Related papers: Notes on Measure and Integration
Lecture notes as per the title. In the first part, the concepts of a measurable space, measurable maps between measurable spaces and that of a measure on a measurable space are introduced, after which the fundamentals of the theory of…
We consider Choquet integrals with respect to dyadic Hausdorff content of non-negative functions which are not necessarily Lebesgue measurable. We study the theory of Lebesgue points. The studies yield convergence results and also a density…
In this article, we propose a general theory of integration of the Riemann and Lebesgue types with respect to arbitrary measures and functions, connected by a continuous bilinear product, with values in abstract vector spaces endowed with a…
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…
Advancements in modern science have led to an increased prevalence of functional data, which are usually viewed as elements of the space of square-integrable functions $L^2$. Core methods in functional data analysis, such as functional…
We report on an original formalization of measure and integration theory in the Coq proof assistant. We build the Lebesgue measure following a standard construction that had not yet been formalized in proof assistants based on dependent…
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,…
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,…
This article begins with a review of quantum measure spaces. Quantum forms and indefinite inner-product spaces are then discussed. The main part of the paper introduces a quantum integral and derives some of its properties. The quantum…
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…
We identify simple universal properties that uniquely characterize the Lebesgue $L^p$ spaces. There are two main theorems. The first states that the Banach space $L^p[0, 1]$, equipped with a small amount of extra structure, is initial as…
We present a new type of integral that is supposed to extend the usability of the Lebesgue integral in certain types of investigations. It is based on the Hausdorff dimension and measure. We examine the basic properties of the integral and…
The paper, that continuous some previous work of Sch\"onherr & Schuricht, treats density measures on ${\mathbb R}^n$ that concentrate in any neighborhood of a Lebesgue null set. Such measures are typical for purely finitely additive…
Multidimensional integration by parts formulas apply under the standard assumption that one of the functions is continuous and the other has bounded Hardy-Krause variation. Motivated by recently developed results in the probabilistic…
This paper presents a point-free version of the Lebesgue integral for simple functions on $\sigma$-locales. It describes the integral with respect to a measure defined on the coframe of all $\sigma$-sublocales, moving beyond the constraints…
Given a non-archimedean real closed field with archimedean value group which contains the reals, we establish for the category of semialgebraic sets and functions a full Lebesgue measure and integration theory such that the main results…
This article gives some properties of intervals in $\mathbb{R}$ and discusses some problems involving intervals for which the concept of outer measure on $\mathbb{R}$ provides a more efficient solution than an elementary approach. The outer…
Lebesgue integration is a well-known mathematical tool, used for instance in probability theory, real analysis, and numerical mathematics. Thus its formalization in a proof assistant is to be designed to fit different goals and projects.…
The Lebesgue property (order-continuity) of a monotone convex function on a solid vector space of measurable functions is characterized in terms of (1) the weak inf-compactness of the conjugate function on the order-continuous dual space,…
Alternative approaches to Lebesgue integration are considered.