Alphabeta Math
DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge 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.

Parameter-dependent improper multiple integrals

Definition

Let D⊆Rn be open, let I⊆R be an interval, and let f:D×I→R. For t∈I, write ft(x):=f(x,t). If every slice ft is locally Riemann integrable and absolutely improperly integrable in the sense of Improper multiple integrals and absolute convergence on open sets, then

F(t):=∫Df(x,t) dx

defines the parameter-dependent improper multiple integral of f on I.

Local domination near t0∈I means that some relative neighborhood J⊆I of t0 and some nonnegative improperly integrable g:D→R satisfy ∣f(x,t)∣≤g(x) for all x∈D and t∈J.

Depends on

Used by

Dependency tree · two levels

5 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