Revisiting Fortet's proof of existence of a solution to the Schr\"odinger system

Since a few years, the Schr"odinger problem captures the attention of a growing community of mathematicians interested in optimal transport problems. The first result of existence of a solution to this problem dates back to 1940, when Fortet published an article on the subject. In these notes, Fortet’s original proof of existence and uniqueness of a solution to the Schr"odinger system is revisited.