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.

Mutual singularity for signed or complex measures

Definition

Depends on

Dependency tree · 2 levels

Sources