Birkhoff ergodic theorem

For a measure-preserving transformation $T$ and $f\in L^1(\mu )$, the ergodic averagesconverge almost everywhere to $\mathbb{E}[f\mid \mathcal{I}]$, the conditional expectation on the invariant sigma-algebra. The limit has the same integral as $f$.

In a probability measure-preserving system whose transformation is an ergodic transformation, a nonnegative measurable $f$ with $\int fd\mu =+\infty$ has $N^{-1}{\sum }_{j=0}^{N-1}f(T^{j}x)\to +\infty$ almost everywhere. Apply the pointwise ergodic theorem to $\min (f,K)$ on a common full-measure set for integer $K$, then bound the lower limit by each truncated integral. The monotone convergence theorem sends those integrals to infinity. The finite-integral case is the ordinary pointwise ergodic theorem.

For a measure-preserving transformation and real integrable $f$, let $S_{k}f={\sum }_{j<k}f\circ T^j$. For $E_N=\{{\max }_{1\le k\le N}{S}_{k}f>0\}$ the lemma asserts ${\int }_{E_N}fd\mu \ge 0$. Passing to the increasing union gives the corresponding infinite-supremum inequality.

Suppose $f_k\to f$ almost everywhere and ${\sup }_k|f_k|\in L^1$ in a probability measure-preserving system. Thenalmost everywhere and in $L^1$, where $\mathcal{I}$ is the invariant sigma-algebra. For the almost-everywhere assertion, bound the terms with index at least $M$ by ${\sup }_{k\ge M}|f_k-f|$, apply the Birkhoff ergodic theorem, and then let $M\to \infty$. The finitely many remaining end terms vanish by the linear growth bound for integrable observables.

In a probability measure-preserving system, $f\in L^1$ implies $f(T^{n}x)/n\to 0$ almost everywhere. For every $\epsilon >0$, the tail integral formula for moments gives ${\sum }_{n\ge 1}\mu (|f|>\epsilon n)\le \Vert f{\Vert }_1/\epsilon$. Invariance and the Borel-Cantelli lemma finish the proof. Consequently division by $n^a$ also gives zero for every $a\ge 1$.

For every $0<a<1$, choose $1<p<1/a$. On the Bernoulli shift over independent uniform coordinates in $(0,1)$, the observable $f(x)=x_0^{-1/p}$ is integrable, but $\mu (f(T^{n}x)>n^a)=n^{-ap}$. The events are independent and their probabilities have divergent sum, so the Borel-Cantelli lemma implies that $f(T^{n}x)/n^a>1$ infinitely often almost surely. No exponent below one gives a universal bound for all integrable observables.

On a finite measure space, the convergence in the Birkhoff ergodic theorem also holds in $L^1$. Prove it first for bounded truncations by the dominated convergence theorem, then use the $L^1$ contraction $\Vert A_{n}h{\Vert }_1\le \Vert h{\Vert }_1$ and Fatou lemma to remove the truncation.