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Continuous compactly supported functions are translation-continuous in $L^{p}$

Statement

Assume the Axiom of Countable Choice.

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

$∥ τ_{h} f - f ∥_{p} ⟶ 0 (h \to 0) .$

Facts & Assumptions

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

[L1]

Continuous functions on compact metric spaces are uniformly continuous (Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous).

[L2]

Bounded sets have finite Lebesgue measure, and translation preserves Lebesgue measure (Lebesgue measure is sigma-finite, and every metrically bounded subset of $R^{n}$ has finite outer measure, Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation).

[L3]

Translation is the convention of Translation of a function on $R^{n}$ , and $C_{c} (R^{n})$ is defined in The spaces $C_{c} (R^{n})$ and $C_{c}^{\infty} (R^{n})$ .

Proof

technique · direct

1.1

Let $K : = supp (f)$ , and choose $R > 0$ so that [L1, L3, given, choose] $K \subseteq B (0, R)$ . Then for $∣ h ∣ \leq 1$ , the support of $τ_{h} f - f$ lies in the compact set $\overline{B (0, R + 1)}$ . By [L1], $f$ is uniformly continuous on that compact set.

L1L3givenchoose

2.1

Let $ε > 0$ . Uniform continuity gives $δ > 0$ such that [L1, L2, step 1.1, choose, algebra] $∣ f (x - h) - f (x) ∣ < ε$ whenever $∣ h ∣ < δ$ and $x, x - h \in \overline{B (0, R + 1)}$ . Hence for $∣ h ∣ < δ$ , $∥ τ_{h} f - f ∥_{p}^{p} \leq ε^{p} λ_{n} (\overline{B (0, R + 1)}) .$ The right-hand side tends to $0$ with $ε$ , and [L2] makes the measure finite.

L1L2step 1.1choosealgebra

3.1

Therefore $∥ τ_{h} f - f ∥_{p} \to 0$ as $h \to 0$ .

step 2.1∎

Depends on

Used by

‖τ_h f - f‖ₚ → 0 in Lᵖ(ℝⁿ) as h → 0, for 1 ≤ p < ∞ Theorem

Dependency tree · two levels

44 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)

Continuous compactly supported functions are translation-continuous in LpLp