DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-29

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.

The distribution function of absolute value

Definition

Let $(X, A, μ)$ be a measure space and let $f : X \to C$ be measurable. The distribution function of $∣ f ∣$ is the map $A_{f} : [0, \infty) \to [0, \infty]$ defined by

$A_{f} (t) : = μ ({x \in X : ∣ f (x) ∣ > t}) .$

This is the tail function that appears in the layer-cake formula.

Depends on

Measures on sigma-algebras

Dependency tree · 2 levels

Sources

Gerald B. Folland, Real Analysis, 2nd ed., Section 6.4