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

The paper presents a general duality theory for vector measure spaces taking its origin in the author's papers written in the 60-s. The main result establishes the direct correspondence between the geometry of a measure in a vector space and the properties of the space of measurable linear functionals on this space viewed upon as closed subspaces of an abstract space of measurable functions. An example of useful new features of this theory is the notion of free measure as well as its applications.