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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.

Integrals, Generalized --- Functions of real variables --- Convex domains --- Vector spaces --- Calculus --- Mathematics --- Physical Sciences & Mathematics --- Operator-valued measures. --- Integrals. --- Measures, Operator-valued --- Mathematics. --- Functional analysis. --- Measure theory. --- Measure and Integration. --- Functional Analysis. --- Lebesgue measure --- Measurable sets --- Measure of a set --- Algebraic topology --- Measure algebras --- Rings (Algebra) --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Math --- Science --- Calculus, Integral --- Measure theory --- Operator theory

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This book presents a wide-ranging approach to operator-valued measures and integrals of both vector-valued and set-valued functions. It covers convergence theorems and an integral representation for linear operators on spaces of continuous vector-valued functions on a locally compact space. These are used to extend Choquet theory, which was originally formulated for linear functionals on spaces of real-valued functions, to operators of this type.

Linear operators. --- Integral representations. --- Operator-Valued Measures, Integrals for Vector- and Set-Valued Functions, Integral Representations for Linear Operators, Choquet Theory for Linear Operators. --- Function spaces. --- Choquet theory. --- Espaces fonctionnels. --- Représentations intégrales. --- Théorie de Choquet. --- Opérateurs linéaires.

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Functional analysis --- Approximatietheorie --- Approximation theory --- Cones (Operator theory) --- Théorie des approximations --- Approximation theory. --- 51 --- Cones of operators --- Operators, Cones of --- Linear operators --- Theory of approximation --- Functions --- Polynomials --- Chebyshev systems --- Mathematics --- 51 Mathematics