I know of six proofs of the Birkhoff ergodic theorem.

• using a maximal inequality (Birkhoff, Riesz, Wiener, Yosida, Kakutani, Garsia...)

• based on martingales and upcrossing inequalities (Bishop 1966)

• using non-standard analysis (Kamae 1982)

• based on variational inequalities (Bourgain 1988)

• using a filling scheme (Chacon)

• Katznelson-Weiss proof (1982) and derivatives (Keane, Petersen)

The last one seems the most easy to remember for me. This may look like a combinatorial trick, but with some thought, it appears quite natural, and I know several results that use some similar idea.

So let $\epsilon>0$ and x in X. We can find an $n(x)$ depending on x such that $$ \overline{\lim}\ {1\over n}\ \Sigma_0^{n-1}\ f(T^k(x)) \leq \lim {1\over n(x)} \sum_{k=0}^{n_(x)-1} f(T^k(x))\ +\ \varepsilon$$ Note that n(x) is finite everywhere, hence bounded on a set R with complement of arbitrary small measure. Then cut the Birkhoff sum according to the sequence $n_{i+1}(x)=n_i(x)+n(T^{n_i}(x))$ if $T^{n_i}(x)$ is in R, $n_{i+1}(x)=n_i(x)+1$ otherwise. A picture should make clear what is going on. The rest of the proof is routine check.

Of course if you are in the business of non-standard analysis, Kamae's proof is both short and enlightening but then you need some work to get the standard statement.

I know of six proofs of the Birkhoff ergodic theorem.

• using a maximal inequality (Birkhoff, Riesz, Wiener, Yosida, Kakutani, Garsia...)

• based on martingales and upcrossing inequalities (Bishop 1966)

• using non-standard analysis (Kamae 1982)

• based on variational inequalities (Bourgain 1988)

• using a filling scheme (Chacon)

• Katznelson-Weiss proof (1982) and derivatives (Keane, Petersen)

The last one seems the most easy to remember for me. This may look like a combinatorial trick, but with some thought, it appears quite natural, and I know several results that use some similar idea.

So let $\epsilon>0$ and $x$ in $X$. We can find an $n(x)$ depending on $x$ such that $$ \overline{\lim}\ {1\over n}\ \Sigma_0^{n-1}\ f(T^k(x)) \leq \lim {1\over n(x)} \sum_{k=0}^{n_(x)-1} f(T^k(x))\ +\ \varepsilon$$ Note that $n(x)$ is finite everywhere, hence bounded on a set $R$ with complement of arbitrary small measure. Then cut the Birkhoff sum according to the sequence $n_{i+1}(x)=n_i(x)+n(T^{n_i}(x))$ if $T^{n_i}(x)$ is in R, $n_{i+1}(x)=n_i(x)+1$ otherwise. A picture should make clear what is going on. The rest of the proof is routine check.

Of course if you are in the business of non-standard analysis, Kamae's proof is both short and enlightening but then you need some work to get the standard statement.