A nice example of Lebesgue integration

Beyond Dominated Convergence: Newer Methods of Integration

Lebesgue's dominated convergence theorem is a crucial pillar of modern analysis, but there are certain areas of the subject where this theorem is deficient. Deeper criteria for convergence of integrals are described in this article.

Mixed norm estimates for certain generalized Radon transforms

In this paper we investigate the mapping properties in Lebesgue-type spaces of certain generalized Radon transforms defined by integration over curves.

不同时间尺度下积分恒等式的另一种证明

A Different Demonstration for Integral Identity Across Distinct Time Scales

针对时间尺度积分中Δ-积分转实数域经典积分能否仅用黎曼和实现的问题，本文给出肯定证明，即可积函数下该转化无需勒贝格积分理论支撑。

Fourier analysis with generalized integration

We generalize the classic Fourier transform operator $\mathcal{F}_{p}$ by using the Henstock-Kurzweil integral theory. It is shown that the operator equals the $HK$-Fourier transform on a dense subspace of $\mathcal{ L}^p$, $1<p\leq 2$. In particular, a theoretical scope of this representation is raised to approximate numerically the Fourier transform of functions on the mentioned subspace. Besides, we show differentiability of the Fourier transform function $\mathcal{F}_{p}(f)$ under more general conditions than in Lebesgue's theory.

Bounded convergence theorem for abstract Kurzweil-Stieltjes integral

In the theories of Lebesgue integration and of ordinary differential equations, the Lebesgue Dominated Convergence Theorem provides one of the most widely used tools. Available analogy in the Riemann or Riemann-Stieltjes integration is the Bounded Convergence Theorem, sometimes called also the Arzela or Arzela-Osgood or Osgood Theorem. In the setting of the Kurzweil-Stieltjes integral for real valued functions its proof can be obtained by a slight modification of the proof given for the Young-Stieltjes integral by Hildebrandt in his monograph from 1963. However, it is clear that the proof by H…

Integration and measures on the space of countable labelled graphs

In this paper we develop a rigorous foundation for the study of integration and measures on the space $\mathscr{G}(V)$ of all graphs defined on a countable labelled vertex set $V$. We first study several interrelated $σ$-algebras and a large family of probability measures on graph space. We then focus on a "dyadic" Hamming distance function $\left\| \cdot \right\|_{ψ,2}$, which was very useful in the study of differentiation on $\mathscr{G}(V)$. The function $\left\| \cdot \right\|_{ψ,2}$ is shown to be a Haar measure-preserving bijection from the subset of infinite graphs to the circle (with …

Lusin's Theorem and Bochner Integration

It is shown that the approximating functions used to define the Bochner integral can be formed using geometrically nice sets, such as balls, from a differentiation basis. Moreover, every appropriate sum of this form will be within a preassigned $ε$ of the integral, with the sum for the local errors also less than $ε$. All of this follows from the ubiquity of Lebesgue points, which is a consequence of Lusin's theorem, for which a simple proof is included in the discussion.

Fourier transform inversion: Bounded variation, polynomial growth, Henstock--Stieltjes integration

In this paper we prove pointwise and distributional Fourier transform inversion theorems for functions on the real line that are locally of bounded variation, while in a neighbourhood of infinity are Lebesgue integrable or have polynomial growth. We also allow the Fourier transform to exist in the principal value sense. A function is called regulated if it has a left limit and a right limit at each point. The main inversion theorem is obtained by solving the differential equation $df(t)-iωf(t)=g(t)$ for a regulated function $f$, where $ω$ is a complex number with positive imaginary part. This …

Simple Formula for Integration of Polynomials on a Simplex

We show that integrating a polynomial of degree t on an arbitrary simplex (with respect to Lebesgue measure) reduces to evaluating t homogeneous polynomials of degree j = 1, 2,. .. , t, each at a unique point $ξ$ j of the simplex. This new and very simple formula can be exploited in finite (and extended finite) element methods, as well as in other applications where such integrals are needed.

Logarithms, constructible functions and integration on non-archimedean models of the theory of the real field with restricted analytic functions with value group of finite archimedean rank

Given a model of the theory of the real field with restricted analytic functions such that its value group has finite archimedean rank we show how one can extend the restricted logarithm to a global logarithm with values in the polynomial ring over the model with dimension the archimedean rank. The logarithms are determined by algebraic data from the model, namely by a section of the model and by an embedding of the value group into its Hahn group. If the archimedean rank of the value group coincides with the rational rank the logarithms are equivalent. We illustrate how one can embed such a l…

容量极大不等式及其应用

Capacitary Maximal Inequalities and Applications

本文引入基于Hausdorff容量和Riesz容量的Hardy-Littlewood极大函数容量类比，证明其在容量积分空间上的强型与弱型有界性，并由此改进Lebesgue型微分定理及Sobolev空间中Lebesgue点的估计。

Gauge-measurable functions

In 1973, E.J. McShane proposed an alternative definition of the Lebesgue integral based on Riemann sums, where gauges are used decide what tagged partitions are allowed. Such an approach does not require any preliminary knowledge of Measure theory. We investigate in this paper a definition of measurable functions also based on gauges. Its relation to the gauge-integrable functions that satisfy McShane's definition is obtained using elementary tools from Real Analysis. We show in particular a dominated integration property of gauge-measurable functions.

New Approach To A Generalized Fractional Integral

The paper presents a new formula for the fractional integration, which generalizes the Riemann-Liouville and Hadamard fractional integrals into a single form, which when a parameter fixed at different values, produces the above integrals as special cases. Conditions are given for such a generalized fractional integration operator to be bounded in an extended Lebesgue measurable space. Semigroup property for the above operator is also proved. Finally, we give a general definition of the Fractional derivatives.

Interval Functions and their Integrals, by Ralph Henstock, Ph.D. thesis, 1948

Ralph Henstock (1923 - 2007) worked in non-absolute integration, including the Riemann-complete or gauge integral which, independently, Jaroslav Kurzweil also discovered in the 1950's. As a Cambridge undergraduate Henstock took a course of lectures, by J.C. Burkill, on the integration of interval functions. Later, under the supervision of Paul Dienes in Birkbeck College, London, he undertook research into the ideas of Burkill (interval function integrands) and Dienes (Stieltjes integrands); and he presented this thesis in December 1948. The thesis contains the germ of Henstock's later work, in…

高阶修正梯形法则在勒贝格空间和亚历山德罗夫空间中的应用

Higher order corrected trapezoidal rules in Lebesgue and Alexiewicz spaces

本文研究了在勒贝格和亚历山德罗夫空间中高阶修正梯形法则的积分公式，通过积分分部得到误差估计，并证明了当f为多项式时的精确性。

The continuous primitive integral in the plane

An integral is defined on the plane that includes the Henstock--Kurzweil and Lebesgue integrals (with respect to Lebesgue measure). A space of primitives is taken as the set of continuous real-valued functions $F(x,y)$ defined on the extended real plane $[-\infty,\infty]^2$ that vanish when $x$ or $y$ is $-\infty$. With usual pointwise operations this is a Banach space under the uniform norm. The integrable functions and distributions (generalised functions) are those that are the distributional derivative $\partial^2/(\partial x\partial y)$ of this space of primitives. If $f=\partial^2/(\part…

An integral with respect to probabilistic-valued decomposable measures

Several concepts of approximate reasoning in uncertainty processing are linked to the processing of distribution functions. In this paper we make use of probabilistic framework of approximate reasoning by proposing a Lebesgue-type approach to integration of non-negative real-valued functions with respect to probabilistic-valued decomposable (sub)measures. Basic properties of the corresponding probabilistic integral are investigated in detail. It is shown that certain properties, among them linearity and additivity, depend on the properties of the underlying triangle function providing (sub)add…

The regulated primitive integral

A function on the real line is called regulated if it has a left limit and a right limit at each point. If $f$ is a Schwartz distribution on the real line such that $f=F'$ (distributional or weak derivative) for a regulated function $F$ then the regulated primitive integral of $f$ is $\int_{(a,b)}f=F(b-)-F(a+)$, with similar definitions for other types of intervals. The space of integrable distributions is a Banach space and Banach lattice under the Alexiewicz norm. It contains the spaces of Lebesgue and Henstock--Kurzweil integrable functions as continuous embeddings. It is the completion of …