I stated von Neumann's mean ergodic theorem (VNMET) in a talk recently and someone in the audience asked what it was good for. The only application I knew of VNMET was to prove Birkhoff's ergodic theorem (BET), which is why I'd stated VNMET in the first place. But I'm pretty sure that VNMET came first, so I doubt it was originally proven with that in mind.
Question 1. How did the theorem (or conjecture) arise in the first place? E.g. was it intended as as mere stepping-stone to BET?
The only applications I've seen (and can find) of ergodic theory to other branches of math (or physics) use BET.
Question 2. What are some applications of VNMET? I'm particularly interested in applications to other branches of mathematics; so I'm looking for something other than the "application" of it to prove BET.
Edit. What I'm really trying to glean with the above questions is an answer to the following question:
Question 3. Is VNMET important/useful beyond proving BET?