Explaining a Mysterious Maximal Inequality — and a Path to the Law of Large Numbers

In 1964 A. Garsia gave a stunningly brief proof of a useful maximal inequality of E. Hopf. The proof has become a textbook standard, but the inequality and its proof are widely regarded as mysterious. Here we suggest a straightforward first step analysis that may dispel some of the mystery. The development requires little more than the notion of a random variable, and, the inequality may be introduced as early as one likes in a graduate probability course. The benefit is that one gains access to a proof of the strong law of large numbers that is pleasantly free of technicalities or tricky ideas.