DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-27

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 interval set function attached to a nondecreasing right-continuous function

Definition

Let $F : R \to R$ be nondecreasing and right-continuous, and adopt the $(a, b]$ convention of This page uses the nondecreasing, right-continuous, $(a, b]$ -interval convention.

For a single h-interval $I$ , define

$μ_{0, F} ((a, b]) : = F (b) - F (a),$

$μ_{0, F} ((a, \infty)) : = \sup_{t > a} (F (t) - F (a)),$

$μ_{0, F} ((- \infty, b]) : = \sup_{t < b} (F (b) - F (t)),$

$μ_{0, F} (R) : = \sup_{a < b} (F (b) - F (a)) .$

For a set $E \in H$ presented as a finite disjoint union $E = ⋃_{i = 1}^{m} I_{i}$ of h-intervals, define

$μ_{0, F} (E) : = \sum_{i = 1}^{m} μ_{0, F} (I_{i}),$

with $μ_{0, F} (\emptyset) : = 0$ for the empty union.

The same set can have more than one disjoint h-interval presentation, so the value needs a well-definedness proof. That is exactly the content of The Stieltjes interval set function is finitely additive on the half-open interval algebra ↗, recorded here in justified_by.

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

Gerald B. Folland, Real Analysis, 2nd ed., Section 1.5 (standard reference, not scraped)