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Integration theory deals with extended real-valued, vector-valued, or operator-valued measures and functions. Different approaches are applied in each of these cases using different techniques. The order structure of the (extended) real number system is used for real-valued functions and measures whereas suprema and infima are replaced with topological limits in the vector-valued case.
A novel approach employing more general structures, locally convex cones, which are natural generalizations of locally convex vector spaces, is introduced here. This setting allows developing a general theory of integration which simultaneously deals with all of the above-mentioned cases.
List of contents
Locally Convex Cones.- Measures and Integrals. The General Theory.- Measures on Locally Compact Spaces.
Summary
Integration theory deals with extended real-valued, vector-valued, or operator-valued measures and functions. Different approaches are applied in each of these cases using different techniques. The order structure of the (extended) real number system is used for real-valued functions and measures whereas suprema and infima are replaced with topological limits in the vector-valued case.
A novel approach employing more general structures, locally convex cones, which are natural generalizations of locally convex vector spaces, is introduced here. This setting allows developing a general theory of integration which simultaneously deals with all of the above-mentioned cases.
Additional text
From the reviews:
“The aim of the present book is to use the theory of locally convex cones for developing a very general and unified theory of integration for extended real-valued, vector-valued, operator-valued and cone-valued countably additive measures and functions. … Providing a very general and nontrivial approach to integration theory, the book is of interest for researchers in functional analysis, abstract integration theory and its applications to integral representations of linear operators. It can be used also for advanced post-graduate courses in functional analysis.” (S. Cobzaş, Studia Universitatis Babeş-Bolyai. Mathematica, Vol. LIV (3), September, 2009)
“This is an interesting book which firstly presents an extension of the theory of locally convex topological vector spaces … . Each chapter finishes with notes and remarks. The book is well written and contains many new results. It is well placed for graduated courses and research work.” (Miguel A. Jiménez, Zentralblatt MATH, Vol. 1187, 2010)
“The present book is a monograph on the general theory of integration for extended real-valued, vector-valued, operator-valued and cone-valued measures and functions. … This book is a nice and useful reference for researchers in abstract integration theory who wish to have a quite comprehensive survey of integration in order structures.” (Antonio Fernández, Mathematical Reviews, Issue 2010 j)
Report
From the reviews:
"The aim of the present book is to use the theory of locally convex cones for developing a very general and unified theory of integration for extended real-valued, vector-valued, operator-valued and cone-valued countably additive measures and functions. ... Providing a very general and nontrivial approach to integration theory, the book is of interest for researchers in functional analysis, abstract integration theory and its applications to integral representations of linear operators. It can be used also for advanced post-graduate courses in functional analysis." (S. Cobzas, Studia Universitatis Babes-Bolyai. Mathematica, Vol. LIV (3), September, 2009)
"This is an interesting book which firstly presents an extension of the theory of locally convex topological vector spaces ... . Each chapter finishes with notes and remarks. The book is well written and contains many new results. It is well placed for graduated courses and research work." (Miguel A. Jiménez, Zentralblatt MATH, Vol. 1187, 2010)
"The present book is a monograph on the general theory of integration for extended real-valued, vector-valued, operator-valued and cone-valued measures and functions. ... This book is a nice and useful reference for researchers in abstract integration theory who wish to have a quite comprehensive survey of integration in order structures." (Antonio Fernández, Mathematical Reviews, Issue 2010 j)
