RemarkRemark: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)

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.

Characteristic-function normalization

Convention

For a positive Borel probability measure $μ$ on $R^{n}$ , define $φ_{μ} (t) = \int e^{i t \cdot x} d μ (x)$ . This integral exists because the integrand is Borel of modulus one and $μ (R^{n}) = 1$ ; use Integrable real and complex functions, and their integrals and $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ . With the negative $2 π$ Fourier convention, the same integral is $φ_{μ} (t) = \hat{μ} (- t / (2 π)),$ because $- 2 π i x \cdot (- t / (2 π)) = i t \cdot x$ . In particular $φ_{μ} (0) = 1$ . This is a sign-and-scale dictionary for a supplied positive probability measure; no measure construction, uniqueness theorem or choice principle is used.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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

Gerald Teschl, Topics in Real and Functional Analysis (2017) (standard reference, not scraped)