Related papers: The Daniell Integral: Integration without measure

The basic properties of the Daniell integral are presented. We do not use the standard approach of introducing auxiliary spaces of the "over-functions" and "under-functions." Instead, we use a simple and direct approach based on…

This is an introduction to measure theory, integration and function spaces, with all the needed preliminaries included, and with some applications included as well. We first discuss some basic motivations, coming from discrete probability,…

The theory of integration over infinite-dimensional spaces is known to encounter serious difficulties. Categorical ideas seem to arise naturally on the path to a remedy. Such an approach was suggested and initiated by Segal in his…

In 1948 Feynman introduced functional integration. Long ago the problematic aspect of measures in the space of fields was overcome with the introduction of volume elements in Probability Space, leading to stochastic formulations. More…

This paper contains a development of the Theory of Lebesgue and Bochner spaces of summable functions. It represents a synthesis of the results due to H. Lebesgue, S. Banach, S. Bochner, G. Fubini, S. Saks, F. Riesz, N. Dunford, P. Halmos,…

We introduce a notion of integration defined from filters over families of finite sets. This procedure corresponds to determining the average value of functions whose range lies in any algebraic structure in which finite averages make…

We propose a new integral based on Taylor measures, study its properties extensively, and we illustrate that it includes many concepts from mathematics as special cases. In particular, the new integral emerges as a generalization of the…

A Daniell-Stone type characterization theorem for Aumann integrals of set-valued measurable functions will be proven. It is assumed that the values of these functions are closed convex upper sets, a structure that has been used in some…

Measure Theory and Integration is exposed with the clear aim to help beginning learners to perfectly master its essence. In opposition of a delivery of the contents in an academic and vertical course, the knowledge is broken into exercises…

An integral on Euclidean space, equivalent to the Lebesgue integral, is constructed by extending the notion of Riemann sums. In contrast to the Henstock--Kurzweil and McShane integrals, the construction recovers the full measure-theoretic…

The paper presents a general duality theory for vector measure spaces taking its origin in the author's papers written in the 1960s. The main result establishes a direct correspondence between the geometry of a measure in a vector space and…

In a series of publications in the early 1990s, L D Nel set up a study of non-normable topological vector spaces based on methods in category theory. One of the important results showed that the classical operations of derivative and…

We provide a draft of a theory of geometric integration of rough differential forms which are generalizations of classical (smooth) differential forms to similar objects with very low regularity, for instance, involving H\"older continuous…

We are going to widen the scope of the previously defined Hausdorff-integral in two ways. First, in the sense, that we develop the theory of the integral on some naturally generalized measure spaces. Second, we extend it to functions taking…

Integration at a point is a new kind of integration derived from integration over an interval in infinitesimal and infinity domains which are spaces larger than the reals. Consider a continuous monotonic divergent function that is…

Using the recently defined concept of Taylor measures, we propose a generalization of Taylor's theorem to measurable, non-analytic functions, that do not require differentiation. We study consequences of the generalization, including the…

We construct a new kind of measures, called projection families, which generalize the classical notion of vector and operator-valued measures. The maximal class of reasonable functions admits an integral with respect to a projection family,…

We define an integral, the distributional integral of functions of one real variable, that is more general than the Lebesgue and the Denjoy-Perron-Henstock-Kurzweil integrals, and which allows the integration of functions with…

Using the Laplace derivative a Perron type integral, the Laplace integral, is defined. Moreover, it is shown that this integral includes Perron integral and to show that the inclusion is proper, an example of a function is constructed,…

The process of integration was a subject of significant development during the last century. Despite that the Lebesgue integral is complete and has many good properties, its inability to integrate all derivatives prompted the introduction…