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.
An uncountable Bernoulli coordinate process
Example
Let be uncountable and give every finite the uniform law on . Assuming AC, the extension theorem produces a probability measure on the cylinder sigma-algebra of , and its coordinates are independent fair bits.
Verification
Given: An uncountable set and the stated finite uniform laws.
The finite uniform laws are compatible under deletion of coordinates, so the standard-Borel extension theorem applies.
The canonical-process corollary identifies each prescribed finite joint law. Thus finite coordinate events are measurable and have their product probabilities; no assertion is made about arbitrary path events.
Depends on
Used by
- A noncylinder path functional may fail to be measurable Counterexample
Dependency tree · two levels
15 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
- Shalizi, Building Processes, Theorem 29 (standard reference, not scraped)