Bochner Integrals and Vector Measures

This project extends known theorems for scalar valued functions to the context of Banach space valued functions. In particular, it contains generalizations of the classical theory of Lebesgue Integrals, complex measures, Radon-Nikodym theorem and Riesz Representation theorem. We explore some properties of functions whose domains are abstract Banach spaces, where the usual derivatives are replaced by Radon-Nikodym derivatives. The first two Chapters are devoted to infinite dimensional measurable functions and the problem of integrating them. Most of the basic properties of Bochner integration are forced on it by the classical Lebesgue integration and the usual definition of measurability. The Radon-Nikodym theorem for Bochner Integral is the subject to Chapter III. The roles of reflexive spaces, separable anti-dual spaces and the Radon-Nikodym property of Banach spaces are also discussed in this Chapter. One of the most interesting aspects of the theory of the Bochner integral centers about the following questions: When does a vector measure F: Ʒ→X arise as a Bochner integral of an L1(S,X) function (i.e. F(E) = ∫E f dm)? And conversely, if f ∈ L1(S,X). Then, is F: Ʒ→X, defined by F(E) = ∫E f dm, a countably additive vector measure, absolutely continuous with respect to the positive measure m? These two questiones are examined by the Radon-Nikodym theorem and the Riesz Representation theorem. It is worth observing, that the relationsip between these theorems are considered to be just a formality of translating a set of basic definitions from one context to another. There are theories of integration similar to the Bochner Integral, that allow us to integrate functions that are only weakly measurable (The Pettis Integral) with respect to a positive measure. Also, the ultimate generality of the Bochner Integral, the Bartle Integral, for integrating vector valued functions with respect to a general vector measure. However, these theories do not occupy a central role in our study and we limit ourselves to only mentioning [1] as an excellent reference