On the definition of likelihood in abstract spaces

The aim of this work is to give a rigorous definition of likelihood without any reference to the peculiarities of Euclidean spaces, and is thus applicable to a larger class of problems with a more complex result space.Here we intend to offer the simplest possible discussion of the ideas on which likelihood methods are based, with some remarks about their links to some classical measure-theoretical concepts, such as Radon—Nikodym derivatives. Since the definition of likelihood relies on the topological structure of the result space, it is necessary to point out the connections that it has with the measure-theoretical one, mostly caused by the fact that singletons, i.e. the actual observable results, are usually measure-zero sets.