Duality and free measures in vector spaces, the spectral theory of actions of non-locally compact groups (Q2313790)

This article, written in memory of V. N. Sudakov, ``presents a general duality theory for vector measure spaces taking its origin in the author's papers written in the 1960s''. Here, a vector measure space \((E,\mathcal{A},\mu)\) is a triple, consisting of a vector space \(E\), the smallest \(\sigma\)-algebra \(\mathcal{A}\) in \(E\) making all functions of a fixed point-separating linear space of homomorphisms of \(E\) to a finite-dimensional vector space measurable and a probability measure \(\mu\) on \(\mathcal{A}\) such that \((\mathcal{A},\mu)\) is isomorphic to a Lebesgue-Rokhlin space. The main tool in this duality theory is the ``measurable dual'' \(\mathrm{Lin}(E,\mu)\) of \((E,\mathcal{A},\mu)\), i.e., the space of measurable functionals \(f\) on \(E\) which are linear on a subspace \(E_f\subseteq E\) of full measure. It is studied which subspaces of the space of all measurable functions on a Lebesgue space can be represented as a space of type \(\mathrm{Lin}(E,\mu)\). A notion, important in this article, is that of a free measure. If \((E,\mathcal{A},\mu)\) is a vector measure space, then \(\mu\) is called free if every measurable functional on \(E\) is linear up to a correction on a set of zero measure (depending on the functional). One of the properties of a free measure stated here says that every vector measure space is a linear isomorphic mod 0 image of a vector space with a free continuous measure. ``The author hopes that this publication will, for one thing, revive the interest in important problems of measure and integration theory in vector spaces and the conceptual setting of these problems. In addition, the author intends to draw attention to new problems closely related to this theory.'' The end of the introduction is dedicated to Sudakov's mathematical achievement and career.