Infinite Measure Spaces Research Articles

Weak compactness and representation in variable exponent Lebesgue spaces on infinite measure spaces

Relative weakly compact sets and weak convergence in variable exponent Lebesgue spaces {L^{p(cdot )}(Omega )} for infinite measure spaces (Omega ,mu ) are characterized. Criteria recently obtained in [14] for finite measures are here extended to the infinite measure case. In particular, it is showed that the inclusions between variable exponent Lebesgue spaces for infinite measures are never L-weakly compact. A lattice isometric representation of {L^{p(cdot )}(Omega )} as a variable exponent space L^{q(cdot )}(0,1) is given.

Ergodic Schrödinger operators in the infinite measure setting

We develop the basic theory of ergodic Schrödinger operators, which is well known for ergodic probability measures, in the case of a base dynamics on an infinite measure space. This includes the almost sure constancy of the spectrum and the spectral type, the definition and discussion of the density of states measure and the Lyapunov exponent, as well as a version of the Pastur–Ishii theorem. We also give some counterexamples that demonstrate that some results do not extend from the finite measure case to the infinite measure case. These examples are based on some constructions in infinite ergodic theory that may be of independent interest.

Degeneracy results for fully nonlinear integral operators

It is shown that integral operators of the fully nonlinear type K(x)(t)=∫Ωk(t,s,x(t),x(s))ds exhibit similar degeneracy phenomena in a large class of spaces as superposition operators F(x)(t)=f(t,x(t)). In particular, K is Fréchet differentiable in Lp only if it is affine with respect to the “x(t)” argument. Similar degeneracy results hold if K satisfies a local Lipschitz or compactness condition. Also vector functions, infinite measure spaces, and a much richer class of function spaces than only Lp are considered. As a side result, degeneracy assertions for superposition operators are obtained in this more general setting, complementing the known results for scalar functions. As a particular example, it is shown that the operators arising in continuous limits of coupled Kuramoto oscillators fail everywhere to be Fréchet differentiable or locally compact.

A new characterization of the Haagerup property by actions on infinite measure spaces

AbstractThe aim of the article is to provide a characterization of the Haagerup property for locally compact, second countable groups in terms of actions on $\unicode[STIX]{x1D70E}$-finite measure spaces. It is inspired by the very first definition of amenability, namely the existence of an invariant mean on the algebra of essentially bounded, measurable functions on the group.

Entropy of infinite systems and transformations

The Kolmogorov-Sinai entropy is a far reaching dynamical generalization of Shannon entropy of information systems. This entropy works perfectly for probability measure preserving (p.m.p.) transformations. However, it is not useful when there is no finite invariant measure. There are certain successful extensions of the notion of entropy to infinite measure spaces, or transformations with infinite invariant measures. The three main extensions are Parry, Krengel, and Poisson entropies. In this survey, we shortly overview the history of entropy, discuss the pioneering notions of Shannon and later contributions of Kolmogorov and Sinai, and discuss in somewhat more details the extensions to infinite systems. We compare and contrast these entropies with each other and with the entropy on finite systems.

Multidimensional scaling on metric measure spaces

Multidimensional scaling (MDS) is a popular technique for mapping a finite metric space into a low-dimensional Euclidean space in a way that best preserves pairwise distances. We overview the theory of classical MDS, along with its optimality properties and goodness of fit. Further, we present a notion of MDS on infinite metric measure spaces that generalizes these optimality properties. As a consequence we can study the MDS embeddings of the geodesic circle $S^1$ into $\mathbb{R}^m$ for all $m$, and ask questions about the MDS embeddings of the geodesic $n$-spheres $S^n$ into $\mathbb{R}^m$. Finally, we address questions on convergence of MDS. For instance, if a sequence of metric measure spaces converges to a fixed metric measure space $X$, then in what sense do the MDS embeddings of these spaces converge to the MDS embedding of $X$?

The validity space of Dunford–Schwartz pointwise ergodic theorem

We show that if a σ-finite infinite measure space (Ω,μ) is quasi-non-atomic, then the Dunford–Schwartz pointwise ergodic theorem holds for f∈L1(Ω)+L∞(Ω) if and only ifμ{|f|≥λ}<∞ for all λ>0.

Optimal Estimation and Control for Lossy Network: Stability, Convergence, and Performance

In this paper, we study the problems of optimal estimation and control, i.e., the linear quadratic Gaussian (LQG) control, for systems with packet losses but without acknowledgment. Such acknowledgment is a signal sent by the actuator to inform the estimator of the incidence of control packet losses. For such system, which is usually called as a user datagram protocol (UDP)-like system, the optimal estimation is nonlinear and its calculation is time-consuming, making its corresponding optimal LQG problem complicated. We first propose two conditions: 1) the sensor has some computation abilities; and 2) the control command, exerted to the plant, is known to the sensor. For a UDP-like system satisfying these two conditions, we derive the optimal estimation. By constructing the finite and infinite product probability measure spaces for the estimation error covariances (EEC), we give the stability condition for the expected EEC, and show the existence of a measurable function to which the EEC converges in distribution, and propose some practical methods to evaluate the estimation performance. Finally, the LQG controllers are derived, and the conditions for the mean square stability of the closed-loop system are established.

Estimation of integrals with respect to infinite measures using regenerative sequences

Letfbe an integrable function on an infinite measure space (S,, π). We show that if aregenerative sequence{Xn}n≥0with canonical measureπcould be generated then a consistent estimator of λ ≡ ∫Sfdπ can be produced. We further show that under appropriate second moment conditions, a confidence interval for λ can also be derived. This is illustrated with estimating countable sums and integrals with respect to absolutely continuous measures on ℝdusing a simple symmetric random walk on ℤ.

Estimation of integrals with respect to infinite measures using regenerative sequences

Letfbe an integrable function on an infinite measure space (S,, π). We show that if aregenerative sequence{Xn}n≥0with canonical measureπcould be generated then a consistent estimator of λ ≡ ∫Sfdπ can be produced. We further show that under appropriate second moment conditions, a confidence interval for λ can also be derived. This is illustrated with estimating countable sums and integrals with respect to absolutely continuous measures on ℝdusing a simple symmetric random walk on ℤ.

The Daugavet property in rearrangement invariant spaces

We study rearrangement invariant spaces with the Daugavet property. The main result of this paper states that under mild assumptions the only nonseparable rearrangement invariant space X X over an atomless finite measure space with the Daugavet property is L ∞ L_{\infty } endowed with its canonical norm. We also prove that a uniformly monotone rearrangement invariant space over an infinite atomless measure space with the Daugavet property is isometric to L 1 L_1 . As an application we obtain that an Orlicz space over an atomless measure space has the Daugavet property if and only if it is isometrically isomorphic to L 1 L_1 .

On the Countable Measure-Preserving Relation Induced on a Homogeneous Quotient by the Action of a Discrete Group

We consider a countable discrete group $G$ acting ergodicaly and a.e. freely, by measure-preserving transformations, on an infinite measure space $(\mathcal X,\nu)$ with $\sigma$-finite measure $\nu$. Let $\Gamma \subseteq G$ be an almost normal subgroup with fundamental domain $F\subseteq \mathcal X $ of finite measure. Let $\mathcal R_G$ be the countable measurable equivalence relation on $\mathcal X$ determined by the orbits of $G$. Let $\mathcal R_G| F$ be its restriction to $F$. We find an explicit presentation, by generators and relations, for the von Neumann algebra associated, by the Feldman-Moore (\cite{FM}) construction, to the relation $\mathcal R_G|_F$. The generators of the relation $\mathcal R_G|_F$ are a set of transformations of the quotient space $F\cong \mathcal X/ \Gamma$, in a one to one correspondence with the cosets of $\Gamma$ in $G$. We prove that the composition formula for these transformations is an averaged version, with coefficients in $L^\infty(F,\nu)$, of the Hecke algebra product formula (\cite{BC}). In the situation $G = PGL_2(\mathbb Z[\frac1p])$, $\Gamma=PSL_2(\mathbb Z)$, $p\geq 3$ prime number, the relation $\mathcal R_G|_F$ is the equivalence relation associated to a free, measure-preserving action of a free group on $(p+1)/2$ generators on $F$ (\cite{Ad},\cite {Hj}). We use the coset representations of the transformations generating $\mathcal R_G|_F$ to find a canonical treeing (\cite{Ga}).

Ergodic theorems for actions of hyperbolic groups

In this note we give a short proof of a pointwise ergodic theorem for measure-preserving actions of word hyperbolic groups, also obtained recently by Bufetov, Khristoforov and Klimenko. Our approach also applies to infinite measure spaces, and one application is to linear actions of discrete groups on the plane.

Rational weak mixing in infinite measure spaces

AbstractRational weak mixing is a measure theoretic version of Krickeberg’s strong ratio mixing property for infinite measure preserving transformations. It requires ‘density’ ratio convergence for every pair of measurable sets in a dense hereditary ring. Rational weak mixing implies weak rational ergodicity and (spectral) weak mixing. It is enjoyed for example by Markov shifts with Orey’s strong ratio limit property. The power, subsequence version of the property is generic.

Slow entropy and differentiable models for infinite-measure preserving ℤk actions

AbstractWe define ‘slow’ entropy invariants for ℤd actions on infinite measure spaces, which measure growth of itineraries at subexponential scales. We use this notion to construct infinite-measure preserving ℤ2 actions which cannot be realized as a group of diffeomorphisms of a compact manifold preserving a Borel measure, in contrast to the situation for ℤ actions, where every infinite-measure preserving action can be realized in this way.

Convergence in distribution of random metric measure spaces (Λ-coalescent measure trees)

We consider the space of complete and separable metric spaces which are equipped with a probability measure. A notion of convergence is given based on the philosophy that a sequence of metric measure spaces converges if and only if all finite subspaces sampled from these spaces converge. This topology is metrized following Gromov’s idea of embedding two metric spaces isometrically into a common metric space combined with the Prohorov metric between probability measures on a fixed metric space. We show that for this topology convergence in distribution follows—provided the sequence is tight—from convergence of all randomly sampled finite subspaces. We give a characterization of tightness based on quantities which are reasonably easy to calculate. Subspaces of particular interest are the space of real trees and of ultra-metric spaces equipped with a probability measure. As an example we characterize convergence in distribution for the (ultra-)metric measure spaces given by the random genealogies of the Λ-coalescents. We show that the Λ-coalescent defines an infinite (random) metric measure space if and only if the so-called “dust-free”-property holds.

Measurable rigidity of actions on infinite measure homogeneous spaces, II

We consider the problems of measurable isomorphisms and join- ings, measurable centralizers and quotients for certain classes of ergodic group actions on infinite measure spaces. Our main focus is on systems of algebraic origin: actions of lattices and other discrete subgroups < G on homoge- neous spaces G/H where H is a sufficiently rich unimodular subgroup in a semi-simple group G. We also consider actions of discrete groups of isometries < Isom(X) of a pinched negative curvature space X, acting on the space of horospheres Hor(X). For such systems we prove that the only measurable isomorphisms, joinings, quotients etc. are the obvious algebraic (or geometric) ones. This work was inspired by the previous work of Shalom and Steger, but uses completely different techniques which lead to more general results. 1. Introduction and Statement of the Main Results

LIMIT LAWS FOR DISTORTED CRITICAL RETURN TIME PROCESSES IN INFINITE ERGODIC THEORY

We consider conservative ergodic measure preserving transformations on infinite measure spaces and investigate the asymptotic behaviour of distorted return time processes with respect to sets satisfying a type of Darling–Kac condition. We identify two critical cases for which we prove uniform distribution laws.

The behavior of best [formula omitted]-approximations as [formula omitted]. A counterexample of convergence

In the space of summable sequences we give an example of a one-dimensional affine subspace C such that the best L p -approximations of 0 from C fail to converge as p ↓ 1 . We thus give an answer to this problem of convergence in infinite measure spaces.

Universal skyscraper templates for infinite measure preserving transformations

We call an ordered set $\mathbf{c} = (c(i): i \in \mathbb{N})$, of nonnegative extended real numbers $c(i)$, a universal skyscraper template if it is the distribution of first return times for every ergodic measure preserving transformation $T$ of an infinite Lebesgue measure space. If ∑ <SUB><I>i</I></SUB>$ c(i)<\infty$, we give a family of examples of ergodic infinite measure preserving transformations that do not admit c as a skyscraper template. <br> If the distribution $\mathbf{c}$ satisfies $\gcd\{i: c(i) >0 \} = 1 $, and if either of the conditions $c(I) = \infty$ (for some integer $I$), or $i n f_i \{c(i) \} > 0$ is satisfied, then $\mathbf{c}$ is a universal skyscraper template.