CorollaryStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

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

Statement

If $f:[a,b]\to\mathbb R$ has bounded variation and $\alpha:[a,b]\to\mathbb R$ is continuous, then $\int_a^b f\,d\alpha$ exists.

Facts & Assumptions

Given: A BV function $f$ and a continuous function $\alpha$ 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 $\int\alpha\,df$ is equivalent to existence of $\int f\,d\alpha$ (Riemann–Stieltjes integration by parts).

[L3]

Bounded variation and continuity are those of Bounded variation and total variation on an interval and Continuity of $f : A \to \mathbb{R}$ at a point of $A$ and on $A$ : the $\varepsilon$ - $\delta$ 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 $\alpha$ is continuous and $f$ is BV, [L1] gives the integral $\int_a^b\alpha\,df$ .

L1L3

2.1

Integration by parts [L2] then gives existence of $\int_a^b f\,d\alpha$ and its value $f(b)\alpha(b)-f(a)\alpha(a)-\int_a^b\alpha\,df$ .

step 1.1L2∎

Depends on

Used by

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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)