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Let $(E,\mathscr E)$ be a measurable space and $Q:E\times \mathscr E\to\Bbb [-1,1]$ be a signed bounded kernel, i.e. $Q_x(\cdot)$ is a finite measure on $(E,\mathscr E)$ for any $x\in E$ and $x\mapsto …

Let $X,Y$ be Polish spaces and $\kappa:X\times \mathcal B(Y)\to[0,1]$ be a Borel-measurable stochastic kernel on $Y$ given $X$. Under which conditions for a probability measure $\nu$ on $Y$ there exis …

Some thoughts. It does not seem likely that you can achieve existence for non-analytic $\eta$, hence I'd suggest trying to show that it is - or finding a counterexample. Let's say $c = 0$ and $u = 1_B …

Let $X$ and $Y$ be two measurable spaces, and let $p$ be a probability measure on $X\times Y$. Denote by $p_X$ the marginal of $p$ on $X$, that is an image of $p$ under projection on $X$. Consider two …

If I am correct, one can proceed as follows. Consider a set $A\subset X\times \mathcal P(X)^2$ given by $$ A = \{(x,p,q):(x,p)\in \Gamma,\rho(p,q)\leq\varepsilon\} $$ then we obtain $\Gamma^\varepsilo …

Let $X$ and $\bar X$ be two standard Borel spaces, and let $A\subseteq X\times\bar X$ be an analytic subset of the product space. Let $P$ be any probability measure such that $P(A) = 1$, and denote by …

Let $X$ and $Y$ be standard Borel spaces and let $J$ be an analytic subset of $X\times \mathcal P(Y)$ where $\mathcal P(\Omega)$ is a set of probability measures on a Borel space $\Omega$ endowed wit …

Let $X = (X_1,X_2)$ and $\hat X = (\hat X_1,\hat X_2)$ be two random variables where $X_i,\hat X_i$ are taking values over the Polish space $E_i$ endowed with their Borel $\sigma$-algebras, where $i=1 …

Before passing to question, let me briefly recap what's importance sampling of random variables is about. Suppose $\xi$ is a real-valued random variable with density $f$, and let $g:\Bbb R\to \Bbb R$ …

Let $\mathcal P(X)$ denote the space of all probability measure defined on a measurable space $X$. We canonically endow the former with its own measurability structure, generated by evaluation maps. L …

Let $E$ be a compact subset of $\mathbb{R}^n$. Let the density function $\phi(x,y)$ be Lipschitz continuous and such that $$ \int\limits_E \phi(x,y)dy=1 $$ for all $x\in E$. Let us consider the non-in …

Let $(E,\mathscr E)$ be a measurable space and $P,\tilde P$ be two stochastic kernels on that space. I wonder how the induced measures $\mathsf P_x$ and $\tilde{\mathsf P}_x$ differ on the space of fi …

KET is often used to construct stochastic processes in continuous time when the state space is $\Bbb R^d$. As far as I am familiar with its proof, it uses standard monotonic class-like arguments toget …

Let $X$ be a standard Borel space: that is, a topological space equivalent to a Borel subset of $\Bbb R$. It is known that for any probability measure $p$ on $X$ and any universally measurable set $A\ …

Let $X$ and $Y$ be standard Borel spaces, and let $A\subseteq X\times Y$ be an analytic set with a full projection on $X$: that is $\pi_X(A) = X$. Suppose that there exists a Borel-measurable kernel $ …