Infinite Measure Spaces Research Articles

Computability of measurable sets via effective topologies

We investigate in the frame of TTE the computability of functions of the measurable sets from an infinite computable measure space such as the measure and the four kinds of set operations. We first present a series of undecidability and incomputability results about measurable sets. Then we construct several examples of computable topological spaces from the abstract infinite computable measure space, and analyze the computability of the considered functions via respectively each of the standard representations of the computable topological spaces constructed.

Computability of measurable sets via effective metrics

We consider how to represent the measurable sets in an infinite measure space. We use sequences of simple measurable sets converging under metrics to represent general measurable sets. Then we study the computability of the measure and the set operators of measurable sets with respect to such representations. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Occupation times of sets of infinite measure for ergodic transformations

Assume that T is a conservative ergodic measure preserving transformation of the infinite measure space (X,A,µ). We study the asymptotic behaviour of occupation times of certain subsets of infinite measure. Specifically, we prove a Darling-Kac type distributional limit theorem for occupation times of barely infinite components which are separated from the rest of the space by a set of finite measure with c.f.-mixing return process. In the same setup we show that the ratios of occupation times of two components separated in this way diverge almost everywhere. These

On the extrapolation estimates

We present elementary proofs of precise Yano’s type extrapolation estimates on infinite measure space recently proved by M. J. Carro.

The Schur and (weak) Dunford-Pettis properties in Banach lattices

AbstractWe study the Schur and (weak) Dunford-Pettis properties in Banach lattices. We show that l1, c0 and l∞ are the only Banach symmetric sequence spaces with the weak Dunford-Pettis property. We also characterize a large class of Banach lattices without the (weak) Dunford-Pettis property. In MusielakOrlicz sequence spaces we give some necessary and sufficient conditions for the Schur property, extending the Yamamuro result. We also present a number of results on the Schur property in weighted Orlicz sequence spaces, and, in particular, we find a complete characterization of this property for weights belonging to class ∧. We also present examples of weighted Orlicz spaces with the Schur property which are not L1-spaces. Finally, as an application of the results in sequence spaces, we provide a description of the weak Dunford-Pettis and the positive Schur properties in Orlicz spaces over an infinite non-atomic measure space.

Criterion on the limits of superprocesses

Starting with super-diffusion processes, the asymptotic behavior of the superprocesses on finite and infinite measure spaces is systematically studied by general branching mechanisms. The complete features of their limits are described and a criterion on their behavior is presented.

Functions of conditionally negative type on Kazhdan groups

An explicit bound is given for functions of conditionally negative type on Kazhdan groups, in terms of a set of generators and the corresponding Kazhdan constants. This is used to estimate how far a set in an infinite measure space can be translated by the action of a Kazhdan group. Some estimates are given for Kazhdan constants.

The strength of the isomorphism property

AbstractIn § 1 of this paper, we characterize the isomorphism property of nonstandard universes in terms of the realization of some second-order types in model theory. In §2, several applications are given. One of the applications answers a question of D. Ross in [this Journal, vol. 55 (1990), pp. 1233–1242] about infinite Loeb measure spaces.

Orlicz spaces which areL p -spaces

Let Ф denote a finite-valued Orlicz function vanishing only at zero and consider the Orlicz spaceL Ф over a non-atomic, σ-finite and infinite measure space with the norm ∥ · ∥ being either the Luxemburg norm or (in the case where $$\mathop {\lim }\limits_{u \to 0} \Phi (u)/(u) = 0$$ and $$\mathop {\lim }\limits_{u \to + \infty } \Phi (u)/(u) = + \infty $$ the Orlicz norm. Ifp ∈ [1, + ∞) and $$||1_A + 1_B ||^P = ||1_A ||^P + ||1_B ||^P $$ for all disjoint setsA andB of positive and finite measure, thenL Ф is the Lebesgue spaceL p .

Second Order Ergodic Theorems for Ergodic Transformations of Infinite Measure Spaces

Second Order Ergodic Theorems for Ergodic Transformations of Infinite Measure Spaces

Second order ergodic theorems for ergodic transformations of infinite measure spaces

For certain pointwise dual ergodic transformations T T we prove almost sure convergence of the log-averages \[ 1 log ⁡ N ∑ n = 1 N 1 n a ( n ) ∑ k = 1 n f ∘ T k ( f ∈ L 1 ) \frac {1}{{\log N}}\sum \limits _{n = 1}^N {\frac {1}{{na\left ( n \right )}}\sum \limits _{k = 1}^n {f \circ {T^k}\left ( {f \in {L_1}} \right )} } \] and the Chung-Erdös averages \[ 1 log ⁡ a ( N ) ∑ k = 1 N 1 a ( k ) f ∘ T k ( f ∈ L 1 + ) \frac {1}{{\log a\left ( N \right )}}\sum \limits _{k = 1}^N {\frac {1}{{a\left ( k \right )}}f \circ {T^k}} \left ( {f \in L_1^ + } \right ) \] towards ∫ f \smallint f , where a ( n ) a\left ( n \right ) denotes the return sequence of T T .

The Converse of the Dominated Ergodic Theorem in Hurewicz Setting

AbstractThe converse of the dominated ergodic theorem in infinite measure spaces is extended to non-singular transformations, i.e. transformations that only preserve the measure of null sets. An inverse weak maximal inequality is given and then applied to obtain related results in Orlicz spaces.

Ergodic transformations and sequences of integers

Using an ergodic transformation defined on an infinite measure space, we discuss complements in ℤ of the setA consisting of finite sums of odd powers of 2.

Infinite tensor products of commutative subspace lattices

Every infinite tensor product of commutative subspace lattices is unitarily equivalent to a certain lattice of projections on L 2 ( X , ν ) {L^2}(X,\nu ) , where ( X , ν ) (X,\nu ) is an infinite product measure space. This representation reflects the structure of the individual component lattices in that the components of the tensor product correspond to the coordinates of the product space. This result generalizes the similar representation for finite tensor products. It is then used to show that an infinite tensor product of purely atomic commutative subspace lattices must be either purely atomic or noncompact, and in the latter case the algebra of operators under which the lattice is invariant has no compact operators.

Order preserving nonexpansive operators inL 1

We study the limit behaviour ofT k f and of Cesaro averagesA n f of this sequence, whenT is order preserving and nonexpansive inL 1 + . IfT contracts also theL ∞-norm, the sequenceT n f converges in distribution, andA n f converges weakly inL p (1<p<∞), and also inL 1 if the measure is finite. “Speed limit” operators are introduced to show that strong convergence ofA n f need not hold. The concept of convergence in distribution is extended to infinite measure spaces.

On multiparameter ergodic and martingale theorems in infinite measure spaces

A unified proof is given of several ergodic and martingale theorems in infinite measure spaces.

Random Sets Without Separability

Suppose $\mathscr{V}$ and $\mathscr{F}$ are sets of subsets of $X$, for some fixed $X$. We apply Konig's lemma from infinitary combinatorics to prove that if $\mathscr{V}$ and $\mathscr{F}$ satisfy some simple closure properties, and $T$ is a Choquet capacity on $X$, then there is a probability measure on $\mathscr{F}$ such that for every $V \in \mathscr{F}, \{F \in \mathscr{F}: F \cap V \neq \varnothing\}$ is measurable with probability $T(V)$. This extends the well-known case when $\mathscr{F}$ and $\mathscr{V}$ are the closed (respectively, open) subsets of a second countable Hausdorff space $X$. The result enables us to define a general notion of random measurable set; for example, we can build a point process with Poisson distribution on any infinite (possibly nontopological) measure space.

Maximal functions for a semiflow in an infinite measure space

Maximal functions for a semiflow in an infinite measure space

Approximations and the spectral properties of measure-preserving group actions

This paper presents some extensions and applications of the method of approximations of ergodic theory (see [6]). Two notions of approximation are defined which are applicable to arbitrary σ-finite-measure-preserving group actions (see §1). Building upon results of [2], [13] and [6], the speeds of such approximations are related to the questions of spectral multiplicity, spectral type and ergodicity (see §3). For the result on spectral multiplicity, there is first established a general result concerning the spectral decomposition of unitary representations (see §2). The last section is devoted to applications—chiefly to certain classes of cylinder transformations which arise in connection with irregularity of distribution (see [12]). These transformations provide examples (on infinite measure spaces) of approximations of all finite multiplicities. The method of approximations is shown to be a natural tool for the study of their spectral properties.

A further note on arrow's impossibility theorem

For arbitrary measure spaces of agents we investigate the existence of social welfare functions which fulfill a suitable non-dictatorship axiom; it turns out that there is an essential difference between finite and σ-finite but infinite measure spaces. Furthermore the condition of universal domain of social welfare functions is weakened.