On Nonmeasurable Functions of Two Variables and Iterated Integrals

2009 ◽  
Vol 16 (4) ◽  
pp. 705-710
Author(s):  
Alexander Kharazishvili
Keyword(s):  
Lebesgue Measure ◽  
Two Dimensional ◽  
Iterated Integrals ◽  
Functions Of Two Variables ◽  
Dimensional Lebesgue Measure

Abstract Following the paper of Pkhakadze [Trudy Tbiliss. Mat. Inst. Razmadze 20: 167–209, 1954], we consider some properties of real-valued functions of two variables, which are not assumed to be measurable with respect to the two-dimensional Lebesgue measure on the plane 𝐑2, but for which the corresponding iterated integrals exist and are equal to each other. Close connections of these properties with certain set-theoretical axioms are emphasized.

STARLIKENESS AND CONVEXITY OF CAUCHY TRANSFORMS ON REGULAR POLYGONS

2020 ◽  
pp. 1-12
Author(s):  
PENG-FEI ZHANG ◽  
XIN-HAN DONG
Keyword(s):  
Lebesgue Measure ◽  
Cauchy Transform ◽  
Two Dimensional ◽  
Regular Polygons ◽  
Cauchy Transforms ◽  
Dimensional Lebesgue Measure

Abstract For $n\geq 3$ , let $Q_n\subset \mathbb {C}$ be an arbitrary regular n-sided polygon. We prove that the Cauchy transform $F_{Q_n}$ of the normalised two-dimensional Lebesgue measure on $Q_n$ is univalent and starlike but not convex in $\widehat {\mathbb {C}}\setminus Q_n$ .

The Measure of Plane Sets

1943 ◽  
Vol 39 (1) ◽  
pp. 51-53
Author(s):  
P. A. P. Moran
Keyword(s):  
Finite Number ◽  
Coordinate System ◽  
Lebesgue Measure ◽  
Upper Bound ◽  
Two Dimensional ◽  
Image Position ◽  
Bounded Sets ◽  
Dimensional Lebesgue Measure

We consider bounded sets in a plane. If X is such a set, we denote by Pθ(X) the projection of X on the line y = x tan θ, where x and y belong to some fixed coordinate system. By f(θ, X) we denote the measure of Pθ(X), taking this, in general, as an outer Lebesgue measure. The least upper bound of f (θ, X) for all θ we denote by M. We write sm X for the outer two-dimensional Lebesgue measure of X. Then G. Szekeres(1) has proved that if X consists of a finite number of continua,Béla v. Sz. Nagy(2) has obtained a stronger inequality, and it is the purpose of this paper to show that these results hold for more general classes of sets.

$\beta$-transformation, natural extension and invariant measure

2000 ◽  
Vol 20 (5) ◽  
pp. 1271-1285 ◽  
Author(s):  
GAVIN BROWN ◽  
QINGHE YIN
Keyword(s):  
Invariant Measure ◽  
Lebesgue Measure ◽  
Natural Extension ◽  
Connected Region ◽  
Two Dimensional ◽  
Constant Multiple ◽  
Unit Square ◽  
Simply Connected Region ◽  
Simply Connected ◽  
Dimensional Lebesgue Measure

For $\beta>1$, consider the $\beta$-transformation $T_\beta$. When $\beta$ is an integer, the natural extension of $T_\beta$ can be represented explicitly as a map on the unit square with an invariant measure: the corresponding two-dimensional Lebesgue measure. We show that, under certain conditions on $\beta$, the natural extension is defined on a simply connected region and an invariant measure is a constant multiple of the Lebesgue measure.We characterize those $\beta$ in terms of the $\beta$-expansion of one, and study the structure and size of the set of all such $\beta$.

Subsets of the Plane with Small Linear Sections and Invariant Extensions of the Two-Dimensional Lebesgue Measure

1999 ◽  
Vol 6 (5) ◽  
pp. 441-446
Author(s):  
A. Kharazishvili
Keyword(s):  
Lebesgue Measure ◽  
Euclidean Plane ◽  
Point Of View ◽  
Two Dimensional ◽  
Linear Sections ◽  
Dimensional Lebesgue Measure

Abstract We consider some subsets of the Euclidean plane 𝐑2, having small linear sections (in all directions), and investigate those sets from the point of view of measurability with respect to certain invariant extensions of the classical Lebesgue measure on 𝐑2.

Fundamental group of tiles associated to quadratic canonical number systems

2008 ◽  
Vol 58 (2) ◽  
Author(s):  
Benoit Loridant
Keyword(s):  
Fundamental Group ◽  
Lebesgue Measure ◽  
Affine Plane ◽  
Quadratic Polynomial ◽  
Two Dimensional ◽  
Closed Disk ◽  
Number Systems ◽  
Complete Set ◽  
Dimensional Lebesgue Measure

AbstractIf A is a 2 × 2 expanding matrix with integral coefficients, and ⊂ ℤ2 a complete set of coset representatives of ℤ2/Aℤ2 with |det(A)| elements, then the set ℐ defined by Aℐ = ℐ + is a self-affine plane tile of ℝ2, provided that its two-dimensional Lebesgue measure is positive.It was shown by Luo and Thuswaldner that the fundamental group of such a tile is either trivial or uncountable.To a quadratic polynomial x 2 + Ax + B, A, B ∈ ℤ such that B ≥ 2 and −1 ≤ A ≤ B, one can attach a tile ℐ. Akiyama and Thuswaldner proved the triviality of the fundamental group of this tile for 2A < B + 3, by showing that a tile of this class is homeomorphic to a closed disk. The case 2A ≥ B + 3 is treated here by using the criterion given by Luo and Thuswaldner.

scholarly journals Lossless Digital Filter Overflow Oscillations; Approximation of Invariant Fractals

1997 ◽  
Vol 07 (11) ◽  
pp. 2603-2610 ◽  
Author(s):  
Peter Ashwin ◽  
W. Chambers ◽  
G. Petkov
Keyword(s):  
Lebesgue Measure ◽  
Digital Filters ◽  
Measure Zero ◽  
Two Dimensional ◽  
Finite Precision ◽  
Present Evidence ◽  
Sizeable Fraction ◽  
Finite Precision Arithmetic ◽  
Dimensional Lebesgue Measure ◽  
Overflow Oscillations

We investigate second order lossless digital filters with two's complement overflow. We numerically approximate the fractal set D of points that iterate arbitrarily close to the discontinuity. For the case of eigenvalues of the associated linear map of the form eiθ with θ/π ∉ Q we present evidence that D has positive two dimensional Lebesgue measure. For θ/π ∈ Q we confirm that D has Lebesgue measure zero. As a by-product we get estimates of the exterior dimension of D. These results imply that if such filters are realized using finite-precision arithmetic then they will have a sizeable fraction of orbits that are periodic with high period overflows.

Consistency conditions for a finite set of projections of a function

1979 ◽  
Vol 85 (1) ◽  
pp. 61-68 ◽  
Author(s):  
K. J. Falconer
Keyword(s):  
Lebesgue Measure ◽  
Convex Subset ◽  
Compact Convex ◽  
Unit Vector ◽  
Compact Convex Subset ◽  
Finite Collection ◽  
Finite Set ◽  
Almost All ◽  
Dimensional Lebesgue Measure ◽  
Unit Vectors

Let H(μ, θ) be the hyperplane in Rn (n ≥ 2) that is perpendicular to the unit vector 6 and perpendicular distance μ from the origin; that is, H(μ, θ) = (x ∈ Rn: x. θ = μ). (Note that H(μ, θ) and H(−μ, −θ) are the same hyperplanes.) Let X be a proper compact convex subset of Rm. If f(x) ∈ L1(X) we will denote by F(μ, θ) the projection of f perpendicular to θ; that is, the integral of f(x) over H(μ, θ) with respect to (n − 1)-dimensional Lebesgue measure. By Fubini's Theorem, if f(x) ∈ L1(X), F(μ, θ) exists for almost all μ for every θ. Our aim in this paper is, given a finite collection of unit vectors θ1, …, θN, to characterize the F(μ, θi) that are the projections of some function f(x) with support in X for 1 ≤ i ≤ N.

scholarly journals Characterizations of the lebesgue measure and product measures related to holomorphic functions having non-negative imaginary or real part

2020 ◽  
Vol 31 (12) ◽  
pp. 2050102
Author(s):  
Mitja Nedic
Keyword(s):  
Lebesgue Measure ◽  
Fourier Coefficients ◽  
Holomorphic Functions ◽  
Special Role ◽  
Borel Measures ◽  
Study Product ◽  
Nevanlinna Functions ◽  
Product Measures ◽  
Lebesgue Measures ◽  
Dimensional Lebesgue Measure

In this paper, we study a class of Borel measures on [Formula: see text] that arises as the class of representing measures of Herglotz-Nevanlinna functions. In particular, we study product measures within this class where products with the Lebesgue measures play a special role. Hence, we give several characterizations of the [Formula: see text]-dimensional Lebesgue measure among all such measures and characterize all product measures that appear in this class of measures. Furthermore, analogous results for the class of positive Borel measures on the unit poly-torus with vanishing mixed Fourier coefficients are also presented, and the relation between the two classes of measures with regard to the obtained results is discussed.

Differentiation of n-Dimensional Additive Processes

1981 ◽  
Vol 33 (3) ◽  
pp. 749-768 ◽  
Author(s):  
M. A. Akcoglu ◽  
A. Del Junco
Keyword(s):  
Vector Space ◽  
Lebesgue Measure ◽  
Cartesian Product ◽  
Initial Point ◽  
Positive Cone ◽  
Real Vector Space ◽  
Real Vector ◽  
One Dimensional ◽  
Additive Processes ◽  
Dimensional Lebesgue Measure

Let n ≧ 1 be an integer and let Rn be the usual n-dimensional real vector space, considered together with all its usual structure. The usual n-dimensional Lebesgue measure on Rn is denoted by λn. The positive cone of Rn is Rn+ and the interior of Rn + is Pn. Hence Pn is the set of vectors with strictly positive coordinates. A subset of Rn is called an interval if it is the cartesian product of one dimensional bounded intervals. If a, b ∊ Rn then [a, b] denotes the interval {u|a ≦ u ≦ b|. The closure of any interval I is of the form [a, b]; the initial point of I will be defined as the vector a. The class of all intervals contained in Rn+ is denoted by . Also, for each u ∊ Pn, let be the set of all intervals that are contained in the interval [0, u] and that have non-empty interiors. Finally let en ∊ Pn be the vector with all coordinates equal to 1.

scholarly journals Integration over an Infinite-Dimensional Banach Space and Probabilistic Applications

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Claudio Asci
Keyword(s):  
Banach Space ◽  
Lebesgue Measure ◽  
Dimensional Banach Space ◽  
Infinite Dimensional ◽  
Change Of Variables ◽  
Infinite Dimensional Banach Space ◽  
Dimensional Lebesgue Measure

We study, for some subsets I of N*, the Banach space E of bounded real sequences {xn}n∈I. For any integer k, we introduce a measure over (E,B(E)) that generalizes the k-dimensional Lebesgue measure; consequently, also a theory of integration is defined. The main result of our paper is a change of variables' formula for the integration.

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