DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)

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.

Relative Entropy of Probability Measures

Definition

Assume AC. If $P$ and $Q$ are probability measures on the same measurable space, define $D (P ∥ Q) = {\begin{cases} \int h \log h d Q, & P ≪ Q and h = d P / d Q, \\ + \infty, & P ≪̸ Q . \end{cases}$ Use $0 \log 0 = 0$ . The Radon–Nikodym derivative $h$ is unique up to $Q$ -almost-everywhere equality, so the integral is independent of its representative. Its negative part is finite because $- u \log u \leq 1 / e$ for $0 \leq u \leq 1$ and $Q$ is a probability; hence $D (P ∥ Q)$ is a well-defined extended real number.

Relative Entropy of Probability Measures

Definition

Depends on

Dependency tree · 2 levels

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