In this chapter we shall develop a general integration theory for cone-valued functions with respect to operator-valued measures. The structure of locally convex cones will allow the use of many of the main concepts of classical measure theory for (extended) real-valued functions. Section 1 introduces measurability for cone-valued functions on a set X with respect to a (weak) σ-ring of subsets of X. This notion does not involve any reference to a particular measure. Bounded operator-valued measures will be defined in Section 3. The introduction of its modulus allows the extension of any given measure to a full locally convex cone containing the given cone and its neighborhood system, thus greatly facilitating the expansion of our concepts. This yields a new understanding of the variation of a measure, not as a separate positive real-valued measure associated with the given one, but as a component of its extension. The development of an integration theory for cone-valued functions with respect to an operator-valued measure follows in Section 4. Section 5 contains the general convergence theorems for sequences of functions and measures, that is variations and adaptations of the dominated convergence theorem. Chapter II concludes with a long list of special cases and examples in Section 6, demonstrating the generality of the approach. These examples include classical real-valued measure theory as well as settings with vector-,cone-, functional- and operator-valued measures and functions.
Keywords
- Convex Cone
- Topological Vector Space
- Disjoint Subset
- Continuous Linear Operator
- Neighborhood System
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
(2009). Measures and Integrals. The General Theory. In: Operator-Valued Measures and Integrals for Cone-Valued Functions. Lecture Notes in Mathematics, vol 1964. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-87565-9_3
Download citation
DOI: https://doi.org/10.1007/978-3-540-87565-9_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-87564-2
Online ISBN: 978-3-540-87565-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)
