feat: add Rokhlin lemma eval problem#308
Open
kim-em wants to merge 1 commit into
Open
Conversation
§109 of Knill's "Some Fundamental Theorems in Mathematics" (Rokhlin
1947, independently Kakutani 1943). Every aperiodic measure-preserving
automorphism of a standard Borel probability space admits, for every
height n and every ε > 0, a measurable tower base B such that B, T B,
…, T^{n-1} B are pairwise disjoint with total measure ≥ 1 - ε. The
[StandardBorelSpace Ω] hypothesis is essential — the countable-
cocountable σ-algebra on ℝ has MeasurableSingletonClass but admits no
nontrivial Rokhlin towers for the integer-shift map. Mathlib has
MeasurePreserving, periodic-point infrastructure, and StandardBorelSpace
but no Rokhlin lemma.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This file contains hidden or bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
Sign up for free
to join this conversation on GitHub.
Already have an account?
Sign in to comment
Add this suggestion to a batch that can be applied as a single commit.This suggestion is invalid because no changes were made to the code.Suggestions cannot be applied while the pull request is closed.Suggestions cannot be applied while viewing a subset of changes.Only one suggestion per line can be applied in a batch.Add this suggestion to a batch that can be applied as a single commit.Applying suggestions on deleted lines is not supported.You must change the existing code in this line in order to create a valid suggestion.Outdated suggestions cannot be applied.This suggestion has been applied or marked resolved.Suggestions cannot be applied from pending reviews.Suggestions cannot be applied on multi-line comments.Suggestions cannot be applied while the pull request is queued to merge.Suggestion cannot be applied right now. Please check back later.
This PR adds an eval problem for the Rokhlin lemma (Rokhlin 1947,
independently Kakutani 1943): every aperiodic measure-preserving
automorphism of a standard Borel probability space admits, for every
height
nand everyε > 0, a measurable tower baseBsuch thatB, T B, …, T^{n-1} Bare pairwise disjoint with total measure≥ 1 − ε. §109 of Knill's Some Fundamental Theorems in Mathematics.The
[StandardBorelSpace Ω]hypothesis is essential: the countable-cocountable σ-algebra on
ℝwith the integer-shift mapx ↦ x + 1is aperiodic and measure-preserving (for the 0/1 measure that sends
countable sets to 0 and cocountable sets to 1), but admits no
nontrivial Rokhlin towers — every cocountable base intersects its own
shift, and every countable base has zero-measure tower. The countable-
cocountable σ-algebra has
MeasurableSingletonClassbut is strictlycoarser than the Borel σ-algebra of any Polish topology on
ℝ,hence not standard Borel.
Mathlib has
MeasurePreserving,IsProbabilityMeasure,Function.periodicPts,Set.PairwiseDisjoint, andStandardBorelSpace, but no Rokhlin lemma (grep -ri 'rokhlin' Mathlib/Dynamics/finds nothing; the onlytowerhits areIsScalarTower). The Challenge ships four small helper definitions(
IsAperiodic,towerFloor,towerUnion,IsRokhlinTower).No formalization found in any major prover.
🤖 Prepared with Claude Code