# Exercise 2.8.4 in Einsiedler and Ward

$$\newcommand{\set}[1]{\\{#1}\\}$$ $$\newcommand{\ab}[1]{\langle #1\rangle}$$ $$\newcommand{\mc}{\mathcal}$$ $$\newcommand{\Z}{\mathbb Z}$$ $$\newcommand{\C}{\mathbf C}$$

Exercise 2.8.4 in Einsiedler and Ward's Ergodic Theory with a View Towards Number Theory asks the following.

Exercise. Give a different proof of the mean ergodic theorem as follows. For a measure preserving system $$(X, \mc X, \mu, T)$$ and a function $$f\in L^2_\mu$$, show that the map $$n\mapsto \ab{T^nf, f}:\Z\to \C$$ is positive definite and apply the Herglotz-Bochner Theorem to translate the problem into one concerned with functions on $$S^1$$, and there use the fact that $$\frac{1}{N} \sum_{n=0}^{N-1} z^n$$ converges for each $$z\in S^1$$.

First let me assume that the system is invertible (otherwise I was facing trouble showing the positive-definiteness of the said map). Applying the Herglotz-Bochner theorem we can find a positive measure $$\rho$$ on $$S^1$$ such that $$\hat \rho(n) = \ab{T^nf, f}$$ for all $$n$$. Let $$f_N = \frac{1}{N}\sum_{n=0}^{N-1}T^nf$$.

I am only able to show the weak$$^\ast$$ convergence of $$(f_N)_{N\geq 1}$$ using the approach mentioned in the problem.

Note that $$\ab{f_N, f} = \frac{1}{N} \sum_{n=0}^{N-1} \hat\rho(n)$$. Using dominated convergence theorem we see that the latter converges to $$\rho(1)$$. It follows, by similar reasoning, that $$\ab{f_N, T^kf}$$ converges to $$\hat\rho(1)$$ for all $$k$$. Now define $$S = \overline{\text{Span}\{T^nf:\ n\in \Z}\}$$. It follows that $$\ab{f_N, g}$$ converges whenever $$g\in S$$. Also, if $$g\in S^\perp$$, then $$\ab{f_N, g}$$ clearly converges to $$0$$. Thus for any $$g\in L^2_\mu$$ we have that $$\ab{f_N, g}$$ converges. This shows the weak* convergence of $$f_N$$.

How do I upgrade to norm convergence?

• Not sure, but Rudin 'Functional analysis' Theorem 12.44 might help. Commented Apr 25, 2023 at 5:02

First notice that $$||f_N||_2=\langle f_N, f_N \rangle = \frac{1}{N^2} \sum_{n,m=0}^{N-1} \langle T^nf,T^mf \rangle,$$ then use the fact that $$T$$ is measure preserving to obtain $$\langle T^nf,T^mf \rangle = \langle T^{n-m}f,f \rangle$$ and then use the Bochner-Herglotz theorem. Indeed, by the theorem there is a finite and positive measure $$\sigma_f$$, so that $$\langle T^{n-m}f,f \rangle = \int_{\mathbb{S}} z^{n-m}\ d \sigma_{f}(z)$$. Therefore, $$||f_N||_2$$ becomes $$\frac{1}{N^2} \sum_{n,m=0}^{N-1} \int_{\mathbb{S}} z^n \cdot \bar{z}^m\ d \sigma_{f}(z) = \big|\big| \frac{1}{N} \sum_{n=0}^{N-1} z^n \big|\big|_{L^2(\sigma_f)}.$$

In the same manner one sees that

$$\big| \big| f_N-f_M \big| \big|_2 = \big|\big| \frac{1}{N} \sum_{n=0}^{N-1} z^n - \frac{1}{M} \sum_{m=0}^{M-1} z^m\big|\big|_{L^2(\sigma_f)}.$$

Then you can use the hint and the dominated convergence theorem to conclude that $$(f_N)$$ is a Cauchy sequence and thus converges.

• I can elaborate further if you like, although it's fairly straightforward.
– User
Commented Jun 22, 2023 at 17:15
• Thanks for the response. Though I do not see your idea. Can you elaborate. Thank you. Commented Jun 22, 2023 at 17:20
• Thanks for adding details. What this (to me) shows is that the $L^2$-norm of $f_N$ converges as $N\to \infty$. I don't see why this shows that $f_N$ itself converges in the $L^2$-sense as $N\to \infty$. Commented Jun 24, 2023 at 2:57
• Oh, yes, you want convergence in norm! You can repeat the same process to show that $(f_N)$ is Cauchy.
– User
Commented Jun 24, 2023 at 12:11
• I'll try it. But usually it is very hard to establish that such averages are Cauchy. A useful tool is here is the van der Corput trick. Commented Jun 25, 2023 at 3:32