Birkhoff pointwise ergodic theorem

Birkhoff遍历定理

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Other names / variants: pointwise ergodic theorem

Translation evidence: Literature usage

Meaning in context

For a probability-preserving $T$ and $f\in L^1(\mu)$, the averages $n^{-1}\sum_{j=0}^{n-1}f(T^jx)$ converge almost everywhere to $E[f\mid\mathcal I]$, with $\mathcal I$ the invariant sigma-algebra; for ergodic $T$ the limit is $\int f\,d\mu$.

在概率空间 $(X,\mathcal B,\mu)$ 上，若可测变换 $T$ 保持 $\mu$ 且 $f\in L^1(\mu)$，则轨道平均几乎处处收敛于 $E[f\mid\mathcal I]$，其中 $\mathcal I$ 为不变σ代数；若 $T$ 遍历，极限为 $\int f\,d\mu$。

Example

For a cyclic permutation of $r$ equally weighted states, the orbit average equals the arithmetic mean of the $r$ observable values.

Usage and distinctions

The nonergodic limit need not be constant; the theorem is pointwise almost everywhere, unlike a bare norm-convergence statement.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Henk Bruin — Notes on Ergodic TheorySection 4, The Ergodic Theorem, Theorem 4 and its proof (invariant limit and ergodic constant case) · Checked 2026-10-14
2. Omri Sarig — Lecture Notes on Ergodic Theory§2.3.1, Theorem 2.3, printed p.40 / PDF p.48 (conditional-expectation limit for probability-preserving transformations) · Checked 2026-10-14

Translation sources

1. 朱子阳 — 算术动力系统：代数数论专题 VI（作者主页讲义）§2，遍历定理段落，Birkhoff遍历定理（PDF第5页） · Checked 2026-10-14

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