Published paper
Role in dependence graphs
Proof-critical source
Winning property of counterexamples to Uniform Littlewood's Conjecture
This paper is included only for the following marked statements:
- Ergodic theorem · PNAS VOR pp. 656–660, theorem and equations (1)–(9)Proves pointwise time-average convergence within its historical smooth invariant-volume flow scope.
- Modern discrete $L^1$ form · Walters, Theorem 1.14, pp. 34–35Records the exact modern probability-preserving-map formulation used by Oseledets, distinct from the historical paper's narrower printed scope.
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Audit summary
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Exact reviewed source
Exact PNAS version of record · official PMC page scans · Proceedings of the National Academy of Sciences 17 (1931), 656–660
George D. Birkhoff. Proof of the ergodic theorem. Proceedings of the National Academy of Sciences of the United States of America 17 (1931), no. 12, 656–660.
Open audited source ↗01Statements2 reported findingsNo statement issues found
The paper proves almost-everywhere existence of time frequencies for the printed class of smooth invariant-volume flows; it does not literally state the later abstract transformation theorem.
Ergodic theoremCorrect with an important scope boundary
Time spent in a measurable region has an almost-everywhere limiting frequency
Version of record pp. 656–660 · equations (1)–(9) and final italic theorem
For a smooth flow defined by analytic differential equations on a closed analytic manifold and preserving a volume integral, the paper proves that, outside a null set, each moving point has a limiting time probability of lying in a measurable region. The result is a pointwise rather than merely mean convergence theorem. The printed article says its abstract lemma extends to function space under suitable restrictions, but it does not spell out the modern theorem for an arbitrary probability-preserving transformation and every function.
Modern discrete formCorrect dependency boundary
The form used by Oseledets is verified at a terminal monograph locator
Peter Walters, An Introduction to Ergodic Theory, GTM 79 (1982), Theorem 1.14, pp. 34–35, DOI 10.1007/978-1-4612-5775-2
Oseledets applies the pointwise theorem to an integrable scalar function over a probability-preserving discrete transformation. Walters's Theorem 1.14 states and proves precisely this measure-theoretic version. The monograph is therefore the terminal verification source for the modern specialization, while this node preserves the exact historical paper and its narrower printed scope.
02Proofs2 reported findingsNo proof issues found
Within its printed smooth-flow scope, the proof closes through the separately represented recurrence note and standard measure foundations.
Pointwise convergence proofCorrect and complete modulo represented source
A maximal-type lemma controls upper and lower crossing-time limits
Version of record pp. 656–659 · lemma and equations (2)–(9)
The paper defines invariant level sets for the limsup and liminf of crossing-time averages, applies the integral inequality supplied by its lemma, and uses a countable choice of level parameters to force equality almost everywhere. A global countable transversal and the same argument for residence times in a region then yield the ratio defining the time probability. The measure-zero qualifications and two-sided-time passage are accounted for.
Recursive source expansionClosed with represented article branch
The preceding recurrence note supplies the only omitted journal argument
Version of record pp. 656–657 · “by the argument of the earlier note” and “the recurrence theorem stated results directly from this lemma”
The key disjoint-tower integral inequality is imported from Birkhoff's immediately preceding PNAS note, Proof of a Recurrence Theorem for Strongly Transitive Systems, and that note is represented as its own audited ordinary node. The remaining limiting, measurability, and countable-section steps are written in the article or are standard measure-theoretic foundations.
03Novelty0 reported findingsNo novelty concerns found
No non-novelty findings.