Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Strict and mod-null invariant sigma-algebras
Definition
For a system in Measure-preserving transformations and systems, set These are respectively the strictly invariant and invariant modulo null sets families. All their members are measurable in the original sigma-algebra; the terminology uses Sigma-algebras and Measure-null sets and almost-everywhere statements relative to a measure. Their sigma-algebra property is proved in Both invariant families are sigma-algebras ↗.
Depends on
Used by
- Invariant conditional-entropy description of the SMB limit Corollary
- Stationary irreducible Markov shift is ergodic Corollary
- Mod-null invariance need not be strict invariance Counterexample
- Ergodic partial sums, time averages, and the invariant L2 subspace Definition
- Ergodicity relative to an invariant measure Definition
- False: an ergodic invariant sigma-algebra has only two sets False statement
- Both invariant families are sigma-algebras Proposition
- Birkhoff limit for a stationary integrable process Theorem
- Finite-measure identification of the Birkhoff limit Theorem
Dependency tree · two levels
6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- E–W Proposition 2.14; Sarig Proposition 1.1 (standard reference, not scraped)