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feat: add Rokhlin lemma eval problem#308

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feat: add Rokhlin lemma eval problem#308
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eval/rokhlin-lemma

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@kim-em kim-em commented May 24, 2026

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 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. §109 of Knill's Some Fundamental Theorems in Mathematics.

The [StandardBorelSpace Ω] hypothesis is essential: the countable-
cocountable σ-algebra on with the integer-shift map x ↦ x + 1
is 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 MeasurableSingletonClass but is strictly
coarser than the Borel σ-algebra of any Polish topology on ,
hence not standard Borel.

Mathlib has MeasurePreserving, IsProbabilityMeasure,
Function.periodicPts, Set.PairwiseDisjoint, and
StandardBorelSpace, but no Rokhlin lemma (grep -ri 'rokhlin' Mathlib/Dynamics/ finds nothing; the only tower hits are
IsScalarTower). The Challenge ships four small helper definitions
(IsAperiodic, towerFloor, towerUnion, IsRokhlinTower).

No formalization found in any major prover.

🤖 Prepared with Claude Code

§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>
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