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

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

Statement

Assume the Axiom of Countable Choice.

Let $1 \leq p < \infty$ and $f \in L^{p} (R^{n})$ . Then

$∥ τ_{h} f - f ∥_{p} ⟶ 0 (h \to 0) .$

Facts & Assumptions

Given: The Axiom of Countable Choice, $1 \leq p < \infty$ , $f \in L^{p} (R^{n})$ , and $ε > 0$ .

[L1]

$C_{c} (R^{n})$ is dense in $L^{p} (R^{n})$ ( $C_{c} (R^{n})$ is dense in $L^{p} (R^{n})$ for $1 \leq p < \infty$ ).

[L2]

Compactly supported continuous functions are translation-continuous in $L^{p}$ (Continuous compactly supported functions are translation-continuous in $L^{p}$ ).

[L3]

Lebesgue measure is translation invariant, so $∥ τ_{h} u ∥_{p} = ∥ u ∥_{p}$ (Translation of a function on $R^{n}$ , Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation).

[L4]

Minkowski's inequality is available (Minkowski's inequality for integrals, including $p = \infty$ ).

Proof

technique · direct

1.1

By [L1], choose $g \in C_{c} (R^{n})$ with [L1, L2, given, choose] $∥ f - g ∥_{p} < ε / 3$ . By [L2], choose $δ > 0$ such that $∣ h ∣ < δ$ implies $∥ τ_{h} g - g ∥_{p} < ε / 3$ .

L1L2givenchoose

2.1

For $∣ h ∣ < δ$ , [L3] and [L4] give [L3, L4, step 1.1, algebra] $∥ τ_{h} f - f ∥_{p} \leq ∥ τ_{h} (f - g) ∥_{p} + ∥ τ_{h} g - g ∥_{p} + ∥ g - f ∥_{p} = 2 ∥ f - g ∥_{p} + ∥ τ_{h} g - g ∥_{p} < ε .$

L3L4step 1.1algebra

3.1

Since $ε > 0$ was arbitrary, $∥ τ_{h} f - f ∥_{p} \to 0$ as [step 2.1] $h \to 0$ .

step 2.1∎

Depends on

Used by

Dependency tree · two levels

30 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)

∥τhf−f∥p→0∥τh​f−f∥p​→0 in Lp(Rn)Lp(Rn) as h→0h→0, for 1≤p<∞1≤p<∞