DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-01

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.

An $L^{1}$ approximate identity on $R^{n}$

Definition

An $L^{1}$ approximate identity on $R^{n}$ is a family $(K_{ε})_{ε > 0}$ of functions in $L^{1} (R^{n})$ , where

$∥ K_{ε} ∥_{1} : = \int_{R^{n}} ∣ K_{ε} (x) ∣ d x,$

such that:

$\int_{R^{n}} K_{ε} (x) d x = 1$ for every $ε > 0$ ;
there is $M < \infty$ with $∥ K_{ε} ∥_{1} \leq M$ for every $ε > 0$ ;
for every $δ > 0$ , $\int_{∣ x ∣ > δ} ∣ K_{ε} (x) ∣ d x ⟶ 0 (ε \to 0^{+}) .$

This is the general notion used later for both compactly supported mollifiers and the Gaussian family.

Depends on

The class $L^1(\mu)$ of integrable functions

Used by

Dependency tree · two levels

2 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

Terence Tao, An Introduction to Measure Theory (standard reference, not scraped)
Richard L. Wheeden and Antoni Zygmund, Measure and Integral: An Introduction to Real Analysis (standard reference, not scraped)

An L1L1 approximate identity on RnRn