DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

How 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.

Finite product mass on words

Definition

Let $A$ be a finite nonempty alphabet with its full power-set $σ$ -algebra (equivalently, with the discrete Borel $σ$ -algebra), let $p$ be a probability vector on $A$ , and let $n \geq 1$ . On $Ω_{n} : = A^{n}$ equip the set of all subsets with its full power-set $σ$ -algebra. Assign each word $x^{n} = (x_{1}, \dots, x_{n})$ the weight $w_{p, n} (x^{n}) : = \prod_{i = 1}^{n} p (x_{i}),$ and for $E \subseteq Ω_{n}$ define $μ_{p, n} (E) : = \sum_{x^{n} \in E} w_{p, n} (x^{n})$ . Write $μ_{p, n}$ as the product mass assignment $p^{\otimes n}$ . For $1 \leq i \leq n$ , the coordinate map is $X_{i} (x^{n}) : = x_{i}$ .

Depends on

Finite distributions, entropy, divergence, and stochastic matrices

Used by

Dependency tree · two levels

2 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

Thomas M. Cover and Joy A. Thomas, Elements of Information Theory, second edition, Chapter 11 (standard reference, not scraped)
Yury Polyanskiy and Yihong Wu, Information Theory: From Coding to Learning, author-hosted book draft (standard reference, not scraped)