Non-additive measures and integrals

Rendiconti del Circolo Matematico di Palermo Series 2, 2019

We study Riemann-Lebesgue integrability of a vector function relative to an arbitrary non-negative set function. We obtain some classical integral properties. Results regarding the continuity properties of the integral and relationships among Riemann-Lebesgue, Birkhoff simple and Gould integrabilities are also established.

2011

Information Sciences, 2014

This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues.

The basic theme of this chapter (and a recurring theme in this book) is that we can use integration of functions to help us better understand the measure of sets.

Lecture Notes in Computer Science, 2012

The aim of this paper is to introduce some classes of aggregation functionals when the evaluation scale is a complete lattice. We focus on the notion of quantile of a lattice-valued function which have several properties of its real-valued counterpart and we study a class of aggregation functionals that generalizes Sugeno integrals to the setting of complete lattices. Then we introduce in the real-valued case some classes of aggregation functionals that extend Choquet and Sugeno integrals by considering a multiple quantile model generalizing the approach proposed in [3].

Transactions of the American Mathematical Society, 1971

The systems introduced by R. Henstock and later by E. J. McShane to provide powerful generalizations of the Riemann integral are used to construct outer measures and upper integrals and to develop a Lebesgue type theory in quite general settings.

2006

We give some Fubini's theorems (interversion of the order of integration and product capacities) in the framework of the Choquet integral for product sigma-algebras. Following Ghirardato this is performed by considering slice-comonotonic functions. Our results can be easily interpreted for belief functions, in the Dempster and Shafer setting.

2010

Journal of Mathematical Analysis and Applications, 1973

Introduction. Let J be a <r-field of subsets of an abstract set M and let m(e) be a non-negative measure function defined on J. The classical Radon-Nikodym theorem [17, p. 36](1) states that, if M is the union of a countable number of sets of finite measure, then a necessary and sufficient condition for a completely additive real function R(e), defined over J, to be a Lebesgue integral (with respect to m(e)) is that R(e) be absolutely continuous relative to m(e). Our purpose is to extend this theorem to functions with values in an arbitrary Banach space and apply the resulting theorem to obtain an integral representation for the general bounded linear transformation on the space of summable functions to an arbitrary Banach space. A number of writers [4, 6, 7, 8, 11, 12, 13, 14] have obtained similar extensions; however they have all imposed restrictions either on the Banach space or on the completely additive functions considered. The theorem proved here is free of all such restrictions. It is evident that any such generalization of the Radon-Nikodym theorem will involve a corresponding generalization of the Lebesgue integral, of which there are many. A variation of an integral studied in detail by B. J. Pettis(2) will be used here. A point function x(p) defined on ¥ to a Banach space X is said to be Pettis integrable [12] provided there exists a function X(e) on J to Ï such that, for each element x of the space 3-adjoint to ï and each element e of J, the function x(x(p)) is Lebesgue integrable on the set e to the value x(X(e)). Whenever X(e) exists, it is completely additive and absolutely continuous relative to m(e). On the other hand, Pettis [12, p. 303] gave an example of a completely additive function which is absolutely continuous but is not an integral in his sense. This shows that the ordinary Pettis integral cannot appear in a general Radon-Nikodym theorem. However, without changing essentially the definition or general properties of the integral, we can enlarge the class of functions admissable for integration (so that it contains certain functions other than point functions) and thus obtain an integral which will serve our purposes. The class of functions which we will admit for integration consists of all multivalued set functions x(e) defined for elements of J having finite, nonzero Except for §5, the contents of this paper were presented to the Society, September 12, 1943. The results in §5 were presented February 27, 1944, under the title Representation of linear transformations on summable functions.

Journal of Applied Mathematics, 2013

We consider the regularity for nonadditive measures. We prove that the non-additive measures which satisfy Egoroff&#39;s theorem and have pseudometric generating property possess Radon property (strong regularity) on a complete or a locally compact, separable metric space.

Demonstratio Mathematica, 1992

We review the development of the theory of integra- tion with respect to a vector measure with values in a Banach space. The starting point is a 1955 paper by Bartle, Dunford and Schwartz where the authors consider the vector version of Riesz&#39;s Theorem on bounded linear functionals on spaces of continuous functions over a com- pact space. Next we address the quest for the right look at the space of such integrable functions. We end by looking at applications of the theory.

Real Analysis Exchange

Recently several authors have established a remarkable property of the variational measures associated with a function. Expressed in classical language, this property asserts that if a function is ACG * on all sets of Lebesgue measure zero then the function must be globally ACG *. This article is an exposition of some ideas related to this property with the intention of bringing it to the attention of a wider audience than these original papers might attract. If f : [a, b] → R then a necessary and sufficient condition for the identity f (x)−f (a) = x a f (t) dt in the sense of the Denjoy-Perron integral is that µ f is σ-finite and absolutely continuous with respect to Lebesgue measure on [a, b].

Journal of Mathematical Sciences, 1998

Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.

Proceedings of the American Mathematical Society, 1995

A Fubini theorem for positive linear functional on the vector lattice of the real-valued functions is given. This result properly contains that of the Riemann-¿í-abstract integral.