Document Type
Lecture
Publication Date
9-16-2010
Abstract
The title above refers to a 1955 paper of Bartle, Dunford, and Schwartz 'Weak Compactness and Vector Measures' in which integration of scalar functions with respect to a vector measure was first defined. Vector measures considered in the original paper were Banach space valued and the notion of weak compactness played the key role in the development of the theory. 55 years later, I am happy to report that the theory can be developed without any constraints on the range space (and so the notion of weak topology is not even available, in general). Let �� be a σ-algebra of subsets of a set ��, �� be a Hausdorff sequentially complete topological vector space, and let ��: �� → �� be a countably additive measure. Denote by L₀(��) the space of (classes of) measurable ℝ-valued functions. Assume that the measure is convexly bounded or, equivalently, all essentially bounded functions are integrable. Then the classical definition of the Bartle-Dunford-Schwartz integral is possible. The space L₁(��) of BDS-integrable functions is a vector lattice, but with its natural topology it is only an A-solid topological vector group and, in general, the Lebesgue Dominated Convergence Theorem is not valid in L₁(��). Let L₁₀(��) be the largest vector subspace of L₁(��) that is solid in L₀(��). L₁₀(��) with its natural topology is a Dedekind σ-complete Hausdorff locally solid vector lattice (F-lattice, if �� is an F-space) having the σ-Lebesgue property (the Dominated Convergence Theorem holds). If �� contains no isomorphic copy of c₀, then L₁₀(��) has the σ-Levi property (the Beppo Levi Theorem holds).
Relational Format
presentation
Recommended Citation
Labuda, Iwo, "Vector Measures Without Weak Compactness" (2010). Analysis Seminar. 19.
https://egrove.olemiss.edu/math_analysis/19
