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

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Linearity and interval additivity of the Riemann–Stieltjes integral

Statement

Whenever the integrals on the right exist,

$\int_a^b(uf+vg)\,d\alpha=u\int_a^b f\,d\alpha+v\int_a^b g\,d\alpha,$ $\int_a^b f\,d(u\alpha+v\beta)=u\int_a^b f\,d\alpha+v\int_a^b f\,d\beta.$

Let $a\le c\le b$ , suppose $\alpha$ has bounded variation, and suppose $f$ is continuous at $c$ . Then integrability on $[a,b]$ is equivalent to integrability on both $[a,c]$ and $[c,b]$ , and

$\int_a^b f\,d\alpha=\int_a^c f\,d\alpha+\int_c^b f\,d\alpha.$

Facts & Assumptions

Given: Functions for which the displayed integrals are defined, scalars $u,v$ , and for additivity a BV integrator $\alpha$ and a cut $c$ where $f$ is continuous.

[L1]

Stieltjes integrability is convergence of all sufficiently fine tagged sums (Riemann–Stieltjes sums, upper and lower sums, and the Riemann–Stieltjes integral).

[L2]

Such a limit is unique (The Riemann–Stieltjes integral is unique).

[L3]

Partitions can be inserted at $c$ and spliced across $c$ (Partition of $[a,b]$ as a finite strictly increasing list $a = t_0 < t_1 < \dots < t_n = b$ , its subintervals and their lengths, its mesh, refinement, and the common refinement of two partitions).

[L4]

Finite sums distribute and split at an index (Finite sums and finite products, by recursion, Laws of finite sums and finite products).

[L5]

Continuity at $c$ makes $f(x)$ close to $f(c)$ when $x$ is close to $c$ (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).

[L6]

Reversal and singleton conventions are those of the oriented integral (The integral with oriented limits: $\int_a^a f := 0$ and $\int_b^a f := -\int_a^b f$ ).

[L7]

The sum of the absolute integrator increments over any partition is at most $\operatorname{Var}_{[a,b]}(\alpha)$ (Bounded variation and total variation on an interval).

[L8]

Every Cauchy sequence of real sums has a finite real limit (The Cauchy criterion from the least-upper-bound property: in a complete ordered field every Cauchy sequence converges).

Proof

technique · direct

1.1

Each tagged sum is exactly linear in $f$ and in $\alpha$ , by distribution in the finite sum. Passing to mesh limits and using uniqueness proves both linearity formulas.

L1L2L4

1.2

For a partition containing $c$ , its Stieltjes sum splits exactly into the sums on the two subintervals. Inserting $c$ into a fine partition changes only the interval containing $c$ . Direct subtraction bounds the difference, for any choices of the old and new tags, by the oscillation of $f$ near $c$ times the sum of the relevant absolute increments of $\alpha$ , hence by that oscillation times $\operatorname{Var}_{[a,b]}(\alpha)$ . Continuity of $f$ at $c$ makes this error tend to zero with the mesh.

L1L3L4L5L7

2.1

If the whole-interval integral exists, take any two sufficiently fine sums on $[a,c]$ and splice each with the same sufficiently fine sum on $[c,b]$ . The two whole-interval sums are close, so their common right part cancels and the left sums are Cauchy. Choose uniform left-hand sums with mesh tending to zero; they form a Cauchy sequence and have a limit by [L8]. Every arbitrary sufficiently fine left-hand sum is close to a sufficiently late uniform one, so the entire left-hand mesh family has that limit. The symmetric argument gives the right integral. Conversely, if both restricted integrals exist, splice their fine sums and use step 1.2 to compare with arbitrary whole-interval sums. The exact split gives the displayed value by [L2]. Endpoint cuts and reversed limits follow from [L6].

step 1.2L1L2L3L6L8∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 80 results over 16 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

W. Rudin, Principles of Mathematical Analysis, Ch. 6, Theorem 6.12 (standard reference, not scraped)
William F. Trench, Introduction to Real Analysis, Ch. 3 (standard reference, not scraped)