radon-Nikodym theorem are cornerstones of the classical measure theory. These theorems were generalized in several settings and several ways. In the first part of the talk I will give a short overwiev of this topic. In the remaining time I will present a Lebesgue-Radon-Nikodym type theorem which generalizes the well-known Ando-decomposition of bounded positive operators, the Darst decomposition of finitely additive nonnegative set functions, and Gudder's Radon-Nikodym theorem for representable functionals.

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