LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-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.

Total variation duality, kernel contraction, and coupling bound

Statement

Let $μ, ν$ be probability measures on $(X, X)$ and let $P$ be a probability kernel on that space. Then $∥ μ - ν ∥_{T V} = \sup_{f : X \to [0, 1] X -measurable} ∣ \int f d μ - \int f d ν ∣,$ $∥ μ P - ν P ∥_{T V} \leq ∥ μ - ν ∥_{T V} .$ If $(U, V)$ is a coupling of $μ$ and $ν$ on a probability space and $E$ is a measurable event on which $U = V$ , then $∥ μ - ν ∥_{T V} \leq P (E^{c}) .$

Facts & Assumptions

Given: Two probability measures $μ, ν$ on the same measurable space, a probability kernel $P$ , and, for the coupling conclusion, random elements $(U, V)$ with laws $μ, ν$ and a measurable event $E$ such that $U = V$ on $E$ .

[F1]

Total variation distance for probability laws: total variation is $\sup_{A \in X} ∣ μ (A) - ν (A) ∣$ .

[F2]

Measure kernel and probability kernel: $P (x, \cdot)$ is a probability measure and $x \mapsto P (x, A)$ is measurable for each $A$ .

[F3]

Tonelli's theorem for nonnegative measurable functions on a sigma-finite product: nonnegative product-measurable functions may be integrated in either order.

Proof

Given: [F1]–[F3] and the hypotheses in the Statement.

1.1F1F3givenalgebra

For measurable $f : X \to [0, 1]$ , put $A_{t} = {x : f (x) > t}$ for $t \in [0, 1]$ . The identity $f (x) = \int_{0}^{1} 1_{A_{t}} (x) d t$ and [F3] give $\int f d μ - \int f d ν = \int_{0}^{1} [μ (A_{t}) - ν (A_{t})] d t$ . Its absolute value is at most $\int_{0}^{1} ∣ μ (A_{t}) - ν (A_{t}) ∣ d t \leq ∥ μ - ν ∥_{T V}$ by [F1]. Conversely, each indicator $1_{A}$ is an admissible $f$ and attains the setwise difference for its set $A$ . Taking suprema proves the dual formula.

2.1F1F2step 1.1given

For each $A \in X$ , the function $x \mapsto P (x, A)$ is measurable and takes values in $[0, 1]$ by [F2]. Apply step 1.1 to that function to obtain $∣ μ P (A) - ν P (A) ∣ \leq ∥ μ - ν ∥_{T V}$ . Taking the supremum over $A$ proves kernel contraction.

3.1F1givenalgebra∎

For any $A \in X$ , $∣ μ (A) - ν (A) ∣ = ∣ E [1_{A} (U) - 1_{A} (V)] ∣ \leq E ∣ 1_{A} (U) - 1_{A} (V) ∣ \leq P (E^{c})$ , because the two indicators agree on $E$ . Taking the supremum over $A$ proves the stated bound.

Depends on

Used by

Dependency tree · two levels

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Sources

Gareth O. Roberts and Jeffrey S. Rosenthal, Coupling and Ergodicity of Adaptive MCMC (standard reference, not scraped)