On abstract Fubini theorems for finitely additive integration

International Journal of Advanced Mathematical Sciences, 2013

We consider the integration of functions of two variables in a measure space. Some definitions, theorems and proves relating to measurable functions and measure space were considered by using Fubini's theorem. Application on the improvement of the Jensen's inequality with respect to the probability measure space is treated.

Transactions of the American Mathematical Society, 1990

If "I ' "2' ... , len are cardinals with "I the cardinality of a nonmeasurable set, and for i = 2, 3, ... , n lei is the cardinality of a set of reals which is not the union of "i-I measure-O sets, then for any nonnegative function f: R n-> R all of the iterated integrals Iu = ! ! ... ! f(x i ' x 2 ' ... , xn)dXU(I) dXU(2) ... dXu(n)' (J E Sn' which exist are equal. If all n! of the integrals exist, then the weaker condition of the case n = 2 implies they are equal. These cardinal conditions are consistent with and independent of ZFC, and follow from the existence of a real-valued measure on the continuum. Other necessary conditions and sufficient conditions for the existence and equality of iterated integrals are also treated.

International Journal of Approximate Reasoning, 2008

Since the seminal paper of Ghirardato, it is known that Fubini Theorem for non-additive measures can be available only for functions defined as "slice-comonotonic". We give different assumptions that provide such Fubini theorems in the framework of product σ-algebras.

Indagationes Mathematicae, 1978

Journal of Function Spaces and Applications, 2011

A Fubini-type theorem is proved, for the Kurzweil-Henstock integral of Riesz-space-valued functions defined on (not necessarily bounded) subrectangles of the "extended" real plane.

International Journal of Approximate Reasoning, 2020

This paper is devoted to generalizations of Fubini theorem, Transformation theorem, and generalized Minkowski inequality for the so called pseudo-integral. The approach is based on the relation of double pseudo-integrals and iterated pseudo-integrals. Since the pseudointegral covers Lebesgue integral, Sugeno's fuzzy integral, and Zhao's (N)fuzzy integral (with a respect to special nonadditive measure: ∨-measure), the obtained results generalize the corresponding previously known results.

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.

Axioms

The usefulness of Fubini’s theorem as a measurement instrument is clearly understood from its multiple applications in Analysis, Convex Geometry, Statistics or Number Theory. This article is an expository paper based on a master class given by the second author at the University of Vigo and is devoted to presenting some Applications of Fubini’s theorem. In the first part, we present Brunn–Minkowski’s and Isoperimetric inequalities. The second part is devoted to the estimations of volumes of sections of balls in Rn.

Journal of Mathematical Analysis and Applications, 2012

We establish integral representation results for suitably pointwise continuous and comonotonic additive functionals of bounded variation defined on Stone lattices. As an application, we prove a comonotonic version of the Daniell-Stone Theorem.

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.

Mathematika, 2000

We use methods from descriptive set theory to derive Fubini-like results for the very general Method I and Method II (outer) measure constructions. Such constructions, which often lead to non--nite measures, include Carath eodory and Hausdor -type measures. We encounter several questions of independent interest, such as the measurability of measures of sections of sets, the decomposition of sets into subsets with good sectional properties, and the analyticity of certain operators over sets. We indicate applications to Hausdor and generalised Hausdor measures and to packing dimensions.

Journal of the Australian Mathematical Society, 2003

The integration of vector (and operator) valued functions with respect to vector (and operator) valued measures can be simplified by assuming that the measures involved take values in the positive elements of a Banach lattice.

Demonstratio Mathematica, 1988

arXiv: Probability, 2018

I prove a theorem about iterated integrals for non-product measures in a product space. The first task is to show the existence of a family of measures on the second space, indexed by the points on of the first space (outside a negligible set), such that integrating the measures on the index against the first marginal gives back the original measure (see Theorem 2.1). At the end, I give a simple application in Optimal Transport.

Mediterranean Journal of Mathematics, 2009

The first author introduced an integration theory of vector functions with respect to an operator-valued measure in complete bornological locally convex topological vector spaces. In this paper some important results behind this Dobrakov-type integration technique in non-metrizable spaces are given.