In an old post, someone said that the Calderon-Zygmund decomposition "follows from a simple stopping time argument". I would like to see this probabilistic argument (should be something simple). I know that the Lebesgue differentiation theorem, wich is important in the usual proof, can be proved using probability theory and then... but is there a faster way?
Also (I do not know too much probability theory) Lebesgue theorem can be proved by a kind of "energy increment" method (it looks like a regularity result): some quantity is "incremented" until we get enough structure and the process can be stopped...(I am thinking also to Roth theorem). Is there a corespondence, or a probabilistic interpretation for this? That "stop" has something to do with "stopping time"? Another thing: is there a direct probabilistic proof of Roth theorem (not via Szemeredi theorem)?
