scholarly journals On the almost everywhere convergence of the ergodic averages

1990 ◽  
Vol 10 (1) ◽  
pp. 141-149
Author(s):  
F. J. Martín-Reyes ◽  
A. De La Torre
Keyword(s):  
Measure Space ◽  
Uniform Boundedness ◽  
Finite Measure ◽  
Necessary And Sufficient Condition ◽  
Ergodic Averages ◽  
Measurable Transformation ◽  
Almost Everywhere ◽  
Finite Measure Space ◽  
Finite Invariant Measure ◽  
Necessary And Sufficient

AbstractLet (X, ν) be a finite measure space and let T: X → X be a measurable transformation. In this paper we prove that the averages converge a.e. for every f in Lp(dν), 1 < p < ∞, if and only if there exists a measure γ equivalent to ν such that the averages apply uniformly Lp(dν) into weak-Lp(dγ). As a corollary, we get that uniform boundedness of the averages in Lp(dν) implies a.e. convergence of the averages (a result recently obtained by Assani). In order to do this, we first study measures v equivalent to a finite invariant measure μ, and we prove that supn≥0An(dν/dμ)−1/(p−1) a.e. is a necessary and sufficient condition for the averages to converge a.e. for every f in Lp(dν).

Moving Weighted Averages

1993 ◽  
Vol 45 (3) ◽  
pp. 449-469 ◽  
Author(s):  
M. A. Akcoglu ◽  
Y. Déniel
Keyword(s):  
Weight Function ◽  
Sufficient Conditions ◽  
Nonnegative Function ◽  
Finite Measure ◽  
Weight Functions ◽  
Ergodic Averages ◽  
Fixed Weight ◽  
Finite Measure Space ◽  
Image Position ◽  
Necessary And Sufficient

AbstractLet ℝ denote the real line. Let {Tt}tєℝ be a measure preserving ergodic flow on a non atomic finite measure space (X, ℱ, μ). A nonnegative function φ on ℝ is called a weight function if ∫ℝ φ(t)dt = 1. Consider the weighted ergodic averagesof a function f X —> ℝ, where {θk} is a sequence of weight functions. Some sufficient and some necessary and sufficient conditions are given for the a.e. convergence of Akf, in particular for a special case in whichwhere φ is a fixed weight function and {(ak, rk)} is a sequence of pairs of real numbers such that rk > 0 for all k. These conditions are obtained by a combination of the methods of Bellow-Jones-Rosenblatt, developed to deal with moving ergodic averages, and the methods of Broise-Déniel-Derriennic, developed to deal with unbounded weight functions.

A continuous generalization of the transversal property

1984 ◽  
Vol 95 (1) ◽  
pp. 21-23
Author(s):  
P. Komjáth
Keyword(s):  
Finite Subset ◽  
Measure Space ◽  
Choice Function ◽  
Finite Measure ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Finite Sets ◽  
Real Numbers ◽  
One To One ◽  
Necessary And Sufficient

A transversal for a set-system is a one-to-one choice function. A necessary and sufficient condition for the existence of a transversal in the case of finite sets was given by P. Hall (see [4, 3]). The corresponding condition for the case when countably many countable sets are given was conjectured by Nash-Williams and later proved by Damerell and Milner [2]. B. Bollobás and N. Varopoulos stated and proved the following measure theoretic counterpart of Hall's theorem: if (X, μ) is an atomless measure space, ℋ = {Hi: i∈I} is a family of measurable sets with finite measure, λi (i∈I) are non-negative real numbers, then we can choose a subset Ti ⊆ Hi with μ(Ti) = λi and μ(Ti ∩ Ti′) = 0 (i ≠ i′) if and only if μ({U Hi: iεJ}) ≥ Σ{λi: iεJ}: for every finite subset J of I. In this note we generalize this result giving a necessary and sufficient condition for the case when I is countable and X is the union of countably many sets of finite measure.

On the Ergodic Averages and the Ergodic Hilbert Transform

1995 ◽  
Vol 47 (2) ◽  
pp. 330-343
Author(s):  
L. M. Fernández-Cabrera ◽  
F. J. Martín-Reyes ◽  
J. L. Torrea
Keyword(s):  
Hilbert Transform ◽  
Measure Space ◽  
Dual Problem ◽  
Finite Measure ◽  
Positive Function ◽  
Ergodic Averages ◽  
Unified Approach ◽  
Measure Preserving Transformation ◽  
Finite Measure Space ◽  
Abstract Method

AbstractLet T be an invertible measure-preserving transformation on a σ-finite measure space (X, μ) and let 1 < p < ∞. This paper uses an abstract method developed by José Luis Rubio de Francia which allows us to give a unified approach to the problems of characterizing the positive measurable functions v such that the limit of the ergodic averages or the ergodic Hilbert transform exist for all f ∈ Lp(νdμ). As a corollary, we obtain that both problems are equivalent, extending to this setting some results of R. Jajte, I. Berkson, J. Bourgain and A. Gillespie. We do not assume the boundedness of the operator Tf(x) = f(Tx) on Lp(νdμ). However, the method of Rubio de Francia shows that the problems of convergence are equivalent to the existence of some measurable positive function u such that the ergodic maximal operator and the ergodic Hilbert transform are bounded from LP(νdμ) into LP(udμ). We also study and solve the dual problem.

Akcoglu's Ergodic Theorem for Uniform Sequences

1980 ◽  
Vol 32 (4) ◽  
pp. 880-884
Author(s):  
James H. Olsen
Keyword(s):  
Ergodic Theorem ◽  
Measure Space ◽  
Finite Measure ◽  
Positive Contraction ◽  
Almost Everywhere ◽  
Finite Measure Space ◽  
Image Position

Let (X, F,) be a sigma-finite measure space. In what follows we assume p fixed, 1 < p < ∞ . Let T be a contraction of Lp(X, F, μ) (‖T‖,p ≦ 1). If ƒ ≧ 0 implies Tƒ ≧ 0 we will say that T is positive. In this paper we prove that if is a uniform sequence (see Section 2 for definition) and T is a positive contraction of Lp, thenexists and is finite almost everywhere for every ƒ ∊ Lp(X, F, μ).

The Individual Ergodic Theorem for Contractions with Fixed Points

1980 ◽  
Vol 23 (1) ◽  
pp. 115-116 ◽  
Author(s):  
James H. Olsen
Keyword(s):  
Fixed Points ◽  
Ergodic Theorem ◽  
Measure Space ◽  
Finite Measure ◽  
Individual Ergodic Theorem ◽  
Almost Everywhere ◽  
Finite Measure Space ◽  
The Individual ◽  
Image Position

Let (X, I, μ) be a σ-finite measure space and let T take Lp to Lp, p fixed, 1<p<∞,‖t‖p≤1. We shall say that the individual ergodic theorem holds for T if for any uniform sequence K1, k2,… (for the definition, see [2]) and for any f∊LP(X), the limitexists and is finite almost everywhere.

scholarly journals Spaces of measurable functions

2013 ◽  
Vol 11 (7) ◽  
Author(s):  
Piotr Niemiec
Keyword(s):  
Hilbert Space ◽  
Measure Space ◽  
Finite Measure ◽  
Metrizable Space ◽  
Equivalence Classes ◽  
Convergence In Measure ◽  
Measurable Functions ◽  
Almost Everywhere ◽  
Finite Measure Space ◽  
Completely Metrizable

AbstractFor a metrizable space X and a finite measure space (Ω, $\mathfrak{M}$, µ), the space M µ(X) of all equivalence classes (under the relation of equality almost everywhere mod µ) of $\mathfrak{M}$-measurable functions from Ω to X, whose images are separable, equipped with the topology of convergence in measure, and some of its subspaces are studied. In particular, it is shown that M µ(X) is homeomorphic to a Hilbert space provided µ is (nonzero) nonatomic and X is completely metrizable and has more than one point.

scholarly journals Interchanging a limit and an integral: necessary and sufficient conditions

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Takashi Kamihigashi
Keyword(s):  
Measure Space ◽  
Sufficient Conditions ◽  
Finite Measure ◽  
Necessary And Sufficient Conditions ◽  
Integrable Functions ◽  
Finite Measure Space ◽  
Pointwise Limit ◽  
Necessary And Sufficient

AbstractLet $\{f_{n}\}_{n \in \mathbb {N}}$ { f n } n ∈ N be a sequence of integrable functions on a σ-finite measure space $(\Omega, \mathscr {F}, \mu )$ ( Ω , F , μ ) . Suppose that the pointwise limit $\lim_{n \uparrow \infty } f_{n}$ lim n ↑ ∞ f n exists μ-a.e. and is integrable. In this setting we provide necessary and sufficient conditions for the following equality to hold: $$ \lim_{n \uparrow \infty } \int f_{n} \, d\mu = \int \lim_{n \uparrow \infty } f_{n} \, d\mu. $$ lim n ↑ ∞ ∫ f n d μ = ∫ lim n ↑ ∞ f n d μ .

A Local Ergodic Theorem on Lp

1974 ◽  
Vol 26 (5) ◽  
pp. 1206-1216 ◽  
Author(s):  
J. R. Baxter ◽  
R. V. Chacon
Keyword(s):  
Measure Space ◽  
Finite Measure ◽  
Ergodic Theorems ◽  
Semi Group ◽  
Almost Everywhere ◽  
Local Case ◽  
Finite Measure Space ◽  
Continuity Assumption ◽  
Local Ergodic Theorem ◽  
Image Position

Two general types of pointwise ergodic theorems have been studied: those as t approaches infinity, and those as t approaches zero. This paper deals with the latter case, which is referred to as the local case.Let (X, , μ) be a complete, σ-finite measure space. Let {Tt} be a strongly continuous one-parameter semi-group of contractions on , defined for t ≧ 0. For Tt positive, it was shown independently in [2] and [5] that1.1almost everywhere on X, for any f ∊ L1. The same result was obtained in [1], with the continuity assumption weakened to having it hold for t > 0.

Pointwise Convergence of Alternating Sequences

1988 ◽  
Vol 40 (3) ◽  
pp. 610-632 ◽  
Author(s):  
M. A. Akcoglu ◽  
L. Sucheston
Keyword(s):  
Banach Space ◽  
Measure Space ◽  
Pointwise Convergence ◽  
Finite Measure ◽  
Linear Contraction ◽  
Negative Function ◽  
Almost Everywhere ◽  
Finite Measure Space ◽  
Complex Valued

Let 1 < p < ∞ and let Lp be the usual Banach Space of complex valued functions on a σ-finite measure space. Let (Tn), n ≧ 1, be a sequence of positive linear contractions on Lp. Hence and , where is the part of Lp that consists of non-negative Lp functions. The adjoint of Tn is denoted by which is a positive linear contraction of Lq with q = p/(p — 1).Our purpose in this paper is to show that the alternating sequences associated with (Tn), as introduced in [2], converge almost everywhere. Complete definitions will be given later. When applied to a non negative function, however, this result is reduced to the following theorem.(1.1) THEOREM. If (Tn) is a sequence of positive contractions of Lp then (1.2) exists a.e. for all.

L1-Sequential Convergence on Stone Spaces and Stone's Theorem on Unitary Semigroups

1975 ◽  
Vol 18 (2) ◽  
pp. 191-193
Author(s):  
J. B. Cooper
Keyword(s):  
Measure Space ◽  
Finite Measure ◽  
Sequential Convergence ◽  
Measurable Functions ◽  
Almost Everywhere ◽  
Finite Measure Space ◽  
Stone Spaces ◽  
Sequence Of Functions ◽  
Stone’S Theorem

The following theorem is a well-known tool in the study of measurable functions:Theorem. Let (M; μ) be a finite measure space and let (xn) be a sequence of functions in L1(M; μ) so that xn→0 in the norm of L1(M; μ). Then there is a sub-sequence so that xnk →0 pointwise almost everywhere on M.

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