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Let $\mathcal X\subseteq\mathbb R^m$ be a Borel measurable set. $\Phi:\mathcal X\to\mathbb R^n$ be a continuous mapping and $\mathcal Y = \Phi(\mathcal X)\subseteq\mathbb R^n$ its image. Let $\mathcal …

I am reading the paper [1]. At page 18, eq 115, it is claimed the following: Given a separable process $(X_t)_{t\in T}$, we have $\lim_{n\to\infty}\mathbb E[\sup_{t\in T}(X_t-X_{\pi_n(t)})]=0$. Here …

$\newcommand{\KL}{\operatorname{KL}}$Let $X$ be a Polish metric space and $P(X)$ the space of probability measures on $X$. Given $\mu, \nu\in P(X)$, recall that $$\KL(\mu\parallel\nu) = \begin{cases}\ …

I think the most general reference is: M. Liero, A. Mielke, and G. Savaré. Optimal entropy-transport problems and a newhellinger–kantorovich distance between positive measures. I …

I know that the weak lower semi-continuity of the KL divergence was proved in [1]. If I remember well, the same property is true for any $f$ divergence (with suitable assumptions on the probability sp …

Consider two Polish metric probability spaces $(\mathcal{A}, \Sigma_\mathcal{A})$ and $(\mathcal{B}, \Sigma_\mathcal{B})$, endowed with their Borel $\sigma$-algebras. Denote as $\mathcal{P}_\mathcal{B …