User barry simon - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T18:01:44Z http://mathoverflow.net/feeds/user/17298 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/73248/applications-of-and-motivation-for-von-neumanns-mean-ergodic-theorem/73358#73358 Answer by Barry Simon for Applications of and motivation for von Neumann's mean ergodic theorem Barry Simon 2011-08-21T20:51:43Z 2011-08-21T20:57:51Z <p>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 $L$.</p>