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 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.