Finitely additive integration

Cardinal functions of partially ordered sets, topological spaces and Boolean algebras; precalibers; ideals of sets.

Journal of Mathematical Analysis and Applications, 1987

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Journal of the London Mathematical Society, 1968

2017

In this paper, we define finitely additive, probability and modular functions over semiring-like structures. We investigate finitely additive functions with the help of complemented elements of a semiring. Then with the help of those observations, we also generalize some classical results in probability theory such as Boole's Inequality, the Law of Total Probability, Bayes' Theorem, the Equality of Parallel Systems, and Poincar\'{e}'s Inclusion-Exclusion Theorem. While we prove that modular functions over a couple of celebrated semirings are almost constant, we show it is possible to define many different modular functions over some semirings such as bottleneck algebras and the semiring (Id(D),+,⋅), where D is a Dedekind domain. Finally, we prove that under suitable conditions a function f is finitely additive iff it is modular and f(0)=0.

2008

There are presented two recent results on integrals based on non-additive measures. First is related to Jensen type inequality for a pseudo-integral, and the second is a connection of integral with aggregation functions with infinite inputs.

Journal of Mathematical Analysis and Applications, 2001

The semiatom is a basic concept in the non-additive measure theory, or the fuzzy measure theory, and has been used for applications of the theory (T.

Bulletin of the Australian Mathematical Society, 1992

The set functions associated with Schr odinger's equation are known to be unbounded on the algebra of cylinder sets. However, there do exist examples of scalar values set functions which are unbounded, yet -additive on the underlying algebra of sets. The purpose of this note is to show that the set functions associated with Schr odinger's equation and not -additive on cylinder sets. In the course of the proof, general conditions implying the non -additivity of unbounded set functions are given.

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.

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.

Glasgow Mathematical Journal, 1975

Baire functions is defined to be the smallest subclass of [-oo, oo] x which contains L and is closed under the formation of monotone, pointwise, sequential limits, so that SS L sf n /f or & L sf n \f=> Segal and Kunze [7], following Loomis [4], used this idea as the basis for a very efficient, elementary presentation of the theory of measure spaces (X, S, n) and their associated integral spaces (X, H?(S, n), \-dfi). Maron [5] then used it to study (not necessarily constructed) abstract integral spaces (X, £C, /) in the absence of any structure on X. In [7], measurable functions and constructed integrals are presented in an "integral oriented" way to illuminate the role of L-Baire functions. In this article we show how to use .L-Baire functions to give quick, informative proofs of the basic properties of measures and their associated integrals (on the class of summable, measurable functions) using the "measure oriented" definitions of Halmos [2]. The idea is to show that the objects defined in the "measure oriented" way coincide with an "integral oriented" counterpart, hence a fortiori have the desired properties (see (2.5), (2.6) and (5.5)). Having done this, it is easy to obtain a very sharp and general Riesz-Markov type theorem (6.3) which describes the 1-1 correspondence between the collection of all (not necessarily complete) a-finite measure spaces (X, S, n) and the collection of all integral spaces (X, SP, I) for which £C satisfies the hypothesis of Stone [8] : / e £C =>f A 1 e £C. To obtain this generality, we use the definition of an integral space (X, £?, I) given in [5] which avoids null sets by allowingi? «= [-oo, oo]*. A discussion of the results obtained and their proofs is given in § 7.

JOURNAL OF ADVANCES IN MATHEMATICS

In this paper, we will represent some applications to various problems of mass theory and integration, by using the concept of local convergences and exhaustive sequences. We will continue the idea of point-wise I -convergence, Ideal exhaustiveness that was introduced by Komisarski [3], and Kostyrko, Sal´at and Wilczy´nski [4]. The equi-integrable introduced in Bohner-type ideal integrals and a new study on the application of symmetric differences have been presented in the theory of mass and continuous functions, continuing the results of Boccuto, Das, Dimitriou, Papanastassiou [2].

Proceedings of the American Mathematical Society, 1998

We prove that quasi-measures on compact Hausdorff spaces are countably additive. Contained in this result is a proof that every quasi-measure decomposes uniquely into a measure and a quasi-measure that has no smaller measure beneath it. We also show that it is consistent with the usual axioms of set-theory that quasi-measures on compact Hausdorff spaces are ℵ 1 \aleph _1 -additive. Finally, we construct an example that places strong restrictions on other forms of additivity.

Proceedings of the American Mathematical Society, 1985

We show that under certain general conditions any finitely additive measure which is defined for all subsets of a set X and is invariant under the action of a group G acting on X is concentrated on a G-invariant subset Y on which the G-action factors to that of an amenable group. The result is then applied to prove a conjecture of S. Wagon about finitely additive measures on spheres.

Journal of Mathematical Analysis and Applications, 1965