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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