Theorem on selectors and its extensions

I will present the analytical proof of the selectors theorem (in June I presented a much more combinatorial proof by Park and Pham). With the analytic approach, the argument is much clearer and shorter with much better constants. Let me remind you that this solves a quite important problem about the characterization of the mean value of suprema of positive Bernoulli-type processes, stated by Talagrand. I also intend to demonstrate how the result can be used to establish a similar mean value characterization of positive empirical processes and infinitely divisible processes. It is a joint work of Rafał Martynek, Rafał Meller and me.