Lebesgue Integral Research Articles

An Extension of the 1-Dim Lebesgue Integral of a Product of Two Functions

In this paper, our main aim is to present a reasonable extension of the 1-dim Lebesgue integral of the product of two functions, in case this Lebesgue integral does not exist (i.e., the integrals of its negative and positive parts are both ∞). This extension works fine quite generally, as shown by several examples, and it is based on general hypotheses guaranteeing the sign of the integral (in the sense of being necessarily <0 or =0 or else >0), without computing its actual value. For this purpose, our method provides much more precise results than the Lebesgue–Stieltjes integration by parts.

The integral: a vision of its evolution through time I

This article is the first part of one that has three parts. In the first one we present a vision of the evolution of the idea of integral, from ancient Greece to the Lebesgue integral. In the second part we will present how the Riemann integral has been investigated towards a unified theory of the integral.

Decomposition Integrals of Set-Valued Functions Based on Fuzzy Measures

The decomposition integrals of set-valued functions with regards to fuzzy measures are introduced in a natural way. These integrals are an extension of the decomposition integral for real-valued functions and include several types of set-valued integrals, such as the Aumann integral based on the classical Lebesgue integral, the set-valued Choquet, pan-, concave and Shilkret integrals of set-valued functions with regard to capacity, etc. Some basic properties are presented and the monotonicity of the integrals in the sense of different types of the preorder relations are shown. By means of the monotonicity, the Chebyshev inequalities of decomposition integrals for set-valued functions are established. As a special case, we show the linearity of concave integrals of set-valued functions in terms of the equivalence relation based on a kind of preorder. The coincidences among the set-valued Choquet, the set-valued pan-integral and the set-valued concave integral are presented.

Generalized fractional calculus in Banach spaces and applications to existence results for boundary value problems

In this paper, we present the definitions of fractional integrals and fractional derivatives of a Pettis integrable function with respect to another function. This concept follows the idea of Stieltjes-type operators and should allow us to study fractional integrals using methods known from measure differential equations in abstract spaces. We will show that some of the well-known properties of fractional calculus for the space of Lebesgue integrable functions also hold true in abstract function spaces. In particular, we prove a general Goebel–Rzymowski lemma for the De Blasi measure of weak noncompactness and our fractional integrals.We suggest a new definition of the Caputo fractional derivative with respect to another function, which allows us to investigate the existence of solutions to some Caputo-type fractional boundary value problems. As we deal with some Pettis integrable functions, the main tool utilized in our considerations is based on the technique of measures of weak noncompactness and Mönch’s fixed-point theorem. Finally, to encompass the full scope of this research, some examples illustrating our main results are given.

Sufficient conditions from countably Lipschitz to be strongly generalized absolutely continuous

The CL-integral is an integral that is defined by using countably Lipschitz condition. The CL-integral is more general than the Lebesgue integral, but it is more specific than Denjoy integral in the wide sense. The Denjoy integral in the wide sense is more general than the Henstock-Kurzweil integral. In this paper, it will be given some conditions such that a CL-integrable function is Henctock-Kurzweil-integrable on [a, b].

Connes' integration and Weyl's laws

This paper deals with some questions regarding the notion of integral in the framework of Connes' noncommutative geometry. First, we present a purely spectral theoretic construction of Connes' integral. This answers a question of Alain Connes. We also deal with the compatibility of Dixmier traces with Lebesgue's integral. This answers another question of Alain Connes. We further clarify the relationship of Connes' integration with Weyl's laws for compact operators and Birman–Solomyak's perturbation theory. We also give a “soft proof” of Birman–Solomyak's Weyl's law for negative order pseudodifferential operators on closed manifold. This Weyl's law yields a stronger form of Connes' trace theorem. Finally, we explain the relationship between Connes' integral and semiclassical Weyl's law for Schrödinger operators, including for (fractional) Schrödinger operators on Euclidean spaces and on noncommutative manifolds. We thus get a neat links between noncommutative geometry and semiclassical analysis.

The double Fourier transform of non-Lebesgue integrable functions of bounded Hardy–Krause variation

AbstractIn the classical Fourier analysis, the representation of the double Fourier transform as the integral off(x,y)exp(−i⟨(x,y),(s1,s2)⟩f(x,y)\exp(-i\langle(x,y),(s_{1},s_{2})\rangleis usually defined from the Lebesgue integral. Using an improper Kurzweil–Henstock integral, we obtain a similar representation of the Fourier transform of non-Lebesgue integrable functions onR2\mathbb{R}^{2}. We prove the Riemann–Lebesgue lemma and the pointwise continuity for the classical Fourier transform on a subspace of non-Lebesgue integrable functions which is characterized by the bounded variation functions in the sense of Hardy–Krause. Moreover, we generalize some properties of the classical Fourier transform defined onLp⁢(R2)L^{p}(\mathbb{R}^{2}), where1<p≤21<p\leq 2, yielding a generalization of the results obtained by E. Hewitt and K. A. Ross. With our integral, we define the space of integrable functionsKP⁢(R2)\mathrm{KP}(\mathbb{R}^{2})which contains a subspace whose completion is isometrically isomorphic to the space of integrable distributions on the plane as defined by E. Talvila [The continuous primitive integral in the plane,Real Anal. Exchange45(2020), 2, 283–326]. A question arises about the dual space of the new spaceKP⁢(R2)\mathrm{KP}(\mathbb{R}^{2}).

Topological and topological linear properties of the Sugeno-Lorentz spaces

The properties of spaces of Sugeno integrable functions are quite different from those of ordinary spaces of Lebesgue integrable functions. In our previous research, given a nonadditive measure μ and constants 0<p<∞ and 0<q<∞, the completeness and separability of the Sugeno-Lorentz spaces were discussed with respect to the Sugeno-Lorentz prenorm (⋅)p,q. The purpose of the paper is to further advance our study of the Sugeno-Lorentz spaces from the perspective of topology. To this end, the Sugeno-Lorentz topologies are defined as the topologies generated by the prenorms (⋅)p,q. Some topological and topological linear properties of the Sugeno-Lorentz spaces are discussed. Of particular interest is that the Sugeno-Lorentz spaces and the Sugeno-Lorentz topologies are unique regardless of p and q.

Applications of statistical Riemann and Lebesgue integrability of sequence of functions

Applications of statistical Riemann and Lebesgue integrability of sequence of functions

A categorical derivation of Lebesgue integration

AbstractWe identify simple universal properties that uniquely characterize the Lebesgue spaces. There are two main theorems. The first states that the Banach space , equipped with a small amount of extra structure, is initial as such. The second states that the functor on finite measure spaces, again with some extra structure, is also initial as such. In both cases, the universal characterization of the integrable functions produces a unique characterization of integration. We use the universal properties to derive some of the basic elements of integration theory. We also state universal properties characterizing the sequence spaces and , as well as the functor taking values in Hilbert spaces.

On a geometric combination of functions related to Prékopa–Leindler inequality

AbstractWe introduce a new operation between nonnegative integrable functions on , that we call geometric combination; it is obtained via a mass transportation approach, playing with inverse distribution functions. The main feature of this operation is that the Lebesgue integral of the geometric combination equals the geometric mean of the two separate integrals; as a natural consequence, we derive a new functional inequality of Prékopa–Leindler type. When applied to the characteristic functions of two measurable sets, their geometric combination provides a set whose volume equals the geometric mean of the two separate volumes. In the framework of convex bodies, by comparing the geometric combination with the 0‐sum, we get an alternative proof of the log‐Brunn–Minkowski inequality for unconditional convex bodies and for convex bodies with n symmetries.

On a New Measure on the Levi-Civita Field $$ \mathcal{R} $$

The Levi-Civita field $$ \mathcal{R} $$ is the smallest non-Archimidean ordered field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order. In an earlier paper [13], a measure was defined on $$ \mathcal{R} $$ in terms of the limit of the sums of the lengths of inner and outer covers of a set by countable unions of intervals as those inner and outer sums get closer together. That definition proved useful in developing an integration theory over $$ \mathcal{R} $$ in which the integral satisfies many of the essential properties of the Lebesgue integral of real analysis. Nevertheless, that measure theory lacks some intuitive results that one would expect in any reasonable definition for a measure; for example, the complement of a measurable set within another measurable set need not be measurable. In this paper, we will give a characterization for the measurable sets defined in [13]. Then we will introduce the notion of an outer measure on $$ \mathcal{R} $$ and show some key properties the outer measure has. Finally, we will use the notion of outer measure to define a new measure on $$ \mathcal{R} $$ that proves to be a better generalization of the Lebesgue measure from $$ \mathbb{R} $$ to $$ \mathcal{R} $$ and that leads to a family of measurable sets in $$ \mathcal{R} $$ that strictly contains the family of measurable sets from [13], and for which most of the classic results for Lebesgue measurable sets in $$ \mathbb{R} $$ hold.

General Lebesgue integral inequalities of Jensen and Ostrowski type and applications

Jensen-Ostrowski-type inequalities for general Lebesgue integral are discussed. Here, we derive information inequalities for Lipschitzian derivatives by using new [Formula: see text]-divergence measure which will be useful in Information Theory. In this way, we establish some particular cases of these information inequalities. We also obtain information inequalities for general Lebesgue integral of differentiable functions whose derivatives in absolute value are [Formula: see text]-convex and its particular instances are established.

A New Theorem for Mixed Partial Derivatives

Using the Newton-Leibniz formula for the Lebesgue integral and Lebesgue’s dominated convergence theorem, this Note offers a sufficient condition for the equality of mixed partial derivatives, improving Peano’s result.

Remarks on the global well-posedness of the axisymmetric Boussinesq system with rough initial data

This work establishes a global well-posedness result for the three dimensional axisymmetric Boussinesq system with critical rough initial data. Specifically, we aim at extending the results in our previous paper [22] to the case where the initial vorticity and density are finite Radon measures. The roadmap towards proving our claim is based upon the strategy in [22, 17]. Namely, performing a fixed point argument to build up a unique local solution, then, showing that the constructed solution is in fact global in time. Nevertheless, crucial arguments from [22], required for the construction of the local solution, need to be carefully analyzed when the datum, and more precisely when the initial density, is not a Lebesgue-integrable function. This is where several techniques of measure theory come into play. Accordingly, we first develop numerous notions on axisymmetric measures, in a general context. These techniques are employed thereafter to understand the coupling between some important quantities of the system and, eventually, to construct the desired solution whenever only the atomic parts of the initial measures are small enough.

Jackson-type theorem in the weak $L_{1}$-space

The weak $L_{1}$-space meets in many areas of mathematics. For example, the conjugate functions of Lebesgue integrable functions belong to the weak $L_{1}$-space. The difficulty of working with the weak $L_{1}$-space is that the weak $L_{1}$-space is not a normed space. Moreover, infinitely differentiable (even continuous) functions are not dense in this space. Due to this, the theory of approximation was not produced in this space. In the present paper, we introduced the concept of the modulus of continuity of the functions from the weak $L_{1}$-space, studied its properties, found a criterion for convergence to zero of the modulus of continuity of the function from the weak $L_{1}$-space, and proved in this space an analogue of the Jackson-type theorem.

On the solvability of a functional Volterra integral equation

In this article, we will investigate the existence of a unique bounded variation solution for a functional integral equation of Volterra type in the space L1(R+) of Lebesgue integrable functions.

Affine-periodic solutions for generalized ODEs and other equations

It is known that the concept of affine-periodicity encompasses classic notions of symmetries as the classic periodicity, anti-periodicity and rotating symmetries (in particular, quasi-periodicity). The aim of this paper is to establish the basis of affine-periodic solutions of generalized ODEs. Thus, for a given real number $T&gt; 0$ and an invertible $n\times n$ matrix $Q$, with entries in $\mathbb C$, we establish conditions for the existence of a $(Q,T)$-affine-periodic solution within the framework of nonautonomous generalized ODEs, whose integral form displays the nonabsolute Kurzweil integral, which encompasses many types of integrals, such as the Riemann, the Lebesgue integral, among others. The main tools employed here are the fixed point theorems of Banach and of Krasnosel'skiĭ. We apply our main results to measure differential equations with Henstock-Kurzweil-Stiejtes righthand sides as well as to impulsive differential equations and dynamic equations on time scales which are particular cases of the former.

About the Resolvent Kernel of Neutral Linear Fractional System with Distributed Delays

The present work considers the initial problem (IP) for a linear neutral system with derivatives in Caputo’s sense of incommensurate order, distributed delay and various kinds of initial functions. For the considered IP, the studied problem of existence and uniqueness of a resolvent kernel under some natural assumptions of boundedness type. In the case when, in the system, the term which describes the outer forces is a locally Lebesgue integrable function and the initial function is continuous, it is proved that the studied IP has a unique solution, which has an integral representation via the corresponding resolvent kernel. Applying the obtained results, we establish that, from the existence and uniqueness of a resolvent kernel, the existence and uniqueness of a fundamental matrix of the homogeneous system and vice versa follows. An explicit formula describing the relationship between the resolvent kernel and the fundamental matrix is proved as well.

Hybrid Transforms of Constructible Functions

We introduce a general definition of hybrid transforms for constructible functions. These are integral transforms combining Lebesgue integration and Euler calculus. Lebesgue integration gives access to well-studied kernels and to regularity results, while Euler calculus conveys topological information and allows for compatibility with operations on constructible functions. We conduct a systematic study of such transforms and introduce two new ones: the Euler-Fourier and Euler-Laplace transforms. We show that the first has a left inverse and that the second provides a satisfactory generalization of Govc and Hepworth's persistent magnitude to constructible sheaves, in particular to multi-parameter persistent modules. Finally, we prove index-theoretic formulae expressing a wide class of hybrid transforms as generalized Euler integral transforms. This yields expectation formulae for transforms of constructible functions associated to (sub)level-sets persistence of random Gaussian filtrations.