False statementConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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C_c(X) is dense in L^infinity(mu) for every Radon measure

Statement

The assertion that $C_{c} (X)$ is dense in $L^{\infty} (μ)$ for every Radon measure is false.

Facts & Assumptions

Given: $X = R$ with Lebesgue measure and the constant function $1$ .

Refutation

technique · counterexample

1.1

For any $g \in C_{c} (R)$ , the complement of $supp g$ has positive measure and $g = 0$ there. Hence $∣ 1 - g ∣ = 1$ on a set of positive measure, so $∥ 1 - g ∥_{\infty} \geq 1$ .

given

2.1

Taking $g = 0$ shows the distance is exactly $1$ . Thus the finite- $p$ density theorem cannot be extended to $p = \infty$ .

step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

15 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

Donald L. Cohn, Measure Theory, 2nd ed., Chapter 7 (standard reference, not scraped)