Measures and functions in locally convex spaces

Abstract

In this dissertation we establish results concerning in locally convex spaces-valued measures and measurable functions. The results are explained in three parts: Firstly, we establish Liapounoff convexity-type results for locally convex space-valued measures defined on fields (of sets) or equivalently on Boolean Algebras. Liapounoff convexity-type theorems concern the compactness and convexity of the closure of the range of a vector measure. We specifically investigate such results for measures defined on fields and fields of sets with the interpolation property. We find that vector measures defined on fields with the interpolation property have properties very similar to the status quo, while similar results may not hold for vector measures defined on general fields. In the latter case we consider vector measures with properties stronger than non-atomicity, specifically, the strong continuity property. We investigate these properties and certain locally convex spaces for which some of the additivity conditions can be relaxed. In the second part of this dissertation, we firstly consider the existence of weak integrals in locally convex spaces specifically, locally convex spaces whose duals are barrelled spaces. Then, inspired by results of J. Diestel we investigate the "improved" properties of the composition of nuclear maps with a locally convex space-valued measures and functions and the properties of nuclear space-valued vector measures and functions. Amongst others we find that the measurability and integrability properties of locally convex space-valued measurable functions are improved with such a composition compared to the functions considered on their own. The third part of this dissertation involves the factorization of measurable functions. We first consider the factorization of Polish space-valued measurable functions along the lines of the famous "Doob-Dynkin's lemma", a result found in (scalar-valued) stochastic processes. This allows us to determine when, for two measurable functions, f and g it is possible to find a measurable function h, such that g= h ○ f. Similar results are established for various classes of measurable functions. We discover similar factorization results for certain multifunctions (set-valued functions) and operator-valued measurable functions. Another consequence is a factorization scheme for operators on L1(µ).