View source for Birkhoff ergodic theorem

{{MSC|37A30|37A05,37A10}}

[[Category:Ergodic theorems, spectral theory, Markov operators]]

One of the most important theorems in [[Ergodic theory|ergodic theory]]. For an endomorphism $ T $ of a $ \sigma $-finite measure space $ (X,\Sigma,\mu) $, Birkhoff’s ergodic theorem states that for any function $ f \in {L^{1}}(X,\Sigma,\mu) $, the limit
$$
\overline{f}(x) \stackrel{\text{df}}{=} \lim_{n \to \infty} \frac{1}{n} \sum_{k = 0}^{n - 1} f \! \left( {T^{k}}(x) \right)
$$
(the time average or the average along a trajectory) exists almost everywhere (for almost all $ x \in X $). Moreover, $ \overline{f} \in {L^{1}}(X,\Sigma,\mu) $, and if $ \mu(X) < \infty $, then
$$
\int_{X} f ~ \mathrm{d}{\mu} = \int_{X} \overline{f} ~ \mathrm{d}{\mu}.
$$

For a [[Measurable flow|measurable flow]] $ (T_{t})_{t \geq 0} $ in a $ \sigma $-finite measure space $ (X,\Sigma,\mu) $, Birkhoff’s ergodic theorem states that for any function $ f \in {L^{1}}(X,\Sigma,\mu) $, the limit
$$
\overline{f}(x) \stackrel{\text{df}}{=} \lim_{t \to \infty} \frac{1}{t} \int_{0}^{t} f({T_{t}}(x)) ~ \mathrm{d}{t}
$$
exists almost everywhere, with the same properties as $ f $.

Birkhoff’s theorem was stated and proved by G.D. Birkhoff {{Cite|B}}. It was then modified and generalized in various ways (there are theorems that contain, in addition to Birkhoff’s theorem, also a number of statements of a somewhat different kind, which are known in probability theory as ergodic theorems (cf. [[Ergodic theorem|Ergodic theorem]]); there also exist ergodic theorems for more general semi-groups of transformations {{Cite|KSS}}). Birkhoff’s ergodic theorem and its generalizations are known as individual ergodic theorems, since they deal with the existence of averages along almost each individual trajectory, as distinct from statistical ergodic theorems — the [[Von Neumann ergodic theorem|von Neumann ergodic theorem]] and its generalizations. (In non-Soviet literature, the term “pointwise ergodic theorem” is often used to stress the fact that the averages are almost-everywhere convergent.)

====References====
{|
|valign="top"|{{Ref|B}}|| G.D. Birkhoff, “Proof of the ergodic theorem”, ''Proc. Nat. Acad. Sci. USA'', '''17''' (1931), pp. 656–660. {{MR|}} {{ZBL|0003.25602}} {{ZBL|57.1011.02}}
|-
|valign="top"|{{Ref|KSS}}|| A.B. Katok, Ya.G. Sinai, A.M. Stepin, “Theory of dynamical systems and general transformation groups with invariant measure”, ''J. Soviet Math.'', '''7''': 6 (1977), pp. 974–1065; ''Itogi Nauk. i Tekhn. Mat. Analiz'', '''13''' (1975), pp. 129–262. {{MR|0584389}} {{ZBL|0399.28011}}
|}

====Comments====
In non-Soviet literature, the term “mean ergodic theorem” is used instead of “statistical ergodic theorem”.

A comprehensive overview of ergodic theorems is found in {{Cite|K}}. Many books on ergodic theory contain full proofs of (one or more) ergodic theorems; see e.g. {{Cite|P}}.

====References====
{|
|valign="top"|{{Ref|K}}|| U. Krengel, “Ergodic theorems”, de Gruyter (1985). {{MR|0797411}} {{ZBL|0575.28009}}
|-
|valign="top"|{{Ref|P}}|| K. Peterson, “Ergodic theory”, Cambridge Univ. Press (1983). {{MR|0833286}} {{ZBL|0507.28010}}
|}