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The paper presents a general duality theory for vector measure spaces takingits origin in the author's papers written in the 1960s. The main resultestablishes a direct correspondence between the geometry of a measure in avector space and the properties of the space of measurable linear functionalson this space regarded as closed subspaces of an abstract space of measurablefunctions. An example of useful new features of this theory is the notion of afree measure and its applications.

Duality and free measures in vector spaces, the spectral theory of actions of non-locally compact groups