False statementConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

FALSE: $L^{p}$ includes into $L^{r}$ on every measure space when $p < r$

Statement

For every measure space and every exponents $1 \leq p < r \leq \infty$ , one has $L^{p} \subseteq L^{r}$ .

Facts & Assumptions

Given: The two power-function counterexamples on $R$ .

[L1]

There is an $L^{1}$ function on $R$ that is not in $L^{2}$ ( $L^{1}$ is not a subset of $L^{2}$ on the line).

[L2]

There is an $L^{2}$ function on $R$ that is not in $L^{1}$ ( $L^{2}$ is not a subset of $L^{1}$ on the line).

Refutation

Proof technique: Refute with the two power-function counterexamples on the line: one witness lies in $L^{1} ∖ L^{2}$ and another lies in $L^{2} ∖ L^{1}$ .

1.1

Taking $p = 1$ and $r = 2$ , [L1] already contradicts the claim.

1.2

The reverse failure [L2] shows why the line supports neither global [L2] inclusion.

2.1

Hence the displayed universal inclusion statement is false.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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, Theorem 8.2 (standard reference, not scraped)

FALSE: LpLp includes into LrLr on every measure space when p<rp<r