False statementConstruction: AI-adaptedVerification: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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FALSE: $L^{p} (μ)$ is separable for every measure $μ$ and every $1 \leq p < \infty$

Statement

False claim. For every measure space $(X, A, μ)$ and every $1 \leq p < \infty$ , the space $L^{p} (μ)$ is separable.

Facts & Assumptions

Given: An uncountable set $X$ with counting measure.

[L1]

The general separability theorem requires the Axiom of Countable Choice, sigma-finiteness, and a countably generated sigma-algebra (If $μ$ is sigma-finite and $A$ is countably generated, then $L^{p} (μ)$ is separable for $1 \leq p < \infty$ ).

[L2]

Counting measure is a measure, and separability means having a countable dense subset (Counting measure on an arbitrary set, Counting measure is a measure, Separability: the existence of an at most countable dense subset).

Refutation

technique · direct

1.1

For each $x \in X$ , let $e_{x} : = 1_{{x}}$ . Since [L2, given, algebra] $μ ({x}) = 1$ , each $e_{x}$ lies in $L^{p} (μ)$ . If $x \neq y$ , then $∥ e_{x} - e_{y} ∥_{p}^{p} = ∣ 1 ∣^{p} + ∣ - 1 ∣^{p} = 2,$ so $∥ e_{x} - e_{y} ∥_{p} = 2^{1 / p}$ .

L2givenalgebra

2.1

Thus the uncountable family ${e_{x} : x \in X}$ is pairwise [L1, L2, step 1.1, algebra] $2^{1 / p}$ -separated. No countable set can be dense in a metric space containing uncountably many disjoint balls of radius $2^{1 / p} / 3$ . So this $L^{p} (μ)$ is not separable, contradicting the claim.

L1L2step 1.1algebra

3.1

Therefore the unrestricted statement is false; [L1] records the correct [L1, step 2.1] hypothesis ledger.

L1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 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)

FALSE: Lp(μ)Lp(μ) is separable for every measure μμ and every 1≤p<∞1≤p<∞