LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-04

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Continuous compactly supported functions are recovered by small ball averages

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Let $g \in C_{c} (R^{n})$ and let $x \in R^{n}$ . Then $\lim_{r \to 0^{+}} A_{r} g (x) = g (x) .$

Facts & Assumptions

Given: The Axiom of Countable Choice, a function $g \in C_{c} (R^{n})$ , and a point $x \in R^{n}$ .

[L1]

The class $C_{c} (R^{n})$ consists of continuous functions on $R^{n}$ with compact support. (The spaces $C_{c} (R^{n})$ and $C_{c}^{\infty} (R^{n})$ )

[L2]

The ball average is $A_{r} g (x) = \frac{1}{λ (B (x, r))} \int_{B (x, r)} g (y) d λ (y) .$ (The average of a locally integrable function over a Euclidean ball)

Proof

technique · direct

1.1

Let $ε > 0$ . Since $g$ is continuous at $x$ by [L1], there is [L1, given] $δ > 0$ such that $∥ y - x ∥_{2} < δ ⟹ ∣ g (y) - g (x) ∣ < ε .$

L1given

2.1

For $0 < r < δ$ , every $y \in B (x, r)$ satisfies the hypothesis of step 1.1, [step 1.1, L2, algebra] so $∣ A_{r} g (x) - g (x) ∣ \leq \frac{1}{λ (B (x, r))} \int_{B (x, r)} ∣ g (y) - g (x) ∣ d λ (y) \leq ε .$

step 1.1L2algebra

3.1

Because step 2.1 holds for every $ε > 0$ , one has [step 2.1] $A_{r} g (x) \to g (x)$ as $r \to 0^{+}$ .

step 2.1∎

Depends on

Used by

Lebesgue differentiation theorem on ℝⁿ Theorem

Dependency tree · two levels

13 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, Exercise 1.6.16 (standard reference, not scraped)
Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed., Theorem 3.18 (standard reference, not scraped)