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?