False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-27

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: Jensen's inequality holds on an infinite measure space without normalization

Statement

For every convex $φ$ and every nonnegative measurable $f$ one has $φ ⁣ (\int f d μ) \leq \int φ (f) d μ,$ even when $μ (X) \neq 1$ .

Facts & Assumptions

Given: The statement above.

[L1]

Jensen's inequality is stated for probability measures, so the normalization $μ (X) = 1$ is part of the theorem (Jensen's integral inequality for a probability measure).

[L2]

Counting measure is a measure on $N$ (Counting measure on an arbitrary set, Counting measure is a measure).

Refutation

technique · direct

1.1

On $(N, P (N), #)$ , let $f : = χ_{{1, 2}}$ and[L2, given, construct] $φ (x) : = x^{2}$ . Then $\int f d # = 2, \int φ (f) d # = 2.$

2.1

Therefore [step 1.1, L1, algebra] ∎ $φ ⁣ (\int f d #) = 4 > 2 = \int φ (f) d #,$ so the displayed inequality fails on this infinite measure space. This is why [L1] requires probability normalization.

Depends on

Used by

Nothing in the library uses this result yet.

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

Richard L. Wheeden and Antoni Zygmund, Measure and Integral, Theorem (7.44) (standard reference, not scraped)