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

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Every $L^{1}$ approximate identity converges to the identity in $L^{p}$ for $1 \leq p < \infty$

Statement

Assume the Axiom of Countable Choice.

Let $(K_{ε})_{ε > 0}$ be an $L^{1}$ approximate identity on $R^{n}$ . If $1 \leq p < \infty$ and $f \in L^{p} (R^{n})$ , then

$∥ f * K_{ε} - f ∥_{p} ⟶ 0 (ε \to 0^{+}) .$

Facts & Assumptions

Given: The Axiom of Countable Choice, an $L^{1}$ approximate identity, an exponent $1 \leq p < \infty$ , and $f \in L^{p} (R^{n})$ .

[L1]

Approximate identities are defined in An $L^{1}$ approximate identity on $R^{n}$ .

[L2]

Translation is continuous in finite $L^{p}$ ( $∥ τ_{h} f - f ∥_{p} \to 0$ in $L^{p} (R^{n})$ as $h \to 0$ , for $1 \leq p < \infty$ ).

[L3]

Minkowski's integral inequality and Young's inequality are available (Minkowski's integral inequality, Young's convolution inequality under Countable Choice).

Proof

technique · direct

1.1L1L3givenalgebra

Because $\int K_{ε} = 1$ , one may write [L1, L3, given, algebra] $f * K_{ε} - f = \int_{R^{n}} K_{ε} (y) (τ_{y} f - f) d y .$ Applying [L3] gives $∥ f * K_{ε} - f ∥_{p} \leq \int ∣ K_{ε} (y) ∣ ∥ τ_{y} f - f ∥_{p} d y .$

2.1L1L2L3step 1.1choosealgebra

Let $η > 0$ be arbitrary. If $∥ f ∥_{p} = 0$ , then $f = 0$ in $L^{p}$ and step [L1, L2, L3, step 1.1, choose, algebra] 1.1 gives $∥ f * K_{ε} - f ∥_{p} = 0 < η$ for every $ε > 0$ . Assume now $∥ f ∥_{p} > 0$ , and put $M : = \sup_{ε} ∥ K_{ε} ∥_{1} < \infty$ . By [L2], choose $δ > 0$ so that $∥ τ_{y} f - f ∥_{p} < η / (2 M)$ whenever $∣ y ∣ < δ$ . By [L1], choose $ε_{0} > 0$ so that $\int_{∣ y ∣ \geq δ} ∣ K_{ε} (y) ∣ d y < \frac{η}{4 ∥ f ∥_{p}} (0 < ε < ε_{0}) .$ For such $ε$ , split the integral from step 1.1 into $∣ y ∣ < δ$ and $∣ y ∣ \geq δ$ . The near part is at most $η / 2$ . For the far part, [L3] gives $∥ τ_{y} f - f ∥_{p} \leq 2 ∥ f ∥_{p}$ , so $\int_{∣ y ∣ \geq δ} ∣ K_{ε} (y) ∣ ∥ τ_{y} f - f ∥_{p} d y \leq 2 ∥ f ∥_{p} \int_{∣ y ∣ \geq δ} ∣ K_{ε} (y) ∣ d y < η / 2.$ Hence $∥ f * K_{ε} - f ∥_{p} < η$ whenever $0 < ε < ε_{0}$ .

3.1step 2.1∎

Because $η > 0$ was arbitrary, $∥ f * K_{ε} - f ∥_{p} \to 0$ as [step 2.1] $ε \to 0^{+}$ , proving the convergence in $L^{p}$ .

Depends on

Used by

Dependency tree · two levels

23 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)

Every L1L1 approximate identity converges to the identity in LpLp for 1≤p<∞1≤p<∞