ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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.

Interpolation of an averaging operator on a probability space

Example

On a probability space define $P f = (\int f d μ) 1_{X}$ for complex finite simple f. This is a complex-linear operator with $L^{1} \to L^{\infty}$ and $L^{2} \to L^{2}$ norms at most one. Consequently $∥ P f ∥_{p^{'}} \leq ∥ f ∥_{p}$ for $1 \leq p \leq 2$ .

Facts & Assumptions

[F1]

The absolute value of an integrable function’s integral is at most the integral of its modulus The modulus of an integral is bounded by the integral of the modulus.

[F2]

Complex Cauchy–Schwarz bounds the pairing with the constant one The complex $L^{2}$ pairing is well-defined and satisfies Cauchy–Schwarz.

[F3]

The core endpoint estimates interpolate to the conjugate-exponent estimate Interpolate L1 to Linfinity and L2 to L2 bounds.

Verification

Given: The objects and hypotheses in the statement.

1.1

Every finite simple function on a probability space is integrable; finite sums in its integral show that P is complex-linear. The constant one has every displayed norm equal to one. Hence $∥ P f ∥_{\infty} = ∣ \int f ∣ \leq \int ∣ f ∣ = ∥ f ∥_{1}$ by the integral triangle inequality. In particular $P 1_{X} = 1_{X}$ , so the bound is attained on this input.

F1given

2.1

Complex Cauchy–Schwarz against the constant one gives $∥ P f ∥_{2} = ∣ ⟨ f, 1 ⟩ ∣ \leq ∥ f ∥_{2} ∥ 1 ∥_{2} = ∥ f ∥_{2}$ . Every probability space is sigma-finite, with the constant exhaustion X, so F3 applies with A=B=1 to give $∥ P f ∥_{p^{'}} \leq ∥ f ∥_{p}$ for interior p. The endpoint estimates are the two calculations above.

F2F3step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

16 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

Teschl Theorem 15.2; explicit averaging specialization (standard reference, not scraped)