Measurability properties of certain paradoxical subsets of the real line

2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Mariam Beriashvili
Keyword(s):  
Invariant Measure ◽  
Lebesgue Measure ◽  
Real Line ◽  
Translation Invariant ◽  
The Real

AbstractThe paper deals with the measurability properties of some classical subsets of the real line ℝ having an extra-ordinary descriptive structure: Vitali sets, Bernstein sets, Hamel bases, Luzin sets and Sierpiński sets. In particular, it is shown that there exists a translation invariant measure μ on ℝ extending the Lebesgue measure and such that all Sierpiński sets are measurable with respect to μ.

Multifractal formalism for the inverse of random weak Gibbs measures

2019 ◽  
Vol 20 (04) ◽  
pp. 2050024
Author(s):  
Zhihui Yuan
Keyword(s):  
Probability Measure ◽  
Lebesgue Measure ◽  
Real Line ◽  
Gibbs Measures ◽  
Borel Probability Measure ◽  
Cantor Set ◽  
Multifractal Formalism ◽  
The Real ◽  
Random Dynamics ◽  
Borel Probability

Any Borel probability measure supported on a Cantor set included in [Formula: see text] and of zero Lebesgue measure on the real line possesses a discrete inverse measure. We study the validity of the multifractal formalism for the inverse measures of random weak Gibbs measures. The study requires, in particular, to develop in this context of random dynamics a suitable version of the results known for heterogeneous ubiquity associated with deterministic Gibbs measures.

Sojourns and extremes of a diffusion process on a fixed interval

1982 ◽  
Vol 14 (4) ◽  
pp. 811-832 ◽  
Author(s):  
Simeon M. Berman
Keyword(s):  
Diffusion Process ◽  
Lebesgue Measure ◽  
Stationary Distribution ◽  
Real Line ◽  
Diffusion Coefficients ◽  
Limit Theorems ◽  
Random Variable ◽  
Fixed Interval ◽  
The Real ◽  
And Diffusion

Let X(t), , be an Ito diffusion process on the real line. For u > 0 and t > 0, let Lt(u) be the Lebesgue measure of the set . Limit theorems are obtained for (i) the distribution of Lt(u) for u → ∞and fixed t, and (ii) the tail of the distribution of the random variable max[0, t]X(s). The conditions on the process are stated in terms of the drift and diffusion coefficients. These conditions imply the existence of a stationary distribution for the process.

Lebesque measure zero subsets of the real line and an infinite game

1996 ◽  
Vol 61 (1) ◽  
pp. 246-249 ◽  
Author(s):  
Marion Scheepers
Keyword(s):  
Lebesgue Measure ◽  
Real Line ◽  
Measure Zero ◽  
Cardinal Number ◽  
Minimal Cardinality ◽  
Combinatorial Characterization ◽  
The Real ◽  
Lebesque Measure ◽  
The Ideal

Let denote the ideal of Lebesgue measure zero subsets of the real line. Then add() denotes the minimal cardinality of a subset of whose union is not an element of . In [1] Bartoszynski gave an elegant combinatorial characterization of add(), namely: add() is the least cardinal number κ for which the following assertion fails:For every family of at mostκ functions from ω to ω there is a function F from ω to the finite subsets of ω such that:1. For each m, F(m) has at most m + 1 elements, and2. for each f inthere are only finitely many m such that f(m) is not an element of F(m).The symbol A(κ) will denote the assertion above about κ. In the course of his proof, Bartoszynski also shows that the cardinality restriction in 1 is not sharp. Indeed, let (Rm: m < ω) be any sequence of integers such that for each m Rm, ≤ Rm+1, and such that limm→∞Rm = ∞. Then the truth of the assertion A(κ) is preserved if in 1 we say instead that1′. For each m, F(m) has at most Rm elements.We shall use this observation later on. We now define three more statements, denoted B(κ), C(κ) and D(κ), about cardinal number κ.

scholarly journals Sums and products of intervals in ordered semigroups

2021 ◽  
Vol 29 (2) ◽  
pp. 187-198
Author(s):  
T. Glavosits ◽  
Zs. Karácsony
Keyword(s):  
Real Line ◽  
Ordered Semigroup ◽  
Translation Invariant ◽  
Famous Theorem ◽  
Ordered Semigroups ◽  
The Real ◽  
Ordered Ring

Abstract We show a simple example for ordered semigroup 𝕊 = 𝕊 (+,⩽) that 𝕊 ⊆ℝ (ℝ denotes the real line) and ]a, b[ + ]c, d[ = ]a + c, b + d[ for all a, b, c, d ∈ 𝕊 such that a < b and c < d, but the intervals are no translation invariant, that is, the equation c +]a, b[ = ]c + a, c + b[ is not always fulfilled for all elements a, b, c ∈ 𝕊 such that a < b. The multiplicative version of the above example is shown too. The product of open intervals in the ordered ring of all integers (denoted by ℤ) is also investigated. Let Ix := {1, 2, . . ., x} for all x ∈ ℤ+ and defined the function g : ℤ+ → ℤ+ by g ( x ) : = max { y ∈ ℤ + | I y ⊆ I x ⋅ I x } g\left( x \right): = \max \left\{ {y \in {\mathbb{Z}_ + }|{I_y} \subseteq {I_x} \cdot {I_x}} \right\} for all x ∈ ℤ+. We give the function g implicitly using the famous Theorem of Chebishev. Finally, we formulate some questions concerning the above topics.

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.

Approximately dual and perturbation results for generalized translation-invariant frames on LCA groups

2018 ◽  
Vol 16 (03) ◽  
pp. 1850017 ◽  
Author(s):  
Yavar Khedmati ◽  
Mads S. Jakobsen
Keyword(s):  
Real Line ◽  
Locally Compact ◽  
Translation Invariant ◽  
The Real ◽  
Generalized Translation ◽  
Generalized Shift ◽  
Lca Groups

The results in this paper can be divided into three parts. First, we generalize the recent results of Benavente, Christensen and Zakowicz on approximately dual generalized shift-invariant frames on the real line to generalized translation-invariant (GTI) systems on locally compact abelian (LCA) groups. Second, we explain in detail how GTI frames can be realized as [Formula: see text]-frames. Finally, the known results on perturbation of [Formula: see text]-frames and results on perturbation of generalized shift-invariant systems are applied and extended to GTI systems on LCA groups.

scholarly journals On the Measurability of Functions with Respect to Certain Classes of Measures

2004 ◽  
Vol 11 (3) ◽  
pp. 489-494
Author(s):  
A. Kharazishvili ◽  
A. Kirtadze
Keyword(s):  
Lebesgue Measure ◽  
Real Line ◽  
The Real

Abstract The concept of measurability of real-valued functions with respect to various classes of measures is introduced and the associated notion of an absolutely nonmeasurable function is investigated. A characterization of such functions is given. Also, it is shown that functions produced by the classical Vitali partition of the real line are measurable with respect to the class of all extensions of the Lebesgue measure on this line.

Sojourns and extremes of a diffusion process on a fixed interval

1982 ◽  
Vol 14 (04) ◽  
pp. 811-832 ◽  
Author(s):  
Simeon M. Berman
Keyword(s):  
Diffusion Process ◽  
Lebesgue Measure ◽  
Stationary Distribution ◽  
Real Line ◽  
Diffusion Coefficients ◽  
Limit Theorems ◽  
Random Variable ◽  
The Real ◽  
Simple Limit ◽  
And Diffusion

Let X(t), , be an Ito diffusion process on the real line. For u &gt; 0 and t &gt; 0, let Lt (u) be the Lebesgue measure of the set . Limit theorems are obtained for (i) the distribution of Lt (u) for u → ∞and fixed t, and (ii) the tail of the distribution of the random variable max[0, t] X(s). The conditions on the process are stated in terms of the drift and diffusion coefficients. These conditions imply the existence of a stationary distribution for the process.

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