| In this work, four integrals are studied, two single valued and the other two set valued. The single valued integrals are generalizations of the Pettis and Bartle-Dunford-Schwartz-Lewis integrals to the case that the underlying measure is only finitely additive. Also in the latter case, the integrals take their values in a locally convex space. The central role played by strong additivity for the absolute continuity of these integrals is exhibited. Vitali type convergence theorems are established for the integrals of each kind. For the Pettis generalization, sufficient conditions for the convergence of Pettis-cauchy sequences are given. The Bartle-Dunford-Schwartz-Lewis section concludes with several theorems on representation of operators on the spaces B(S,{dollar}Sigma{dollar}), C(S), and L{dollar}sb1{dollar}.; Next, two set valued integrals are studied. The first is a new set valued integral for functions taking their values in the space of bounded convex subsets of a complete Hausdorff locally convex space. This integral is a direct generalization of the Bochner integral and is seen to coincide with the Aumann and Debrew integrals under proper conditions. Preliminary to the introduction of the integral is a study of infinite series of sets in the space of bounded convex subsets and a study of set valued set functions. Here also, the concept of strong additivity is visited. The integral itself follows a standard development parallel to that of the Bochner integral. The comparison to the Aumann and Debrew integrals follows and then the introduction of a weak integral. Finally, a different sort of weakening of the defining hypothesis is explored to enlarge the class of integrable functions. The second set valued integral was introduced by J. K. Brooks in 1968 and here we give a kind of monotone convergence theorem for that integral. |
