
TL;DR
This paper introduces a generalized integral for real-valued functions with respect to measures valued in extended positive cones of partially ordered vector spaces, extending classical results like the monotone convergence theorem.
Contribution
It defines a new integral concept in the setting of partially ordered vector spaces, establishing key theorems and exploring its relation to classical and operator-valued measures.
Findings
The integral extends classical Lebesgue integration to vector-valued measures.
Monotone convergence, Fatou's lemma, and dominated convergence theorems are proved in this setting.
The integral's domain can be larger than that of existing vector measure integrals.
Abstract
We define an integral of real-valued functions with respect to a measure that takes its values in the extended positive cone of a partially ordered vector space . The monotone convergence theorem, Fatou's lemma, and the dominated convergence theorem are established; the analogues of the classical - and -spaces are investigated. The results extend earlier work by Wright and specialise to those for the Lebesgue integral when equals the real numbers. The hypothesis on that is needed for the definition of the integral and for the monotone convergence theorem to hold (-monotone completeness) is a rather mild one. It is satisfied, for example, by the space of regular operators between a directed partially ordered vector space and a -monotone complete partially ordered vector space, and by every JBW-algebra. Fatou's lemma and the…
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Taxonomy
TopicsAdvanced Banach Space Theory · Holomorphic and Operator Theory · Advanced Operator Algebra Research
