Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Is there any demonstration purely dynamic of the Birkhoff Ergodic Theorem, i.e, without the Maximal Ergodic Theorem??

I ask this question because I never understood the intuition behind The Maximal Ergodic Theorem.

share|cite|improve this question
There is an elementary proof due to Katznelson and Weiss which I found instructive. I do not know if that proof counts as "purely dynamic". – Martin Dec 30 '12 at 0:48
I have deleted some unconstructive comments. @HarryPotter: I would advise you to stay on-topic as you are learning your way around this site. "Joke" posts regarding your pseudonym (like the one I just deleted on a different question) will be considered spam. – Zev Chonoles Dec 30 '12 at 1:24
@Martin Could you give the main ideas of Katznelson and Weiss' proof in an answer? – Davide Giraudo Feb 21 '13 at 9:25

I know a proof of aubadditive ergodic theorem, without using the Birkhoff ergodic theorem. Thus, the Birkhoff ergodic theorem can be regarded as a Corollary of subadditive ergodic theorem.

Given the system $(M,f,\mu)$, we call a sequence $\varphi_n: M \to \Bbb R$ subadditive if $\varphi_{m+n} \leq \varphi_m + \varphi_n \circ f^m$.

The statement of the subadditive ergodic theorem is as follows:

Let $\varphi_n : M \to \Bbb R$, $n\ge 1$ be a subadditive sequence of measurable functions such that $\varphi_1^{+}\in L^1(\mu)$. Then the sequence $(\varphi_n/n)_n$ converges for $\mu$-almost every point to a measurable function $\varphi:M \to [-\infty,+\infty)$. Moreover, $\varphi^{+}\in L^1(\mu)$ and

$$ \int \varphi\,d\mu = \lim_n \frac{1}{n} \int \varphi_n\,d\mu = \inf_{n}\frac{1}{n} \int \varphi_n \,d\mu\in [-\infty , + \infty). $$

One easily sees that the Birkhorf ergodic theorem follows from the above theorem, since the sequence $\varphi_n=\sum_{i=0}^{n-1}\varphi\circ f^i$ is a subadditive function.

The idea of the proof is to consider the number sequence $a_n=\int \varphi_nd\mu$ and thus, by subadditivity, define $$L=\lim_n\frac{a_n}{n}=\inf_n\frac{a_n}{n}$$ Then define $$\varphi_-(x)=\liminf \frac{a_n}{n}; \varphi_+(x)=\limsup \frac{a_n}{n}$$ Finally, one tries to show that $$\int\varphi_-(x)d\mu\geq L\geq \int\varphi_+(x)dmu$$ and finishes the proof.

The proof following the above idea is given by A.Avila and J.Bochi, Proof of the subadditive ergodic theorem.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.