CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-08-11

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.

A bounded-variation integrand is Riemann–Stieltjes integrable against every continuous integrator

Statement

If $f : [a, b] \to R$ has bounded variation and $α : [a, b] \to R$ is continuous, then $\int_{a}^{b} f d α$ exists.

Facts & Assumptions

Given: A BV function $f$ and a continuous function $α$ on $[a, b]$ .

[L1]

A continuous integrand is Stieltjes integrable against a BV integrator (A continuous integrand is Riemann–Stieltjes integrable against every bounded-variation integrator).

[L2]

Existence of $\int α d f$ is equivalent to existence of $\int f d α$ (Riemann–Stieltjes integration by parts).

[L3]

Bounded variation and continuity are those of Bounded variation and total variation on an interval and Continuity of $f : A \to R$ at a point of $A$ and on $A$ : the $ε$ - $δ$ condition, its agreement with $\lim_{x \to c} f (x) = f (c)$ at a limit point, and continuity at an isolated point.

Proof

technique · direct

1.1

Since $α$ is continuous and $f$ is BV, [L1] gives the integral $\int_{a}^{b} α d f$ .

L1L3

2.1

Integration by parts [L2] then gives existence of $\int_{a}^{b} f d α$ and its value $f (b) α (b) - f (a) α (a) - \int_{a}^{b} α d f$ .

step 1.1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

25 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

W. Rudin, Principles of Mathematical Analysis, Ch. 6, Theorem 6.9 (standard reference, not scraped)
William F. Trench, Introduction to Real Analysis, Exercise 3.2.10 (standard reference, not scraped)