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I have a couple of questions regarding ergodicity for Markov processes in continuous time. (In particular, the first question seems like it should be particularly basic, and yet I haven't managed to f …

It seems to me that a considerably simpler proof [see below] of Birkhoff's ergodic theorem can be obtained for bounded observables than for more general $L^1$ observables. Therefore, I feel like it wo …

Let $X$ be a second countable Hausdorff topological space, let $T \colon X \to X$ be a Borel-measurable map, define the $\sigma$-algebra $\mathcal{I}=\{A \in \mathcal{B}(X) : T^{-1}(A)=A\}$, and for e …

Let $(X,\Sigma)$ be a standard measurable space, let $\rho$ be a probability measure on $(X,\Sigma)$, and let $\mathcal{E}$ be a sub-$\sigma$-algebra of $\Sigma$. We will say that a stochastic kernel …

Let $M$ be a Riemannian manifold with volume measure $\lambda$, let $f \colon M \to M$ be a continuous map, and let $\mu$ be a probability measure on $M$ with compact support. Definition. The …

Hello all. Suppose $X$ is a Polish space, $\mu$ is a Borel probability measure on $X$, and $T:X \to X$ is a continuous $\mu$-preserving map which is not ergodic. Does there necessarily exist a Borel …

I've found the answer - it's NO! The paper I found addressing the question is the following: http://projecteuclid.org/euclid.aop/1175287757 ("0-1 Laws for Regular Conditional Probabilities") A simple …

I have the answers to my two questions. (I've actually had them for a while; apologies for the delay in posting.) I will give them in reverse order: The answer to Q2 is yes; the structure of the proo …

Let $(X,\Sigma)$ be a measurable space [which we can assume to be a standard Borel space if we wish]. Let $\mathcal{S}$ be a set of probability measures on $(X,\Sigma)$. [If we wish, we can assume th …

THE QUESTION Let $(X,\mathcal{X})$ be a standard Borel space, $T \colon X \to X$ a measurable map, and $\mu$ a $T$-mixing probability measure. Is it necessarily the case that for all $A \in \mathcal …

This question was asked on Math.SE here, but received no replies after several months. So I have posted it here, though with somewhat revised structuring of the question. Let $V$ be a real vector s …

[I've decided to rewrite the question, to make the essential point clearer.] Let $\,\mathbb{R}^{[0,\infty)}:=\{(x_t)_{t \geq 0} : x_t \in \mathbb{R} \ \, \forall t\}$. We say that a set $Y \subset \m …

Suppose we have a compact metric space $X$, a continuous map $f \colon X \to X$, and Borel probability measures $\mu$ and $\nu$ on $X$ such that the set $$ X_{f,\mu} := \left\{ x \in X \, : \, \frac{1 …

Let $(G,\circ)$ be a Polish group, with identity $e$. Let $\Omega$ be the set of continuous functions $\omega:\mathbb{R} \to G$ such that $\omega(0)=e$. For each $t \in \mathbb{R}$, define the project …

I asked this question on Math Stack Exchange at https://math.stackexchange.com/questions/4739742/; it received 4 upvotes, but no comments or answers even after a 450-point bounty. The question: Is i …