Question about probability spaces $(X,\mathcal B,\mu)$ with a measurable map $T\colon X\to X$ preserving the measure, that is $\mu(T^{—1}A)=\mu(A)$ for all $A$ measurable.

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1answer
45 views

Using the Baire Category Theorem to prove $\mu$ is trivial

Suppose we have a probability measure space $(X,\mathcal A,\mu,T)$ where $T$ is measure-preserving. Then if for every $A,B\in\mathcal A$ we have $\mu\left(A\cap T^{-n}B\right)=\mu(A)\mu(B)$ for all $n\...
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0answers
23 views

Gauss measure is not a pushforward of product measure

Let $ N = \{1,2,3, \ldots \}$. We define $\varphi : N^N \mapsto [0,1]$ as $$ \varphi \left( (a_n)_{n \in_N} \right) = [0;a_1, a_2, \ldots ]$$ Where the expression on the right is a infinite continued ...
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1answer
34 views

Condition expectation of functions: $E(fg\mid\mathcal{A})=gE(f\mid\mathcal{A})$ when $|g|<\infty$ a.e.

Let $(X,\mathcal{B},\mu)$ be a probability space, $\mathcal{A}\subset\mathcal{B}$ a sub-$\sigma$-algebra, then by an easy application of the Radon-Nikodym Theorem, letting $\nu(A)=\int_A\, f\,\mathrm{...
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1answer
25 views

For an invertible measure preserving system, $\lim_NA_f^T(N)=\lim_N A_f^{T^{-1}}(N)$

For an invertible measure preserving system, show that $\lim_NA_f^T(N)=\lim_N A_f^{T^{-1}}(N)$. Here we consider the measure preserving system $(X,\mathcal A,\mu,T)$ where $T$ is invertible and $\mu$...
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0answers
19 views

Extension of erdodic theorem with WLLN

Suppose you have a ergodic (or irreducible) Markov chain $(A_t)_{t\geq0}$ in continuous time. denote by $\pi$ the invariant distribution of $A$. If $f$ is a function of $A_s$ which is integrable w.r.t....
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0answers
22 views

When is the weak limit of operators invertible?

Suppose $T_i$ are invertible operators in $L^{2}(X)$ for X a Lebesgue Probability Space. Is the following true? 1. If the $T_i$ converge weakly to $S$, then $S$ is not necessarily invertible. ...
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0answers
65 views

What does it mean for a function to be invertible 'almost everywhere'

It seems to me that the correct definition of a measure-theoretic inverse for a function f is a function g such that $f \circ g$ and $f \circ g$ are the identity almost everywhere. The problem I have ...
2
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0answers
33 views

Weak convergence of finite measure preserving transformations

I am reading King's paper "The commutant is the weak closure of the powers, for rank-1 transformation" and I am not able to show that: (0.1) "If the $T_i$ are invertible measure preserving ...
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1answer
35 views

Showing an ergodic toral automorphism is not measurably isomorphic to an ergodic circle rotation

The question as listed in the title is the question statement, only I do not want to use that one is mixing and the other is not. Is it true that measurably isomorphic spaces are either both mixing ...
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0answers
23 views

Hilbert space mean ergodic theorem application

Let $(u_n)_{n \geq 0}$ be a bounded sequence in a Hilbert space. We define $$ s_h = \limsup \frac 1 N \sum_{n=o}^{N-1} \langle u_{n+h} , u_n \rangle $$ Show that, if $ \lim \frac 1H \sum_{h=o}^{H-1} ...
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2answers
55 views

Measure-preserving mapping

Let $(X, \mu, T)$ be a mesure-preserving mapping. Let $A \subset X$ be a measurable subset such that any point in $A$ eventually comes back to $A$. We define space $(A, \mu_A)$, $ \mu_A ( B) = \mu (B) ...
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1answer
31 views

Measure on torus invariant under multiplication

Let $T: [0,1] \rightarrow [0,1]$ be a multiplication by $ \beta >1$ mod $1$. Show that $h(x) d x$ is $T$-invariant where $$h (x) = \sum_{n \geq 0} \beta^{-n} \chi _{[0,T^n (1)]} (x)$$ ($\chi$ is ...
2
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1answer
47 views

Exercise 2.1.1 from Einsiedler and Ward

I am studying Ergodic Theory for the first time, and am using the book "Ergodic Theory with a view towards Number Theory" by Einsiedler and Ward. I got stuck at the very first exercise problem, ...
1
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1answer
76 views

Hyperbolicity without ergodicity?

I have a question concerning the ergodic properties of hyperbolic Hamiltonian flows. Let $\Phi_{H}^{t}$ be a Hamiltonian flow on a symplectic manifold $\mathcal{M}$. If $\Phi_{H}^{t}$ is Anosov on a ...
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1answer
18 views

Conditioning on invariant sigma algebra with respect to ergodic measure

So this question arose to me while applying the Ergodic theorem. If $X$ is a finite state (in $ \{1,\dots,d\}$) continuous-time Markov chain, which is ergodic, then $X$ has a unique invariant ...
3
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2answers
40 views

Hausdorff dimension via ergodic theory

This is definitely a soft question, but it was recently mentioned to me that one can study the dimension of fractals via ergodic methods. I'm familiar with ergodic theory on about the level of ...
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0answers
37 views

Converges of measures.

Good afternoon, we have the following: Let $(Y,d)$ is a general metric space, $\mathcal{M}(Y)$ is the set of finite Borel measures on $Y$ and $C_B(Y)$ denotes the Banach space of bounded continuous ...
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1answer
66 views

A Question Regarding Markov Chains and Ergodicity

Suppose the Markov chain with Probability Transition Matrix, $P$ = ($p{_x}{_y}$) is ergodic and $p{_m}(x, y) > 0$ for all states $x$ and $y$. If $n ≥ m$, show that $p_n(x, y) > 0$ for all ...
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1answer
28 views

Approximate eigenvalues of an ergodic invertible transformation

Consider a non-atomic probability space $(X,\mathcal{B}, m)$. Let $T: X \to X $ be an ergodic invertible measure preserving transformation.Let $U_T$ be the Koopman operator associated with $T$. Show ...
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0answers
47 views

Ergodic actions of orthogonal group $O(2)$

I am looking for explicit ergodic action of $O(2)$ on a von Neumann algebra $M$. ($O(2)$=orthogonal group of $2\times 2$ matrix)
2
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1answer
51 views

application of Birkhoff Ergodic Theorem

Let $(\Omega, \mathcal{F} , P)$ be a probability space and let $T$ and $S$ be ergodic, measure-preserving transformations of $(\Omega, \mathcal{F} , P)$. Let $X : \Omega → \mathbb{R}$ be a bounded ...
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1answer
48 views

A question of determining when the entropy is maximum.

Y ={ 1, 2,...,r} We are given that X is the set of two sided sequences with entries from Y and T is the two sided shift on X, and m is a T invariant probability measure on X. If $p_i = m(\{x \...
2
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1answer
39 views

Non-atomic, ergodic measure which is left and right shift invariant.

Given a one-sided shift space, say $X = \prod\limits_{n=1}^\infty \mathbb Z_2$. Denote the left shift by $T$: $T(x_1 x_2 x_3\cdots) = x_2x_3 \cdots$. There are lots of examples of $T$-invariant ...
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1answer
30 views

Are $L_\infty$ functions measurable/integrable?

Lemma 2.6 of "Ergodic Theory with a view towards Number Theory" (Einsiedler-Ward) involves: $$ \int f d\mu $$ where $f \in L^{\infty}$. Actually it is a calligraphic $L$ and I'd love if you would ...
3
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1answer
37 views

What is known about the space of measure-preserving transformations?

I started reading about measure-preserving transformations, the ergodic theorems and mixing, but I was also wondering what is known about the space of measure-preserving transformations. The books ...
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1answer
42 views

ergodicity in $\mathbb{Z}^d$

Fix $d \geq 1$ and let $E(\mathbb{Z}^d)$ denote the set of all edges of the graph $\mathbb{Z}^d$. Let us consider a measure preserving system $(\mathbb{R}^{E(\mathbb{Z}^d)}, B^{E(\mathbb{Z}^d)}_\...
0
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2answers
24 views

measure preserving system

Let $T$ be a measure-preserving transformation on a probability space $(\Omega, \mathcal{F}, P)$ and let $A \in \mathcal{F}$ such that $P(A) > 0$. (i) Show that there exists $n \geq 1 $such that $...
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1answer
21 views

Measure preserving ergodic map commutes with complementation?

This is probably trivial (in which case I apologize), but it's late and I would really like a quick proof/counterexample for this (for a different problem that I'm doing): if $(X,\mathcal{M},\mu,T)$ ...
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1answer
18 views

Initial point and initial distribution of the Markov chains

I am reading about Markov chains on a general state space and the ergodicity theory. Some of the ergodic theorems are presented when we consider n-step transition probability conditional on initial ...
0
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1answer
36 views

Why is the shift map ergodic?

Given a finite set $S$, in the space of strings $\Sigma=S^{(\omega)}$ equipped with the Bernoulli measure $\mu$, I want to know why the shift map $\sigma:\Sigma\rightarrow \Sigma $, define as $\sigma(\...
0
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2answers
28 views

Simple eigenvalue of Koopman operator

Let $T : X \to X$ be a measure-preserving transformation and $U_T : L^2(X, \mu) \to L^2(X, \mu)$ , $(U_T f) (x) = f(Tx).$ What does it mean a $\bf{simple}$ eigenvalue of $U_T$? $\lambda \in \mathbb{...
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0answers
15 views

If $1$ is a simple eigenvalue of $T$, then $T$ is an ergodic measure-preserving transformation

Let $T : X \to X$ be a measure-preserving transformaton and $U_T : L^2 (X, \mu) \to (X, \mu)$, $$(U_T f) (x) = f(Tx).$$ I have to show that if $1$ is a simple eigenvalue of $U_T$, then $T$ is ergodic....
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1answer
22 views

Different ergodic probability measures are mutually singular

Can someone, please, give me a hint on how to demonstrate that different ergodic probability measures are mutually singular? Thank you!
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1answer
60 views

Ergodicity of stochastic process

If one can show that the process converges to a stationary process in probability, does it mean that the process is ergodic?
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1answer
31 views

Example for a non-ergodic stationary process

Let $(X_n)_{n \in \mathbb{N}}$ be a (strictly) stationary process and let $T$ denote the left-shift on $\mathbb{R}^\mathbb{N}$, i.e. $T((x_n)_{n \in \mathbb{N}}) = (x_{n + 1})_{n \in \mathbb{N}}$. ...
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0answers
16 views

Theorem of Daniell-Stone (uniqueness when assuming compactness)

Theorem of Daniell-Stone. Let $L$ be a $\sigma$-continuous abstract integral on a Stone lattice V of real-valued functions on $\Omega$ and let $\mathcal{A}(V)$ denote the set of all $V$-open sets. ...
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1answer
36 views

Strengthening Poincaré Recurrence

Let $(X, B, \mu, T)$ be a measure preserving system. For any set $B$ of positive measure, $E = (n \in \Bbb N |\; \mu(B \; \cap \;T^{-n}B) > 0)$ is syndetic. This exercise comes from Einseidler ...
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0answers
67 views

Ergodicity of stochastic recursive process

Does anyone know how one can show ergodicity for a recursive stochastic process determined by the following equation: \begin{equation} X_n = f(\varepsilon_{n-1},Y_{n-1})X_{n-1}, \end{equation}...
0
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1answer
25 views

Criterium in ergodic theory.

Given a topological space $X$ with a probability measure $\mu$ and a continuous transformation $T:X \rightarrow X$ which preserve measure. If a set $A$ with $1>\mu(A)>0$ is such that the ...
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1answer
38 views

Approximating Borel Measure with Atomic Measures

I see some posts that are related to this one, e.g. Borel Measures: Atoms (Summary) I have a sort of particular question: I have one professor saying the following is true, while another says it's ...
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1answer
37 views

Connection between Ergodic Theory and Markov Chains

Could someone suggest a good reference where the connection between Ergodic Theory and (ergodic) Markov Chains is nicely explained ?
3
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1answer
69 views

Does the Central Limit Theorem Imply the strong Law of Large Numbers?

Assume that $(X_{k})_{k\geq 0}$ is a stationary (or weakly stationary) process defined on the same probability space $(\Omega,\mathcal{F},\mathbb{P})$. Can we assert from the convergence in ...
0
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0answers
50 views

Ergodic Theorem for Markov Chain with one closed communicating class and several transient states

It is known, that if a markov chain with a finite state space has only one closed aperiodic communicating class and several transient states, then there is a unique stationary distribution for this MC....
3
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0answers
34 views

Writing a Hilbert C*-submodule of $L^2(X)$ as an integral sum over Hilbert subundles

In Ergodic Theory, some (though not all) presentations of compact extensions use Hilbert bundles. Given $(X,\mathcal{B},\mu,T)$ and a sub $\sigma$-algebra $\mathcal{G}\subseteq\mathcal{A}$ one has an ...
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1answer
23 views

Showing the ergodicity of a rotation on the unit sphere

Consider the rotation $R_\alpha(z) = \alpha z, R_\alpha : S^1 \to S^1$. Show that $R_\alpha$is ergodic with respect to Haar measure on $S^1$ $\iff$ $\alpha$ is not a root of unity. I don't know how ...
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1answer
58 views

Ergodicity vs Weak Mixing

I have been trying to prove the following: Let $(\Omega,\mathcal{F})$ be a measurable space endowed with probability measure $\mathbb{P}$. Suppose $\tau : \Omega \to \Omega$ is a measure preserving ...
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1answer
28 views

Birkhoff Ergodic theorem for two measures

Suppose $(X,\mathcal{B}, \mu, T)$ and $(X,\mathcal{B}, \nu, T)$ are both ergodic ppt. I'm a bit confused how the B Ergodic Theroem works since the LHS of the equation doesn't involve $\mu$ or $\nu$, ...
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1answer
29 views

Wandering set definition

I've seen two apparently different definitions and am wondering which is correct. A set $W$ is wandering if $\{T^{-k}W; k\in \mathbb{N}_0\}$ (resp. $\{T^{k}W; k\in \mathbb{N}_0\}$) are pairwise ...
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0answers
184 views

What's the probability distribution of a deterministic signal or how to marginalize dynamical systems? (functional integrals in probability theory)

In many signal processing calculations, the (prior) probability distribution of the theoretical signal (not the signal + noise) is required. In random signal theory, this distribution is typically a ...
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1answer
27 views

Using Furstenberg's skew-product for $\alpha n^2$ equidistributed.

I am having trouble proving that if $\alpha_1,\alpha_2$ are irrational, then the sequence $(\alpha_1n,\alpha_2n^2)$ is equidistributed in $\mathbb{T}^2$. It is straightforward to use Furstenberg's ...