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Let $X,Y$ be measurable spaces and $F\subseteq X\times Y$. We say that $f:X\to Y$ is a uniformization map for $F$ if $(x,f(x))\in F$ for each $x\in \pi_X(F)$ where $\pi_X$ is the left projection map. …

Let $X$ be a standard Borel space, so that the space of Borel probability measures on $X$ is also a standard Borel space. We denote it by $\mathcal P(X)$. In this paper for any family of probability m …

When talking about a single random variable, knowing only its distribution, the construction of a probability space is quite easy. Namely, let $(X,\mathscr A)$ be a measurable space and let $\mathsf Q …

Just in the case someone will be interested in a problem of such a kind. Very nice methods are developed by Prof. Kendall E. Atkinson. I read some of his papers and also used his toolbox for MATLAB wh …

There is an equation $$ w(x) = g(x)+\int\limits_0^M w(y)f(x-y)\,dy $$ where $f\geq 0$, $f\in C^\infty(\mathbb R\setminus\{c\})$ for some point $c$ and $\int\limits_{-\infty}^\infty f(t)\,dt\leq 1$. Wi …

Many thanks to Gerald Edgar and D. Kelleher for their answers. I am (and was) a little familiar with the Choquet's theory and tried to find the answer to OP and similar questions there (mostly in "Le …

A coupling of two probability measures $P,\tilde P$ on a Borel space $X$ is any probability measure on $X^2$ whose one-dimensional marginals are $P$ and $\tilde P$. In particular, for any such couplin …

Just for the sake of the question having an answer, the following paper by A. Maitra and W. Sudderth gives a very nice characterization of the value function for the i.o. event in the framework of gam …

Let us consider a Markov Decision Process (MDP) with a Borel state space $X$. Often, the optimization problems over MDP involve optimization of some objectives dependent on the reward function $$ r: …