Riemann and Lebesgue - Bochner integrals of functions defined on a non - void set X with values in a linear convex space E

Abstract

In this thesis we study Riemann and Lebesgue - Bochner integrals of functions defined on a non - void set X with values in a linear convex space E. This study has been done by many authors in the case of functions with values in Banach spaces [4], [6], [7], [8], [10]. The extension of the results from Banach spaces to linear convex spaces is not always straightforward and needs new ideas [2], [12]. This thesis consists of three chapters.

We give in the first chapter relevant background material needed for writing this thesis. Chapter I consists of three sections. In section I.1. we give notations and results on measure spaces. and step functions. In section I.2. we give some needed results on convergence of sequences in linear convex metric spaces. Finally, in section I.3. we give some adopted notations used throughout this thesis. In chapter II we study Riemann integrals of functions defined on a segment [a,b] with values in linear convex space. In section II.1 we give the classical theory of Riemann integral of real valued functions. In section II.2 we study Riemann integral of normed valued functions. We discuss mainly Lebesgue's criteria stating A function & : [a,b] IR is Riemann integrable iff it is bounded andalmost everywhere continuous. We give the example of Grave's (7] showing that in many Banach spaces the condition of the continuity almost everywhere is not necessary. We also give different proofs(see (7]) that the conditions of Lebesgue's criteria are sufficient for Riemann integrability in the case of normed valued functions. Also, we prove the beautiful result (10] showing that Lebesgue's criteria is true in the case of the functions e : [a,b)l'. In

section II.3 we prove that the conditions of Lebesgue's criteria are also sufficient for Riemann integrability of functions with values in locally convex sequentially complete metric spaces. Finally, in chapter III we • survey results on Lebesgue - Bochner integral for functions defined on any non-void set X with values in a linear locally convex sequentially complete m tric space E (2], [13]. In section III. 1 we give the definition of the space L1 (ll,E) and prove that L1(1.L,E) is a linear convex semimetric space. In

section III .2 we prove the completeness of L1(1l, E ) and prove that it is also a locally convex metric sequentially complete space. In section III.3. we prove Lebesgue dominating theorem for L1(ll,E).

Finally, in section III.4 we prove Fubini's theorem.