Martingale Convergence Theorem

Stream: Brownian motion

[Wojciech Czernous]

Hello! As I am still working on the proof of continuous version of MCT, by showing first: 1) the Doob & Snell's upcrossing inequality and 2) Bernstein & Lévy's maximum inequalities (9.18 and 9.16 in Kallenberg 2021, respectively). Precisely, I've started working on 1. during the ItaLean2025 and I continue now. Could I please have a task for 1 and claim it? Say, "Doob's upcrossing inequality for a countable index" ?