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Here is a quote from Lectures on Ergodic Theory by Halmos:

I cannot resist the temptation of concluding these comments with an alternative "proof" of the ergodic theorem. If $f$ is a complex valued function on the nonnegative integers, write $\int f(n)dn=\lim > \frac{1}{n}\sum_{j=1}^nf(n)$ and whenever the limit exists call such functions integrable. If $T$ is a measure preserving transformation on a space $X$, then $$ > \int\int|f(T^nx)|dndx=\int\int|f(T^nx)|dxdn=\int\int|f(x)|dxdn=\int|f(x)|dx<\infty. > $$ Hence, by "Fubini's theorem" (!), $f(T^nx)$ is an integrable function of its two arguments, and therefore, for almost every fixed $x$ it is an integrable function of $n$. Can any of this nonsense be made meaningful?

Can any of this nonsense be made meaningful using nonstandard analysis? I know that Kamae gave a short proof of the ergodic theorem using nonstandard analysis in A simple proof of the ergodic theorem using nonstandard analysis, Israel Journal of Mathematics, Vol. 42, No. 4, 1982. However, I have to say that I am not satisfied with his proof. It is tricky and not very illuminating, at least for me. Besides, it does not look anything like the so called proof proposed by Halmos. Actually, Kamae's idea can be made standard in a very straightforward manner. See for instance A simple proof of some ergodic theorems by Katznelson and Weiss in the same issue of the Israel Journal of Mathematics. By the way, Kamae's paper is 7 pages and Katznelson-Weiss paper is 6 pages.

To summarize, is there a not necessarily short but conceptually clear and illuminating proof of the ergodic theorem using nonstandard analysis, possibly based on the intuition of Halmos?

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I would think if the limit above actually exists then for an infinite integer $n$, the standard part of the average of $n$ terms would be equal to the limit. If you can get this into a form where you've got $\sum_{j=1}^n \sum_{k=1}^m$ for infinite integers $n$ and $m$, where the thing being summed is standard (in the sense used in non-standard analysis) then you shouldn't even need Fubini's theorem. –  Michael Hardy Jun 1 '11 at 18:34
    
As far as your last question is concerned, there is a related thread mathoverflow.net/questions/28997/… –  Martin Sleziak Jul 9 '11 at 15:37

2 Answers 2

up vote 11 down vote accepted

I feel the answer is "no", at least while staying true to the spirit of Halmos's text. Halmos's "proof", if valid, would imply something far stronger (and false), namely that $\lim_{N \to \infty} \frac{1}{N} \sum_{n=1}^N f(T_n x)$ converged for almost every x, where $T_1, T_2, \ldots$ are an arbitrary sequence of measure-preserving transformations. This is not true even in the case when X is a two-element set. So any proof of the ergodic theorem must somehow take advantage of the group law $T^n T^m = T^{n+m}$ in some non-trivial way.

That said, though, nonstandard analysis does certainly generate a Banach limit functional $\lambda: \ell^\infty({\bf N}) \to {\bf C}$ which one does induce something resembling an integral, namely the Cesaro-Banach functional

$f \mapsto \lambda( (\frac{1}{N} \sum_{n=1}^N f(n) )_{N \in {\bf N}} ).$

This does somewhat resemble an integration functional on the natural numbers, in that it is finitely additive and translation invariant. But it is not countably additive, and tools such as Fubini's theorem do not directly apply to it; also, this functional makes sense even when the averages don't converge, so it doesn't seem like an obvious tool in order to demonstrate convergence.

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A (not so) related question: it is known that some theorems in ergodic theory (particularly recurrence theorems) are equivalent to purely number-theoretic statements on the integers, via Furstenberg's correspondence principle and the like. This may be a very naive question to ask, but can this kind of correspondence be exploited to obtain pointwise theorems in ergodic theory (particularly the Birkhoff pointwise ergodic theorem) by number-theoretic methods? –  Mark Jul 24 '11 at 10:00
    
@Mark - In some cases (usually in the field of equidistribution, and less in the additive NT field, which Furstenberg's correspondence deals with) it is possible to obtain distribution results which are essentially equivalent to ergodic theorems,for example - Weyl's equidistribution theorem essentially equivalent to unique ergodicity of irrational rotation on the circle. Vinogradov's theorem about equidistribution of irrational rotations at primes implies a special instance of Bourgain's theorem about means on arithmetical sets. Another example is equidistribution of Hecke points on \Gamma/G. –  Asaf Jul 24 '11 at 16:35
    
Right. That's not what I had in mind, though. I was rather referring to the various pointwise convergence theorems we have for abstract measure-preserving systems, such as the Birkhoff ergodic theorem and its analogues for other sequences of natural numbers. –  Mark Jul 24 '11 at 17:18
    
@Mark: A finitary analogue of the Birkhoff pointwise ergodic theorem via the correspondence principle was analysed by Avigad, Gerhardy, and Towsner at arxiv.org/abs/0706.1512 –  Terry Tao Jul 24 '11 at 17:23
    
@Terry: Thanks. –  Mark Jul 25 '11 at 11:11

There is a short proof in Katok's book. Introduction to the modern theory of dynamical systems [Book]

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Yes, but it has nothing to do with nonstandard analysis. –  Sonat Suer Jul 12 '11 at 8:11

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