Lebesgue Integral Research Articles

Lebesgue Measure and Integration on Subsets of R^d

Henri Lebesgue, a French mathematician, discovered centuries ago that the Riemann Integral does not work well on unbounded functions. It prompts him to consider another way to integration known as Lebesgue Integral. This paper discusses the Riemann integral's shortcomings and introduce a more thorough concept of integration, the Lebesgue integral, repeated integration. There is also some debate about the Lebesgue measure, which determines the Lebesgue integral. Some examples are given, such as F_σ -set, G_δ -set and Cantor function. In this article, we first look into a unified theory for d-dimensional volume based on the concept of a measure, and then we will use that theory to build a stronger and more flexible theory for integration.

On a class of oscillatory integrals and their application to the time dependent Schrödinger equation

In this paper a class of oscillatory integrals is interpreted as a limit of Lebesgue integrals with Gaussian regularizers. The convergence of the regularized integrals is shown with an improved version of iterative integration by parts that generates additional decaying factors and hence leads to better integrability properties. The general abstract results are then applied to the Cauchy problem for the one dimensional time dependent Schrödinger equation, where the solution is expressed for Cn-regular initial conditions with polynomial growth at infinity via the Green's function as an oscillatory integral.

Exponential Sampling Type Kantorovich Max-Product Neural Network Operators

This article deals with the study of approximation behavior of the exponential sampling type Kantorovich max-product neural network operators based on sigmoidal activation function. We study point-wise and uniform convergence theorems in the space of continuous functions for these operators and determine the rate of convergence by means of logarithmic modulus of continuity. We also investigate the approximation of p -th ( 1 ≤ p < ∞ ) Lebesgue integrable functions by the aforementioned operators and study the rate of convergence exploiting Peetre’s K − functional.

The A-integral and Calderon-Zygmund operators

It is known that the function obtained by the action of the Calderon-Zygmund operator on a Lebesgue integrable function can be non-Lebesgue integrable. In this paper, we prove that the function obtained by the action of the Calderon-Zygmund operator on a Lebesgue integrable function is A-integrable and derive an analogue of the Riesz equality.

Approximating Choquet integral in generalized measure theory: Choquet-midpoint rule

In generalized measure theory, Choquet integral is a generalization of Lebesgue integral and mathematical expectation. Approximating Choquet integral in the continuous case on real line is not very easy. So, we need mainly to estimate Choquet integral with respect to non-additive measures. There are few studies on the approximating Choquet integral in the continuous case on real line. In approximation theory, there are many interesting properties of midpoint rule. As a subject for research, there are no results on the midpoint rule for Choquet integral. The main objective of this paper is to propose some applications of midpoint rule for approximating continuous Choquet integral. The choquet-midpoint rule helps us to numerically solve Choquet integrals, in particular, the singular and unbounded integrals. Several numerical examples are considered to illustrate the application of our proposed methodology.

Representation of random variables as Lebesgue integrals

Representation of random variables as Lebesgue integrals

The Boundedness of Generalized Fractional Integral Operators on Small Morrey Spaces

The small Morrey space is the set of locally Lebesgue integrable functions with norm defined supremum over radius of ball . This paper aims to prove the boundedness properties of the generality of fractional integral operators in small Morrey spaces using Hedberg-type inequality. The first, in this paper will be discuss to prove Hedberg-type inequality on small Morrey spaces using dyadic decomposition, H lder inequality, and doubling condition. Furthermore, by using the inequality, it can be proven that the boundedness of generalized fractional integral operators on small Morrey spaces.

Sufficient conditions for the Lebesgue integrability of Mellin transforms

This work is devoted to the study of the integrability of Mellin transforms under some sufficient and necessary conditions in the space X c p of Lebesgue measurable functions f on R + = ( 0 , + ∞ ) such that ∫ 0 + ∞ | u c f ( u ) | p d u u < ∞ , c ∈ R ( 1 < p ≤ 2 ) . These results are inspired by a Móricz-type result in classical Fourier analysis, which generalizes a well-known theorem of Titchmarsh.

On the properties of one auxiliary function for calculating the global extremum

In this paper, we investigate the properties of one Auxiliary Function (AF) for calculating the global optimum of a function of many variables. The considered AF is constructed by transforming the objective function using the Lebesgue integral and is a function of one variable. Despite the fact that in previous works this AF was used to find the global minimum of smooth functions on convex closed sets using the algorithm of dividing a segment in half, its properties have not been studied. Here, this AF is used to calculate the global extremum of continuous functions defined on bounded closed sets of a multidimensional Euclidean space. The basic properties of the AF for any continuous objective function are established, such as non-negativity, positive uniformity, uniform continuity, differentiability and strict convexity, moreover, derivatives up to a certain order are found. The optimality criterion is formulated. The main criterion of optimality is that the value of a free variable, at which the AF and its derivatives are equal to zero up to some order, coincides with the global minimum of the objective function. It follows from this optimality criterion that to calculate the global minimum of the objective function, it is sufficient to find the zero of the AF or its derivative up to some order.

Comparative study of riemann integral and lebesgue integral in calculus

The aim of this research paper is to provide a comprehensive comparison between Riemann integral and Lebesgue integral. Integration is described as the inverse process of differentiation, which is used to determine the original function. Riemann integration is a specific type of definite integral applied to find the exact area under a function graph between two limits in a closed interval. Lebesgue integration, on the other hand, provides a more generalized framework for integration theory. Integrals are essential in mathematical modelling and analysis tools. Studying and comparing the similarities and differences between these two integration methods can help us better understand the essence and properties of integrals, so as to more accurately apply integration methods to solve practical problems. This paper provides a systematic analysis of the basics, definitions, concepts, and properties of Riemann integral and Lebesgue integral. Reasoning, proofs, and examples are consolidated to explain the properties and characteristics of these two integral methods. Finally, this paper explores the strengths and limitations of each integration methods, summarising their advantages and applicability to practical problems and providing insights into their respective computational methods and applicability in different contexts.

A proof of the additivity of rough integral

On the basis of fractional calculus, we introduce an explicit formulation of the integral of controlled paths along Hölder rough paths in terms of Lebesgue integrals for fractional derivatives. The additivity with respect to the interval of integration, a fundamental property of the integral, is not apparent under the formulation because the fractional derivatives depend heavily on the endpoints of the interval of integration. In this paper, we provide a proof of the additivity of the integral under the formulation. Our proof seems to be simpler than those provided in previous studies and is suitable for utilizing the fractional calculus approach to rough path analysis.

An s-first return examination on s-sets

The purpose of this work is to analyze an integral of the s-Riemann type, where the gauge is a positive constant but the points involved in the s-Riemann sums are not randomly chosen. We demonstrate that, under this novel approach, every Hs\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {H}}^{s}$$\\end{document}-Lebesgue integrable function is integrable.

Riemann Integral on Fractal Structures

In this work we start developing a Riemann-type integration theory on spaces which are equipped with a fractal structure. These topological structures have a recursive nature, which allows us to guarantee a good approximation to the true value of a certain integral with respect to some measure defined on the Borel σ-algebra of the space. We give the notion of Darboux sums and lower and upper Riemann integrals of a bounded function when given a measure and a fractal structure. Furthermore, we give the notion of a Riemann-integrable function in this context and prove that each μ-measurable function is Riemann-integrable with respect to μ. Moreover, if μ is the Lebesgue measure, then the Lebesgue integral on a bounded set of Rn meets the Riemann integral with respect to the Lebesgue measure in the context of measures and fractal structures. Finally, we give some examples showing that we can calculate improper integrals and integrals on fractal sets.

On the combination of Lebesgue and Riemann integrals in theory of convolution equations

On the combination of Lebesgue and Riemann integrals in theory of convolution equations

Symmetry and asymptotic solutions for a magnetohydrodynamics Darcy–Forchheimer flow with a p-Laplacian operator

Fluid flows under a p-Laplacian operator formulation have been considered recently in connection with the modeling of non-Newtonian fluid processes. To a certain extent, the main reason behind the interest in p-Laplacian operators is the possibility of determining experimental values for the constant p appearing in them. The goal of the present study is to introduce the analysis of solutions of a one-dimensional porous media flow arising in magnetohydrodynamics with generalized initial data under a Lebesgue integrability condition. We present a weak formulation of this problem, and we consider boundedness and uniqueness properties of solutions and also its asymptotic relation with the standard parabolic p-Laplacian equation. Then, we explore solutions arising from classical symmetries (including an explicit kink solution in the p = 3 case) along with asymptotic stationary and non-stationary solutions. The search for stationary solutions is based on a Hamiltonian approach. Finally, non-stationary solutions are obtained by using an exponential scaling resulting in a Hamilton–Jacobi type of equation.

Quantum structure of the surface layer of metals

In this paper, we propose a model for determining the thickness of the surface layer of metals and the quantum structure of this layer. An atomically smooth metal is represented as a diagram: nanolayer → mesolayer → bulk phase, which differ from each other in the nature of size effects. There is no size effect in the bulk phase. The thickness of the surface layer of metals R(I) has a size from 1 nm to 10 nm, except for cesium, i.e. they represent a nanostructure. It is shown that the energy levels En of the nanolayer are determined by one fundamental parameter - the lattice constant of the metal à. As soon as the parameter à stops changing, the spectrum of quantum states passes into a continuous spectrum. The nanolayer R(I) is a step function in En, which can be easily reduced to the Lebesgue integral, which plays an important role in quantum theory. Quantum threads (quantum planes) in the nanolayer R(I) can be interpreted as solitons, crowdions, discrete breathers turning into nanocracks. Quantum threads in the R(I) nanolayer and the nanolayer itself can be represented as a nanoparticle with a diameter of R(I) and pre-melting occurs in it in a stepwise manner. It is finally shown that all metal monolayers differ significantly from each other.

Inequalities in Riemann–Lebesgue Integrability

In this paper, we prove some inequalities for Riemann–Lebesgue integrable functions when the considered integration is obtained via a non-additive measure, including the reverse Hölder inequality and the reverse Minkowski inequality. Then, we generalize these inequalities to the framework of a multivalued case, in particular for Riemann–Lebesgue integrable interval-valued multifunctions, and obtain some inequalities, such as a Minkowski-type inequality, a Beckenbach-type inequality and some generalizations of Hölder inequalities.

Convergence of random elements via fuzzy integrals

Can convergence of random variables, random vectors or, more generally, random elements of a metric space be studied using fuzzy integrals, in particular the seminormed integral? Among the positive results we present, it is shown that for large families of t-seminorms convergence in distribution and convergence in probability can be characterized via seminormed integrals. This suggests that, while fuzzy integrals were devised for working with non-additive fuzzy measures, some of their theoretical properties within the realm of probability measures can be comparable to those of the Lebesgue integral.

Extension of the Continuity Theorems of Lebesgue Integration

The Dominated Convergence Theorem, Fatou's Lemma, and the Monotone Convergence Theorem are collectively referred to as the continuity theorems of Lebesgue integration. These ubiquitous results are stated and proved in the literature for sequences of measurable functions. It is well known that these results do not hold for arbitrary nets of measurable functions. These results, as well as the Levy Continuity Theorem, Riesz's characterization of $\mathcal{L}_p$ convergence in terms of almost everywhere convergence and convergence of norms, and a stronger version of Scheffe's Theorem, are derived for nets indexed by countably accessible directed sets.

Order of Approximation by a New Univariate Kantorovich Type Operator

In order to approximate Lebesgue integrable functions on [0, 1], a sequence of linear positive integral operators of Kantorovich type Lσ&lt;sσ&gt;f (x) with a parameter sσ is introduced. The estimates for rates of approximation for functions with a specific smoothness are proved using the appropriate modulus of continuity.