On Measures in Locally Convex Spaces

Abstract

The deep and close connection between absolutely summing (or, more generally, p-absolutely summing) mappings and measure theory is well known (see, e.g., [1 – 3]). However, the related considerations have only been concerned with Banach spaces. (In general, both absolutely summing mappings and measures in nonnormed spaces are studied very poorly.) The basic result of this paper, Theorem 3.1, appears purely topological, for no measures are mentioned in its statement. However, in reality, this result is promising in the study of the relation mentioned above in the general-topological case. This statement is not unfounded, because Theorem 3.1 has already made it possible to obtain a new generalization of the Sazonov theorem, to prove the existence of a Radon¨CNikodym density for vector measures in a fairly general situation, (and, as a corollary, the existence of a logarithmic gradient of a differentiable measure), and to write the Gauss-Ostrogradskii formula in an efficient (for applications; see [3]) scalar form (see Theorems 4.1, 6.4, 7.3, and 7.8, respectively). Without going into details, we also mention that Theorem 3.1 is important for the calculus of variations, Lagrange problem, and boundary value problems on nonmetrizable spaces (these problems are not considered in this paper, but some results are given in [4, 5]).