Search results for "Lebesgue integration"
showing 10 items of 19 documents
A Lebesgue-type decomposition for non-positive sesquilinear forms
2018
A Lebesgue-type decomposition of a (non necessarily non-negative) sesquilinear form with respect to a non-negative one is studied. This decomposition consists of a sum of three parts: two are dominated by an absolutely continuous form and a singular non-negative one, respectively, and the latter is majorized by the product of an absolutely continuous and a singular non-negative forms. The Lebesgue decomposition of a complex measure is given as application.
Estimates for the constants of Landau and Lebesgue via some inequalities for the Wallis ratio
2014
Generalized Lebesgue points for Sobolev functions
2017
In this article, we show that a function $f\in M^{s,p}(X),$ $0<s\leq 1,$ $0<p<1,$ where $X$ is a doubling metric measure space, has generalized Lebesgue points outside a set of $\mathcal{H}^h$-Hausdorff measure zero for a suitable gauge function $h.$
On a multiplication and a theory of integration for belief and plausibility functions
1987
Abstract Belief and plausibility functions have been introduced as generalizations of probability measures, which abandon the axiom of additivity. It turns out that elementwise multiplication is a binary operation on the set of belief functions. If the set functions of the type considered here are defined on a locally compact and separable space X , a theorem by Choquet ensures that they can be represented by a probability measure on the space containing the closed subsets of X , the so-called basic probability assignment. This is basic for defining two new types of integrals. One of them may be used to measure the degree of non-additivity of the belief or plausibility function. The other o…
On a normal form of symmetric maps of [0, 1]
1980
A class of continuous symmetric mappings of [0, 1] into itself is considered leaving invariant a measure absolutely continuous with respect to the Lebesgue measure.
A Constructive Minimal Integral which Includes Lebesgue Integrable Functions and Derivatives
2000
In this paper we provide a minimal constructive integration process of Riemann type which includes the Lebesgue integral and also integrates the derivatives of differentiable functions. We provide a new solution to the classical problem of recovering a function from its derivative by integration, which, unlike the solution provided by Denjoy, Perron and many others, does not possess the generality which is not needed for this purpose.The descriptive version of the problem was treated by A. M. Bruckner, R. J. Fleissner and J. Foran in [2]. Their approach was based on the trivial observation that for the required minimal integral, a function F is the indefinite integral of f if and only if F'…
On the integration of Riemann-measurable vector-valued functions
2016
We confine our attention to convergence theorems and descriptive relationships within some subclasses of Riemann-measurable vector-valued functions that are based on the various generalizations of the Riemann definition of an integral.
Generality of Henstock-Kurzweil type integral on a compact zero-dimensional metric space
2011
ABSTRACT A Henstock-Kurzweil type integral on a compact zero-dimensional metric space is investigated. It is compared with two Perron type integrals. It is also proved that it covers the Lebesgue integral.
Everywhere differentiability of viscosity solutions to a class of Aronsson's equations
2017
For any open set $\Omega\subset\mathbb R^n$ and $n\ge 2$, we establish everywhere differentiability of viscosity solutions to the Aronsson equation $$ =0 \quad \rm in\ \ \Omega, $$ where $H$ is given by $$H(x,\,p)==\sum_{i,\,j=1}^na^{ij}(x)p_i p_j,\ x\in\Omega, \ p\in\mathbb R^n, $$ and $A=(a^{ij}(x))\in C^{1,1}(\bar\Omega,\mathbb R^{n\times n})$ is uniformly elliptic. This extends an earlier theorem by Evans and Smart \cite{es11a} on infinity harmonic functions.
A Riemann-Type Integral on a Measure Space
2005
In a compact Hausdorff measure space we define an integral by partitions of the unity and prove that it is nonabsolutely convergent.