DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-30

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.

Mutual singularity for signed or complex measures

Definition

Let $μ$ and $ν$ be signed measures or complex measures on the same measurable space $(X, A)$ . They are mutually singular, written $μ ⊥ ν,$ when there are measurable sets $P, N \in A$ with $P \cap N = \emptyset, P \cup N = X,$ such that:

every measurable subset of $N$ has $μ$ -value $0$ ;
every measurable subset of $P$ has $ν$ -value $0$ .

For positive measures this is equivalent to the usual condition $μ (N) = 0$ and $ν (P) = 0$ , because positivity turns vanishing on the ambient set into vanishing on all measurable subsets.

Depends on

Used by

Jordan decomposition of a signed measure into unique mutually singular positive parts Theorem

Dependency tree · two levels

6 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

John K. Hunter, Measure Theory, Definition 6.19 (standard reference, not scraped)
Richard F. Bass, Real Analysis for Graduate Students, Chapter 12 (standard reference, not scraped)