In this article, we will introduce some important properties of the relative entropy, including lower semi-continuity, convexity, compactness of sublevel sets based on the Donsker-Varadhan variational formula.

Theorem 1 (Lusin’s theorem). Suppose that $X$ and $Y$ are Polish spaces, that $\mu$ is a finite Borel measure on $X$, that $f:X\to Y$ is Borel measurable, and that $\varepsilon>0$. Then there exists a compact subset $K$ of $X$, with
$$
\mu(X\setminus K)<\varepsilon,
$$
such that the restriction of $f$ to $K$ is continuous.

Definition 1. Suppose that $f$ is a function on the Borel subsets of a metric space $X$ taking values in $[0,\infty]$. We say $f$ is locally finite if for each $x \in X$ there exists a neighborhood $N$ of $x$ with $f(N) < \infty$.

Definition 1. A mapping $f$ from the Borel sets of a metrizable space $(X,\tau)$ to $[0,\infty]$ is tight if $f(K)<\infty$ for each compact $K$ in $X$ and
$$
f(A)=\sup\{f(K):K \text{ compact},\ K\subseteq A\},
\quad \text{for each } A\in \mathcal{B}(X).
$$

Definition 1. If $(X,\tau)$ is a topological space, then the Borel $\sigma$-field $\mathcal{B}$ of $X$ is the $\sigma$-field generated by the open sets of $X$ and we say a measure $\mu$ defined on $(X,\mathcal{B})$ is a Borel measure.

Introduction

Markov Chain

Let $(\Omega,\mathcal{F})$ be a measurable space, $P:\Omega\times \mathcal{F}\to [0,1]$ be a transition kernel, that is,

• for every $x\in \Omega$, $P(x,\cdot)$ is a probability measure on $\mathcal{F}$;
• for every $A\in \mathcal{F}$, $x\mapsto P(x,A)$ is measurable.

The Wasserstein distance has wide applications in problems such as optimal transport. It characterizes the distance between two probability measures. In German, Wasser means water, and Stein means stone (although I have not carefully verified whether Wasserstein is a person’s name; after all, many German names are of this kind, for example: Einstein = one stone). This article will mainly answer the following two questions:

• On what set is the Wasserstein distance defined?
• Why is the Wasserstein distance a distance? That is, why does it satisfy the three axioms of a metric?

In this article, we only consider the Wasserstein distance on the $d$-dimensional Euclidean space $\mathbb{R}^d$. Since $\mathbb{R}^d$ is a special Polish space, the conclusions of this article can also be extended to general Polish spaces.

How can one define a product probability measure on infinitely many probability spaces? That is, given a family of probability spaces $(\Omega_t,\mathcal F_t)$, $t\in T$, suppose that for every nonempty finite subset $S\subset T$, a probability measure $\mathbb P_S$ has already been defined on $\prod_{t\in S}\mathcal F_t$. Then how can we define a “suitable” probability measure $\mathbb P$ on the product measurable space

$$
\left(\prod_{t\in T}\Omega_t,\prod_{t\in T}\mathcal F_t\right)
$$

such that its marginal distribution measures are exactly $\mathbb P_S$?

Unless otherwise specified, all classes of sets in this article refer to classes of subsets of $\Omega$.

The Monotone Class Theorem

Stochastic Processes and Paths

Definition 1: Let $(\Omega,\mathscr{F},\mathbb{P})$ be a probability space, and let $(X_t)_{t\in T}$ be a family of random variables on $(\Omega,\mathscr{F},\mathbb{P})$, that is, for every $t\in T$, we have
$$
X_t: (\Omega,\mathscr{F})\to (\mathbb{R},\mathcal{B}(\mathbb{R}))
$$
measurable. Then
$$
\begin{aligned}
X:T\times\Omega&\to \mathbb{R}\\\
(t,\omega)&\mapsto X_t(\omega)
\end{aligned}
$$
is called a stochastic process. For fixed $\omega\in \Omega$, we call
$$
\begin{aligned}
X_\omega:T&\to \mathbb{R}\\\
t&\mapsto X(t,\omega)
\end{aligned}
$$
a path, and call the collection of all paths
$$
\mathbb{D} := \{X_\omega\ ;\ \omega\in \Omega\}
$$
the path space.

Remark: Each element $X_\omega$ of the path space is a mapping from $T$ to $\mathbb{R}$.