LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generated

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.

Measurable information contraction and Pinsker bound

Statement

Assume AC. Let $P, Q$ be probability measures on a measurable space and let $T$ be a measurable map to another measurable space. Then $D (T_{*} P ∥ T_{*} Q) \leq D (P ∥ Q), ∥ P - Q ∥_{T V} : = \sup_{A} ∣ P (A) - Q (A) ∣ \leq \sqrt{D (P ∥ Q) / 2} .$ The entropy convention is the directional KL definition of Kullback–Leibler divergence, and the total-variation convention is the event-supremum convention of Total variation distance for probability laws. The inequality remains valid when $D = + \infty$ .

Facts & Assumptions

Given: AC and probability measures on the stated measurable spaces.

[F1]

Relative Entropy Variational, Chain, and Conditioning Rules: for probability measures, $D (P ∥ Q) = \sup_{g \in B_{b}} (P (g) - \log Q (e^{g}))$ , where $B_{b}$ denotes bounded real measurable functions.

[F2]

Pinsker’s inequality with event-supremum total variation: under AC and the event-supremum convention $\sup_{A} ∣ P (A) - Q (A) ∣$ , $∥ P - Q ∥_{T V} \leq \sqrt{D (P ∥ Q) / 2}$ for all probabilities, including the extended-entropy case.

[F3]

Kullback–Leibler divergence and Total variation distance for probability laws: fix the direction $P ∥ Q$ and the event-supremum normalization; no factor of two is inserted in the definition of total variation.

Proof

Given: The hypotheses and supplier conventions in Facts & Assumptions.

1.1F1F3algebra

For each bounded measurable target test $g$ , the pullback $g \circ T$ is bounded and measurable. Thus $P (g \circ T) = T_{*} P (g)$ and $Q (e^{g \circ T}) = T_{*} Q (e^{g})$ . The variational formula [F1] bounds the target objective by the supremum over all bounded source tests, which is $D (P ∥ Q)$ . Taking the target supremum proves data processing.

2.1F2F3step 1.1∎

Apply the imported theorem [F2] with the precise event-supremum convention [F3]. It gives the displayed square-root constant $1 / \sqrt{2}$ without changing normalization. When $D = + \infty$ , the right side is $+ \infty$ and the inequality is automatic.

Measurable information contraction and Pinsker bound

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources