DefinitionDefinition: Literature-sourcedProof: Not applicablejudge 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.

The absolutely continuous part and the singular part in the Lebesgue decomposition

Definition

Under the hypotheses of Every sigma-finite signed measure admits a Lebesgue decomposition relative to a sigma-finite positive measure, write the unique decomposition of $ν$ relative to $μ$ as $ν = ν_{a} + ν_{s},$ where $ν_{a} ≪ μ$ and $ν_{s} ⊥ μ$ . The measure $ν_{a}$ is the absolutely continuous part of $ν$ with respect to $μ$ , and $ν_{s}$ is the singular part of $ν$ with respect to $μ$ .

Depends on

Used by

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 F. Bass, Real Analysis for Graduate Students, Chapter 13 (standard reference, not scraped)
Sheldon Axler, Measure, Integration & Real Analysis, 9.35 (standard reference, not scraped)