A Riesz representation theorem for cone-valued functions

A Riesz Representation Theorem for Cone-valued Functions

The theory of locally convex cones, as developed in [3], deals with ordered cones that are not necessarily embeddable in vector spaces. A topological structure is introduced using order theoretical concepts. We will review some of the main concepts and globally refer to [3] for details and proofs. An ordered cone is a set endowed with an addition and a scalar multiplication for nonnegative real...

The Riesz representation theorem

The Bounded Convergence Theorem for Riesz Space-Valued Choquet Integrals

The bounded convergence theorem on the Riesz space-valued Choquet integral is formalized for a sequence of measurable functions converging in measure and in distribution. 2010 Mathematics Subject Classification: Primary 28B15; Secondary 28A12, 28E10

The Egoroff Theorem for Riesz Space-valued Monotone Measures

In 1974, Sugeno introduced the notion of fuzzy measure and integral to evaluate nonadditive or non-linear quality in systems engineering. In the same year, Dobrakov independently introduced the notion of submeasure from mathematical point of view to show that most of the theory of countably additive measures remain valid for such measures. Fuzzy measures and submeasures are both special kinds o...

A Formal Proof Of The Riesz Representation Theorem

This paper presents a formal proof of the Riesz representation theorem in the PVS theorem prover. The Riemann Stieltjes integral was defined in PVS, and the theorem relies on this integral. In order to prove the Riesz representation theorem, it was necessary to prove that continuous functions on a closed interval are Riemann Stieltjes integrable with respect to any function of bounded variation...