In this article, we will show some sufficient conditions for optimality and stability.

In this article, we will prove the duality result for Kantorovich problem.

In this article, we will introduce some results for $c$-concavity and cyclical monotonicity, which will be used to prove the duality result in the next article.

In this article, we will consider $X=Y=\Omega \subset \mathbb{R}^d$ with $\Omega$ compact and $c(x,y)=h(x-y)$ for $h$ strictly convex.

In this article, we will talk about the dual problem Kantorovich.

In this article, we will talk about the problems of Monge and Kantorovich and the corresponding existence of solutions.

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.

Whether in the past, the present, or the future, cash payments remain extremely important in every country and every place, even in today’s society where digital payment is widespread. The privacy, security, and independence from network infrastructure that cash offers are irreplaceable. Cash not only helps diversify risk, but can also serve as a way to curb spending.

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 $f(N) < \infty$.