Kurzweil-Henstock Integral in Riesz spaces

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In this chapter we introduce the theory of the Kurzweil-Henstock integral for real-valued functions, defined on a bounded interval of the real line.

The main properties are illustrated, the Fundamental Theorem of Calculus and some convergence theorems are proved; moreover some examples and exercises are given.

In this chapter we deal with the fundamental properties of lattice groups and Riesz spaces. We introduce the concepts of order and (D)-convergence, weak σ-distributivity and Egorov property and prove some related results. We deal also with order bounded and order continuous linear functionals in the setting of Riesz spaces. Finally we introduce the Maeda-Ogasawara-Vulikh representation theorem.

In this chapter we present the basic properties and results on the Kurzweil-Henstock integral for Riesz spacevalued functions, defined on a bounded subinterval of the real line. We prove the uniform convergence theorem, and introduce also the Kurzweil-Stieltjes integral and some of its elementary properties.

We introduce the theory of the double integrals for Riesz space-valued mappings, defined on a bounded subrectangle of the Euclidean plane, and prove some versions of the Fubini theorems. We deal also with some concepts of continuity for Riesz space-valued functions, related with these kinds of results.

We deal with the Kurzweil-Henstock integral for functions defined in an abstract compact topological space and taking values in Riesz spaces.

We introduce the theory and the fundamental properties, and in particular we prove the monotone convergence theorem.

We continue to investigate the topics of Chapter 5, following the approach there introduced, and prove some versions of the Henstock Lemma, the Beppo Levi and the Lebesgue dominated convergence theorem.

Note that the involved measures and the domain of our functions can be even unbounded.

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In this chapter we introduce some integrals for real-valued maps with respect to Riesz space-valued set functions, which are not necessarily finitely additive, but in general can be simply only increasing. First we deal with the Sipos (symmetric) integral and prove the monotone and Lebesgue dominated convergence theorems, Fatou’s lemma and the submodular theorem.

Moreover we introduce the Choquet (asymmetric) integral, giving in particular some applications to the weak and strong laws of large numbers in the context of Riesz spaces.

In this chapter we deal with the strong Luzin ((SL)-) integral, related with the existence of primitives of functions in the weak sense. This integral is a variant of the Kurzweil-Henstock integral, which coincides with it in the real case, but is in general slightly different in the context of Riesz spaces, because some pathologies can occur. We also prove some versions of Hake and monotone convergence type theorems and of the Fundamental Theorem of Calculus, together with the basic properties.

In this chapter we begin with investigating the Pettis, Bochner, Gelfand, Dunford, McShane and Kurzweil- Henstock integrals in the context of Banach spaces, and give some comparison results.

Furthermore, we introduce the Pettis-Kurzweil-Henstock integral for Riesz space-valued functions, giving a Hake-type convergence theorem and a version of the Levi monotone convergence theorem.

This chapter contains an introduction to the theory of MV-algebras and states, together with the notion of observable.

It is proved that every probability MV-algebra is weakly σ-distributive and some applications to intuitionistic fuzzy sets (IF-sets) are given.

Furthermore we show that the probability theory of IF-sets can be considered as a particular case of the probability theory on a suitable MV-algebra.

In this chapter we introduce the concept of independence of states, and give a version of the weak law of large numbers. Moreover, we deal with conditional expectation in the context of Riesz spaces, and give three constructions.

Finally we present further results about probability theory in the context of IF-sets and in particular we deal with joint observables.

In this chapter we present a Kurzweil-Henstock-type integral for metric semigroup-valued functions, defined on (possibly unbounded) subintervals of the extended real line. An example of a metric semigroup which is not a group is the set of all fuzzy numbers.

Besides the elementary properties we prove a version of the Henstock lemma and some convergence theorems in this setting.

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