DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-21

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.

Measure spaces

Definition

A measure space is a triple $(X, A, μ)$ in which $X$ is a set, $A$ is a sigma-algebra on $X$ , and $μ$ is a measure on $(X, A)$ (Measures on sigma-algebras). The members of $A$ are the measurable sets of the space.

Depends on

Measures on sigma-algebras

Used by

Almost uniform convergence Definition
Banach-valued simple function and integral Definition
Compact-group isotypic projection Definition
Convergence in measure Definition
Extended-real-valued measurable functions Definition
Finite, sigma-finite, and semifinite measures Definition
Measure-null sets and almost-everywhere statements relative to a measure Definition
Measure-preserving transformations and systems Definition
Probability measures and probability spaces Definition
The function space Lᵖ(μ) for 0 < p < ∞ Definition
The one-dimensional torus and its normalized Haar integral Definition
A circle representation with an averaged orthogonal weight form Example
Isotypic Haar projections specialize to finite character sums Example
Newton shell theorem from harmonic mean values Example
A positive rank-one Haar average is a nonzero compact intertwiner Lemma
Averaging a Hermitian form unitarizes a finite-dimensional compact-group representation Lemma
Haar averaging projects contractively onto the bounded intertwiners Lemma
Isotypic projections are mutually orthogonal equivariant projections Theorem

Dependency tree · two levels

4 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

S. Axler, Measure, Integration & Real Analysis, Definition 2.56 (standard reference, not scraped)