A Pathological Lebesgue-Measurable Function

1963 ◽  
Vol s1-38 (1) ◽  
pp. 1-4 ◽  
Author(s):  
L. A. Rubel
Keyword(s):  
Measurable Function ◽  
Lebesgue Measurable Function

Some Admissible Estimators in Extreme Value Densities

1975 ◽  
Vol 18 (1) ◽  
pp. 105-110
Author(s):  
R. Singh
Keyword(s):  
Real Number ◽  
Loss Function ◽  
Measurable Function ◽  
Random Variable ◽  
Extreme Value ◽  
Lebesgue Measurable Function ◽  
Single Observation ◽  
Squared Error ◽  
Parametric Functions ◽  
Image Position

Let X be a random variable having the extreme value density of the form(1)where r is assumed to be a positive Lebesgue measurable function of x and the function q is defined byfor all θ in Ω = (0, ∞). It is further assumed that q(θ) approaches zero as θ → ∞.In this note we are concerned with estimating parametric functions g(θ) of the form [1/q(θ)]α, α any real number. The loss function is assumed to be squared error and the estimators are assumed to be functions of a single observation X.

On a class of operators in the hyperfinite $\mathrm{II}_1$ factor

2017 ◽  
Vol 120 (2) ◽  
pp. 249
Author(s):  
Zhangsheng Zhu ◽  
Junsheng Fang ◽  
Rui Shi
Keyword(s):  
Measurable Function ◽  
Von Neumann Algebra ◽  
Irrational Number ◽  
Irrational Rotation ◽  
Identity Operator ◽  
Lebesgue Measurable Function ◽  
Von Neumann ◽  
C Algebra ◽  
Subspace Problem ◽  
Invariant Subspace Problem

Let $R$ be the hyperfinite $\mathrm {II}_1$ factor and let $u$, $v$ be two generators of $R$ such that $u^*u=v^*v=1$ and $vu=e^{2\pi i\theta } uv$ for an irrational number $\theta$. In this paper we study the class of operators $uf(v)$, where $f$ is a bounded Lebesgue measurable function on the unit circle $S^1$. We calculate the spectrum and Brown spectrum of operators $uf(v)$, and study the invariant subspace problem of such operators relative to $R$. We show that under general assumptions the von Neumann algebra generated by $uf(v)$ is an irreducible subfactor of $R$ with index $n$ for some natural number $n$, and the $C^*$-algebra generated by $uf(v)$ and the identity operator is a generalized universal irrational rotation $C^*$-algebra.

On stability of solutions of integral equations in the class of measurable functions

2021 ◽  
pp. 44-54
Author(s):  
Wassim Merchela
Keyword(s):  
Integral Equation ◽  
Uniform Convergence ◽  
Metric Space ◽  
Measurable Function ◽  
Sufficient Conditions ◽  
Stability Of Solutions ◽  
Lebesgue Measurable Function ◽  
Lipschitz Property ◽  
Measurable Functions ◽  
The Stability

Consider the equation G(x)=(y,) ̃ where the mapping G acts from a metric space X into a space Y, on which a distance is defined, y ̃ ∈ Y. The metric in X and the distance in Y can take on the value ∞, the distance satisfies only one property of a metric: the distance between y,z ∈Y is zero if and only if y= z. For mappings X → Y the notions of sets of covering, Lipschitz property, and closedness are defined. In these terms, the assertion is obtained about the stability in the metric space X of solutions of the considered equation to changes of the mapping G and the element y ̃. This assertion is applied to the study of the integral equation f(t,∫_0^1▒K (t,s)x(s)ds,x(t))= y ̃(t),t ∈[0,1], with respect to an unknown Lebesgue measurable function x: [0,1] ∈R. Sufficient conditions are obtained for the stability of solutions (in the space of measurable functions with the topology of uniform convergence) to changes of the functions f,K,(y.) ̃

On Singular Normal Linear Integral Equations

1970 ◽  
Vol 13 (2) ◽  
pp. 199-203 ◽  
Author(s):  
Charles G. Costley
Keyword(s):  
Integral Equations ◽  
Measurable Function ◽  
List Type ◽  
Type Number ◽  
Lebesgue Measurable Function ◽  
Linear Integral ◽  
Image Position ◽  
Linear Integral Equations

In this work we consider the equation1where K(x, y) is singular in the sense that it does not properly belong to L2 and f(x) is an arbitrary L2 function.A Lebesgue measurable function K(x, y) of two variables, having real values on [0.1] × [0.1] is called a singular normal kernel of(i)There exists approximating kernels Km(x, y) satisfying(ii)(iii)(iv)

On Krylov’s estimates for optional semimartingales

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
M. Abdelghani ◽  
A. Melnikov ◽  
A. Pak
Keyword(s):  
Sobolev Space ◽  
Measurable Function ◽  
Diffusion Processes ◽  
Stochastic Integrals ◽  
Controlled Diffusion ◽  
Convergence Of Solutions ◽  
Change Of Variables ◽  
General Class Of Functions ◽  
Class Of Functions ◽  
Controlled Diffusion Processes

Abstract The estimates of N. V. Krylov for distributions of stochastic integrals by means of the L d {L_{d}} -norm of a measurable function are well-known and are widely used in the theory of stochastic differential equations and controlled diffusion processes. We generalize estimates of this type for optional semimartingales, then apply these estimates to prove the change of variables formula for a general class of functions from the Sobolev space W d 2 {W^{2}_{d}} . We also show how to use these estimates for the investigation of L 2 {L^{2}} -convergence of solutions of optional SDE’s.

scholarly journals Convergence of ergodic averages for many group rotations

2015 ◽  
Vol 36 (7) ◽  
pp. 2107-2120
Author(s):  
ZOLTÁN BUCZOLICH ◽  
GABRIELLA KESZTHELYI
Keyword(s):  
Topological Group ◽  
Abelian Group ◽  
Measurable Function ◽  
Dual Group ◽  
Ergodic Averages ◽  
Abelian Topological Group ◽  
Compact Abelian Topological Group ◽  
Image Position ◽  
Rotation Set ◽  
Group Rotations

Suppose that $G$ is a compact Abelian topological group, $m$ is the Haar measure on $G$ and $f:G\rightarrow \mathbb{R}$ is a measurable function. Given $(n_{k})$, a strictly monotone increasing sequence of integers, we consider the non-conventional ergodic/Birkhoff averages $$\begin{eqnarray}M_{N}^{\unicode[STIX]{x1D6FC}}f(x)=\frac{1}{N+1}\mathop{\sum }_{k=0}^{N}f(x+n_{k}\unicode[STIX]{x1D6FC}).\end{eqnarray}$$ The $f$-rotation set is $$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{f}=\{\unicode[STIX]{x1D6FC}\in G:M_{N}^{\unicode[STIX]{x1D6FC}}f(x)\text{ converges for }m\text{ almost every }x\text{ as }N\rightarrow \infty \}.\end{eqnarray}$$We prove that if $G$ is a compact locally connected Abelian group and $f:G\rightarrow \mathbb{R}$ is a measurable function then from $m(\unicode[STIX]{x1D6E4}_{f})>0$ it follows that $f\in L^{1}(G)$. A similar result is established for ordinary Birkhoff averages if $G=Z_{p}$, the group of $p$-adic integers. However, if the dual group, $\widehat{G}$, contains ‘infinitely many multiple torsion’ then such results do not hold if one considers non-conventional Birkhoff averages along ergodic sequences. What really matters in our results is the boundedness of the tail, $f(x+n_{k}\unicode[STIX]{x1D6FC})/k$, $k=1,\ldots ,$ for almost every $x$ for many $\unicode[STIX]{x1D6FC}$; hence, some of our theorems are stated by using instead of $\unicode[STIX]{x1D6E4}_{f}$ slightly larger sets, denoted by $\unicode[STIX]{x1D6E4}_{f,b}$.

On the divergence of Fourier series in the general Haar system

2021 ◽  
pp. 1-10
Author(s):  
Martin Grigoryan ◽  
Artavazd Maranjyan
Keyword(s):  
Fourier Series ◽  
Measurable Function ◽  
Haar System ◽  
Bounded Measurable Function

For any countable set $D \subset [0,1]$, we construct a bounded measurable function $f$ such that the Fourier series of $f$ with respect to the regular general Haar system is divergent on $D$ and convergent on $[0,1]\backslash D$.

scholarly journals I-lacunary summability function of order α

Filomat ◽  
2017 ◽  
Vol 31 (1) ◽  
pp. 69-76
Author(s):  
Ekrem Savaş
Keyword(s):  
Measurable Function ◽  
Statistical Convergence ◽  
Open Problems ◽  
Summability Methods

In this paper, we further generalize recently introduced summability methods in [23](where ideals of N were used to extend certain important summability methods) and introduce new notions, namely, I-statistical convergence of order ?, where 0 < ? < 1 by taking nonnegative real-valued Lebesque measurable function in the interval (1,?). We mainly investigate their relationship and also make some observations about these classes. The study leaves a lot of interesting open problems

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