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?
Best Answer
von Neumann long argued that for physics, his result suffices (see, e.g., Proc. Nat. Acad. Sci. U.S.A. 18 (1932), 263–266,). There is not only truth to that but also to the fact that his result suffices for some of the mathematical applications. Moreover, as von Neumann emphasized [in the above], there is one aspect of his result that is stronger than Birkhoff’s. If one defines $$Av(n,L) (\omega; f) = \frac{1}{n} \sum_{j=L}^{n+L-1} f(T_j(\omega))$$ then as $n \rightarrow\infty$, in $L^2$, $Av(n,L) ( · ; f)$ converges uniformly in $L$ (as can be seen by looking at either the von Neumann or Hopf proofs), but the pointwise convergence need not be uniform in $L$.