Related papers: Lebesgue integration on $\sigma$-locales: simple f…
We study Henstock-type integrals for functions defined in a compact metric space $T$ endowed with a regular $\sigma$-additive measure $\mu$, and taking values in a Banach lattice $X$. In particular, the space $[0,1]$ with the usual Lebesgue…
We prove that every local derivation on a complex semisimple finite-dimensional Leibniz algebra is a derivation.
For a classical weight function $\rho$ defined on a simply connected open subset $\Omega$ of $\mathbb{R}^2$ (either bounded or unbounded) with piecewise $C^1$ boundary, we prove density and compact embedding of a matrix-weighted Sobolev…
When $X$ is locally compact, a quasi-integral (also called a quasi-linear functional) on $ C_c(X)$ is a homogeneous, positive functional that is only assumed to be linear on singly-generated subalgebras. We study simple and almost simple…
For essentially bounded functions on a (closed) compact Riemannian manifold, the noncommutative residue and the Dixmier trace formulation of the noncommutative integral are shown to equate to a multiple of the Lebesgue integral. The…
Motivated by polynomial approximations of differential forms, we study analytical and numerical properties of a polynomial interpolation problem that relies on function averages over interval segments. The usage of segment data gives rise…
The main purpose of this paper is to investigate the behaviour of fractional integral operators associated to a measure on a metric space satisfying just a mild growth condition, namely that the measure of each ball is controlled by a fixed…
We present a replacement for traditional Riemann integrals in undergraduate calculus, which supplements naive precalculus and at the same time opens a way to more sophisticated theories such as Lebesgue integration.
The concept of generalized partial-slice monogenic functions has been recently introduced to include the two theories of monogenic functions and of slice monogenic functions over Clifford algebras. The main purpose of this article is to…
In the present paper, we study a set that can be treated as a generalised set of subsums for a geometric series. This object was discovered independently in various mathematical aspects. For instance, it is closely related to various…
This paper introduces a direct differentiation-based framework that unifies the derivation of influence functions across parametric, nonparametric, and semiparametric models. We show that the Riesz representer of the functional derivative…
This is a companion to our previous paper. Here, we derive local dimension-free estimates for volumes of sub- and super-level sets of analytic functions of several variables.
We obtain a functional model for an arbitrary Abelian locally von Neumann algebra acting on a representing locally Hilbert space under the assumption that the index directed set is countable, in terms of locally essentially bounded…
We estimate the Lebesgue constants for Lagrange interpolation processes on one or several intervals by rational functions with fixed poles. We admit that the poles have accumulation points on the intervals. To prove it we use an analog of…
Lebesgue integration of derivatives of strongly-oscillatory functions is a recurring challenge in computational science and engineering. Integration by parts is an effective remedy for huge computational costs associated with Monte Carlo…
We show that the set of Lebesgue integrable functions in $[0,1]$ which are nowhere essentially bounded is spaceable, improving a result from [F. J. Garc\'{i}a-Pacheco, M. Mart\'{i}n, and J. B. Seoane-Sep\'ulveda. \textit{Lineability,…
The aim of this paper is to provide characterizations of the Lebesgue-almost everywhere continuity of a function f : [a, b] $\rightarrow$ R. These characterizations permit to obtain necessary and sufficient conditions for the Riemann…
The purpose of this paper is to study the lower semicontinuity with respect to the strong $L^1$-convergence, of some integral functionals defined in the space SBD of special functions with bounded deformation. Precisely, let $U$ be a…
A generalized It${\hat {\rm o}}$ formula for time dependent functions of two-dimensional continuous semi-martingales is proved. The formula uses the local time of each coordinate process of the semi-martingale, left space and time first…
In this paper we give a detailed measure theoretical analysis of what we call sum-level sets for regular continued fraction expansions. The first main result is to settle a recent conjecture of Fiala and Kleban, which asserts that the…