In Stein's "Real analysis" in the beginning of the 1.1 "The Hardy-Littlewood maximal function" he writes " It seems that they were led to the study of this function by toying with the question of how a batsman's score in cricket may best be distributed to maximize his satisfaction."
How this question may led to the maximal function? The question is of course not about cricket. It is about intuition behind the maximal function.
I don't know does this question make a sense. But say at least something which will help me to grasp the idea.
The explanation of this example is given in "The Art of Mathematics" by B. Bollobás in probleb 85 "Satisfied Cricketers: the Hardy-Littlewood Maximal Theorem".