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Hello all. I have the following (perhaps basic) question: Let $X$ be a separable metric space. Does there necessarily exist a countable set $\mathcal{C}$ of Borel sets in $X$ such that any two probab …

Well, the question was asked a long time ago, so my answer might not be of much help to the asker any more; but perhaps for the sake of future readers I'll write an answer anyway. Since densities are …

Background of the question. One of the problems that arises with Wiener-driven (and more general Lévy-driven) models of noisy systems is that, due to the extremely rapid decay of tails of infinitely d …

Let $\Omega=\{-1,1\} \times \{-1,1\}$ with the discrete $\sigma$-algebra and uniform measure, let $X(\omega_1,\omega_2)=\omega_2$, and let $$ \mathcal{A}_n \ = \ \left\{ \begin{array}{l l} \sigma(\{\o …

Let $(X_n)_{n \geq 0}$ be an i.i.d. sequence of $\{0,1\}$-valued random variables $X_n \sim \mathrm{Bernoulli}(\frac{1}{2})$, i.e. a sequence of independent tosses of a fair coin. Does there exist a …

Preliminary notations: For a compact metrisable space $X$, $\mathcal{B}(X)$ is the Borel $\sigma$-algebra on $X$. $\overline{\mathcal{B}}(X)$ is the universal completion of $\mathcal{B}(X)$. $\mathca …

Let $\mathbf{\Omega}=(\Omega,\mathcal{F},(\mathcal{F}_t)_{t \in [0,\infty)},\mathbb{P})$ be a filtered probability space satisfying the Usual Conditions. Let $P \colon [0,\infty) \times \mathbb{R} \t …

Yes. As in the comments: take $X=\mathbb{S}^1$; and let $\nu$ be the law of the random set constructed by taking a positive-Lebesgue-measure Cantor set $K \subset \mathbb{S}^1$ and rotating $K$ throug …

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 …

Having read Mateusz Kwaśnicki's answer, I will now write it in my own way: Lemma. Let $S_\infty$ and $T$ be separable metric spaces, and let $(S_j)_{j \in \mathbb{N}}$ be a sequence of Borel subset …

Given a compact metrisable topological space $X$, we write $\mathcal{N}(X)$ for the set of non-empty closed nowhere dense subsets of $X$, which is a Polish space under the topology induced by the Haus …

This is a question in the theory of random dynamical systems. Let $(X,d)$ be a compact metric space, let $(I,\mathcal{I},\nu)$ be a probability space, and let $(f_\alpha)_{\alpha \in I}$ be an $I$-ind …

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 $(M_n)_{n \geq 1}$ be a uniformly bounded martingale over a probability space $(\Omega,\mathcal{F},\mathbb{P})$. Define the probability measure $\mu$ on $\mathbb{R}^\mathbb{N}$ to be the law of $( …

Apologies if this question is too basic for MathOverflow. For a smooth Wiener-driven SDE on a non-compact manifold $M$ taking the form $$ dX_t = b(X_t) dt + \sum_{i=1}^k \sigma_i(X_t) \ast dW_t^i $$ w …