Here is a short post to advertise a proof of the weak bound for the Hardy-Littlewood maximal function. The proof was told to me by Terry Harris, a fellow graduate student at UIUC, and can be found on his webpage. In short, he replaces the use of Vitali’s covering lemma with a clever greedy algorithm. Incidentally his proof gives the better constant , though this is well known, see for instance exercise 42 in these notes of Tao. One cute related geometric question is can one improve the constant in Vitali’s covering lemma to ? This is open for .

Kevin Hughes pointed out to me that this proof basically appears in this paper of Carlsson.