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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.

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

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

A unit-mass smooth bump generates an L1L1 approximate identity

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

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