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.

Ratio and Stochastic Ergodic Theorems for Superadditive Processes

1979 ◽  
Vol 31 (2) ◽  
pp. 441-447 ◽  
Author(s):  
Humphrey Fong
Keyword(s):  
Linear Operator ◽  
Measure Space ◽  
Measure Zero ◽  
Positive Linear Operator ◽  
Finite Measure ◽  
Ergodic Theorems ◽  
Finite Measure Space ◽  
Image Position

1. Introduction. Let (X, , m) be a σ-finite measure space and let T be a positive linear operator on L1 = L1(X, , m). T is called Markovian if(1.1)T is called sub-Markovian if(1.2)All sets and functions are assumed measurable; all relations and statements are assumed to hold modulo sets of measure zero.For a sequence of L1+ functions (ƒ0, ƒ1, ƒ2, …), let(ƒn) is called a super additive sequence or process, and (sn) a super additive sum relative to a positive linear operator T on L1 if(1.3)and(1.4)

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 Classes of operators on vector valued integration spaces

1977 ◽  
Vol 24 (2) ◽  
pp. 129-138 ◽  
Author(s):  
R. J. Fleming ◽  
J. E. Jamison
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Measure Space ◽  
Separable Hilbert Space ◽  
Hermitian Operator ◽  
Finite Measure ◽  
Finite Measure Space ◽  
Image Position ◽  
Vector Valued ◽  
Uniformly Bounded

AbstractLet Lp(Ω, K) denote the Banach space of weakly measurable functions F defined on a finite measure space and taking values in a separable Hilbert space K for which ∥ F ∥p = ( ∫ | F(ω) |p)1/p < + ∞. The bounded Hermitian operators on Lp(Ω, K) (in the sense of Lumer) are shown to be of the form , where B(ω) is a uniformly bounded Hermitian operator valued function on K. This extends the result known for classical Lp spaces. Further, this characterization is utilized to obtain a new proof of Cambern's theorem describing the surjective isometries of Lp(Ω, K). In addition, it is shown that every adjoint abelian operator on Lp(Ω, K) is scalar.

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).

On the relative merits of correlated and importance sampling for Monte Carlo integration

1965 ◽  
Vol 61 (2) ◽  
pp. 497-498 ◽  
Author(s):  
John H. Halton
Keyword(s):  
Importance Sampling ◽  
Measure Space ◽  
Finite Measure ◽  
Monte Carlo Integration ◽  
Close Approximation ◽  
Unbiased Estimators ◽  
Finite Measure Space ◽  
Correlated Sampling ◽  
Approach Zero ◽  
Image Position

Given a totally finite measure space (S, S, μ) and two μ-integrable, non-negative functions f(x) and φ(x) defined in S, such that whenthenwe define correlated sampling as the technique of estimatingby sampling an estimator functionwhere ξ is uniformly distributed in S with respect to μ (i.e. for any T ∈ S, p(T) = μ(T)/μ(S) is the probability that ξ lies in T): and importance sampling as estimating L by sampling the estimator functionwhere η is distributed in S with probability density φ(x)/ΦThen, clearly,It follows that υ(ξ) and ν(η) are both unbiased estimators of L, and that their variances can both be made to approach zero arbitrarily closely by making φ(x) a sufficiently close approximation to f(x).

scholarly journals FAILURE OF THE POINTWISE AND MAXIMAL ERGODIC THEOREMS FOR THE FREE GROUP

2015 ◽  
Vol 3 ◽  
Author(s):  
TERENCE TAO
Keyword(s):  
Free Group ◽  
Weak Type ◽  
Markov Operator ◽  
Finite Measure ◽  
Maximal Inequality ◽  
Ergodic Theorems ◽  
Averaging Operators ◽  
Finite Measure Space ◽  
Image Position ◽  
L Log L

Let $F_{2}$ denote the free group on two generators $a$ and $b$. For any measure-preserving system $(X,{\mathcal{X}},{\it\mu},(T_{g})_{g\in F_{2}})$ on a finite measure space $X=(X,{\mathcal{X}},{\it\mu})$, any $f\in L^{1}(X)$, and any $n\geqslant 1$, define the averaging operators $$\begin{eqnarray}\displaystyle {\mathcal{A}}_{n}f(x):=\frac{1}{4\times 3^{n-1}}\mathop{\sum }_{g\in F_{2}:|g|=n}f(T_{g}^{-1}x), & & \displaystyle \nonumber\end{eqnarray}$$ where $|g|$ denotes the word length of $g$. We give an example of a measure-preserving system $X$ and an $f\in L^{1}(X)$ such that the sequence ${\mathcal{A}}_{n}f(x)$ is unbounded in $n$ for almost every $x$, thus showing that the pointwise and maximal ergodic theorems do not hold in $L^{1}$ for actions of $F_{2}$. This is despite the results of Nevo–Stein and Bufetov, who establish pointwise and maximal ergodic theorems in $L^{p}$ for $p>1$ and for $L\log L$ respectively, as well as an estimate of Naor and the author establishing a weak-type $(1,1)$ maximal inequality for the action on $\ell ^{1}(F_{2})$. Our construction is a variant of a counterexample of Ornstein concerning iterates of a Markov operator.

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.

The Individual Weighted Ergodic Theorem for Bounded Besicovitch Sequences

1982 ◽  
Vol 25 (4) ◽  
pp. 468-471
Author(s):  
James H. Olsen
Keyword(s):  
Linear Operator ◽  
Ergodic Theorem ◽  
Measure Space ◽  
Finite Measure ◽  
Complex Numbers ◽  
Finite Measure Space ◽  
The Individual ◽  
Image Position ◽  
Weighted Ergodic Theorem

AbstractLet (X, , μ) be a σ-finite measure space, p fixed, 1 < p < ∞, T a linear operator of Lp(X,μ), {αi} a sequence of complex numbers. Ifexists and is finite a.e. we say the individual weighted ergodic theorem holds for T with the weights {αi}In this paper we show that if {αi} is a bounded Besicovitch sequence and T is a Dunford-Schwartz operator (i.e.: ||T||1≤1, ||T||∞≤1) then the individual weighted ergodic theorem holds for T with the weights {αi}.

On Characterizations of Conditional Expectation

1973 ◽  
Vol 16 (2) ◽  
pp. 161-163
Author(s):  
A. N. Al-Hussaini
Keyword(s):  
Linear Operator ◽  
Conditional Expectation ◽  
Measure Space ◽  
Finite Measure ◽  
Finite Case ◽  
Finite Measure Space ◽  
Image Position

In the following (Ω, α, μ) is a totally σ-finite measure space except where noted. For a sub-σ-algebra β ⊂ α, the conditional expectation E{f|β} off given β is a function measurable relative to β, such thatIn [5] R.G.Douglas proved, among other things the following, in the finite case:Suppose μ(Ω)=l. Then a linear operator T on L1(Ω, α,μ) is a conditional expect ion if and only if1.11.21.3The point of this note is to characterize conditional expectation in the σ-finite case (Theorems 2, 3).

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