PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-01

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A unit-mass smooth bump generates an $L^{1}$ approximate identity

Statement

Assume the Axiom of Countable Choice.

Let $φ \in C_{c}^{\infty} (R^{n})$ satisfy $\int φ = 1$ , and let $φ_{ε} (x) = ε^{- n} φ (x / ε)$ . Then $(φ_{ε})_{ε > 0}$ is an $L^{1}$ approximate identity.

Facts & Assumptions

Given: The Axiom of Countable Choice, a unit-mass smooth bump, and its rescalings.

[L1]

An $L^{1}$ approximate identity and a mollifier family are defined in An $L^{1}$ approximate identity on $R^{n}$ and The mollifier family generated by a unit-mass smooth bump.

[L2]

Linear change of variables preserves Lebesgue measure in the expected way, and the integral is linear on $L^{1}$ (A linear map $T$ of $R^{n}$ sends Lebesgue measurable sets to Lebesgue measurable sets, with $λ_{n} (T [E]) = ∣ \det T ∣ λ_{n} (E)$ when $T$ is invertible and $T [E]$ Lebesgue null when it is not, The Lebesgue integral is linear on $L^{1} (μ)$ ).

Proof

technique · direct

1.1L1L2givenalgebra

By [L2], the change of variables $u = x / ε$ gives [L1, L2, given, algebra] $\int φ_{ε} (x) d x = \int φ (u) d u = 1$ and likewise $∥ φ_{ε} ∥_{1} = \int ∣ φ_{ε} (x) ∣ d x = \int ∣ φ (u) ∣ d u = ∥ φ ∥_{1} .$ So the mass is $1$ and the $L^{1}$ norms are uniformly bounded.

2.1step 1.1choosealgebra

Let $R > 0$ satisfy $supp (φ) \subseteq \overline{B (0, R)}$ . [step 1.1, choose, algebra] Then $supp (φ_{ε}) \subseteq \overline{B (0, ε R)}$ . Hence for every $δ > 0$ and every $ε < δ / R$ , $\int_{∣ x ∣ > δ} ∣ φ_{ε} (x) ∣ d x = 0.$ So the tails concentrate at the origin.

3.1L1step 1.1step 2.1∎

Steps 1.1 and 2.1 verify the three defining clauses in [L1], so [L1, step 1.1, step 2.1] $(φ_{ε})$ is an $L^{1}$ approximate identity.

Depends on

Used by

Compactly supported smooth functions are dense in W^k,p(Rⁿ) Corollary
Hilbert transform does not map L-infinity to L-infinity Counterexample
Hilbert transform is not strong type (1,1) Counterexample
Hilbert transform of an interval indicator Example
Mollifying 1_[0,1] at two scales Example
The weak Dirichlet Poisson problem on an interval Example
A function with nonnegative test pairings is nonnegative a.e. Lemma
Cotlar's inequality for maximal truncations Lemma
Simultaneous L1 and L2 smooth approximation Lemma
Smooth compactly supported functions of an open set are dense in L² Lemma
Ambient smooth restrictions are dense on bounded Cᵏ domains Theorem
C_c^∞(ℝⁿ) is dense in Lᵖ(ℝⁿ) for 1 ≤ p < ∞ Theorem
Existence of bounded Kruzhkov entropy solutions Theorem
Local smooth approximation in integer-order Sobolev spaces Theorem
Locally integrable functions embed in distributions Theorem
Oleinik's one-sided estimate characterizes bounded entropy solutions Theorem
Plancherel support for SL2(R) Theorem
The distributional Riesz functional of a subharmonic function is a positive Radon measure Theorem
The Hamilton--Jacobi correspondence in one dimension Theorem
The Mihlin–Hörmander Fourier multiplier theorem Theorem
Zero weak gradient gives componentwise constants Theorem

Dependency tree · two levels

39 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

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

A unit-mass smooth bump generates an L1L1 approximate identity