Compact Variation, Compact Subdifferentiability and Indefinite Bochner Integral

The notions of compact convex variation and compact convex subdifferential for the mappings from a segment into a locally convex space (LCS) are studied. In the case of an arbitrary complete LCS, each indefinite Bochner integral has compact variation and each strongly absolutely continuous and compact subdifferentiable a.e. mapping is an indefinite Bochner integral. 0. Introduction and preliminaries As it is well known, the main difference between properties of a Bochner integral ([1]–[6]) on a segment and the classical Lebesgue integral is that the class of indefinite Bochner integrals (B) ∫ x a f(t) dt is essentially smaller than the one of absolutely continuous mappings. Recall ([2], Theorems 3.7.11, 3.8.5, 3.8.6) that, in the case of Banachvalued mappings, the class of indefinite Bochner integrals coincides with the one of absolutely continuous and a.e. differentiable mappings. This problem was resulted, in particular, in the notion of a Radon-Nikodym property (RNP). Each absolutely continuous mapping taking values in a space with RNP (or Radon-Nikodym space) is an indefinite Bochner integral. This class of spaces plays an important role in the modern theory of Banach spaces and locally convex spaces, especially in connection with probability theory, harmonic analysis and topology ([7]– [9]). In this paper, a rather different way is chosen. Not restricting, as far as possible, the class of spaces under consideration, a description of the absolute continuous mapping that are indefinite Bochner integrals is given by means of the nonsmooth analysis. To this end, the notion of a (set-valued) convex variation of a mapping from a segment into a LCS is introduced. The use of properties of compact variation (i.1) and compact subdifferential [10, 12] permitted us to prove the main results of the paper, which include representing every (strongly) absolutely continuous and compact subdifferentiable a.e. mapping into an arbitrary LCS as an indefinite Bochner integral (Theorems 3.1), proving that the class of indefinite Bochner integrals coincides with the class of (strongly) absolutely continuous and differentiable a.e. mappings in the case of Frechet spaces (Theorem 3.2), and showing a presence of a compact variation at each indefinite Bochner integral in an arbitrary complete LCS (Theorem 3.3). Note, in particular, that Theorem 3.2 generalizes the above-mentioned result from the case of Banach spaces ([2], Theorems 3.7.11, 3.8.5, 3.8.6) to the case of Frechet spaces. Note also that in the case of an arbitrary (not Frechet) LCS, the indefinite Bochner integral, being absolutely continuous, can nevertheless be nowhere differentiable (see Example 2.2). 2000 Mathematics Subject Classification. Primary 26A45, 28B05, 46J52; Secondary 28C20, 46B22.