scholarly journals Reconstruction of the One-Dimensional Lebesgue Measure

2020 ◽  
Vol 28 (1) ◽  
pp. 93-104
Author(s):  
Noboru Endou
Keyword(s):  
Lebesgue Measure ◽  
Real Space ◽  
Outer Measure ◽  
Finite Additivity ◽  
One Dimensional ◽  
The Real ◽  
Measurable Set ◽  
The One ◽  
System 1 ◽  
Dimensional Lebesgue Measure

SummaryIn the Mizar system ([1], [2]), Józef Białas has already given the one-dimensional Lebesgue measure [4]. However, the measure introduced by Białas limited the outer measure to a field with finite additivity. So, although it satisfies the nature of the measure, it cannot specify the length of measurable sets and also it cannot determine what kind of set is a measurable set. From the above, the authors first determined the length of the interval by the outer measure. Specifically, we used the compactness of the real space. Next, we constructed the pre-measure by limiting the outer measure to a semialgebra of intervals. Furthermore, by repeating the extension of the previous measure, we reconstructed the one-dimensional Lebesgue measure [7], [3].

scholarly journals Escaping Orbits Are also Rare in the Almost Periodic Fermi–Ulam Ping-Pong

2021 ◽  
Vol 21 (1) ◽  
Author(s):  
Henrik Schließauf
Keyword(s):  
Lebesgue Measure ◽  
Measure Zero ◽  
Initial Condition ◽  
Almost Periodic ◽  
Periodic Forcing ◽  
One Dimensional ◽  
Forcing Function ◽  
Ping Pong ◽  
The One ◽  
Escaping Orbits

AbstractWe study the one-dimensional Fermi–Ulam ping-pong problem with a Bohr almost periodic forcing function and show that the set of initial condition leading to escaping orbits typically has Lebesgue measure zero.

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.

Elementary volume and measurability properties of additive functions

2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Tamar Kasrashvili ◽  
Aleks Kirtadze
Keyword(s):  
Functional Equation ◽  
Invariant Measure ◽  
Real Line ◽  
Elementary Volume ◽  
Additive Functions ◽  
Translation Invariant ◽  
One Dimensional ◽  
The One ◽  
Dimensional Lebesgue Measure ◽  
Theoretical Standpoint

AbstractThe paper is concerned with some aspects of the theory of elementary volume from the measure-theoretical standpoint. It is shown that there exists a nontrivial solution of Cauchy's functional equation, nonmeasurable with respect to every translation invariant measure on the real line, extending the one-dimensional Lebesgue measure.

The Unruh effect without space–time

2020 ◽  
Vol 17 (04) ◽  
pp. 2050057
Author(s):  
Michele Arzano
Keyword(s):  
Real Line ◽  
Thermal Effects ◽  
Thermal State ◽  
Space Time ◽  
Unruh Effect ◽  
Affine Transformations ◽  
One Dimensional ◽  
The Real ◽  
Simple Quantum ◽  
The One

We show how the characteristic thermal effects found for a quantum field in space–time geometries admitting a causal horizon can be found in a simple quantum system living on the real line. The analysis we present is essentially group theoretic in nature: a thermal state emerges naturally when comparing representations of the group of affine transformations of the real line. The freedom in the choice of different notions of translation generators is the key to the one-dimensional Unruh effect we describe.

TRANSMUTATION OPERATORS AND PALEY–WIENER THEOREM ASSOCIATED WITH A CHEREDNIK TYPE OPERATOR ON THE REAL LINE

2010 ◽  
Vol 08 (04) ◽  
pp. 387-408 ◽  
Author(s):  
MOHAMED ALI MOUROU
Keyword(s):  
Real Line ◽  
Difference Operators ◽  
Generalized Convolution ◽  
One Dimensional ◽  
The Real ◽  
First Order ◽  
Wiener Theorem ◽  
The Fourier Transform ◽  
The One ◽  
Transmutation Operators

We consider a singular differential-difference operator Λ on the real line which generalizes the one-dimensional Cherednik operator. We construct transmutation operators between Λ and first-order regular differential-difference operators on ℝ. We exploit these transmutation operators, firstly to establish a Paley–Wiener theorem for the Fourier transform associated with Λ, and secondly to introduce a generalized convolution on ℝ tied to Λ.

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