CounterexampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passaudited 2026-08-11

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A common jump can destroy Riemann–Stieltjes integrability

Example

Let $a < c < b$ and put $f = α = H_{c}$ , the unit step that is zero left of $c$ and one at and right of $c$ . Both functions are BV, but $\int_{a}^{b} f d α$ does not exist.

Facts & Assumptions

Given: The two identical unit-step functions.

[L1]

A Stieltjes integral must be the same limit for every sufficiently fine choice of partition and tags (Riemann–Stieltjes sums, upper and lower sums, and the Riemann–Stieltjes integral, The Riemann–Stieltjes integral is unique).

[L2]

For a nondecreasing function every partition increment is nonnegative, so the absolute values in the variation sum may be removed and the finite sum telescopes to the endpoint increment (Bounded variation and total variation on an interval, Finite sums and finite products, by recursion, Laws of finite sums and finite products).

Verification

technique · counterexample

1.1

By [L2], both $f$ and $α$ have total variation one. For every $h > 0$ small enough, choose a partition containing $c - h$ and $c$ . The only nonzero integrator increment occurs on $[c - h, c]$ .

2.1

Tag that interval first at $c - h$ and then at $c$ . The corresponding sums are respectively $f (c - h) = 0$ and $f (c) = 1$ , although both partitions have mesh tending to zero after the other intervals are refined. Thus no common mesh limit exists, contradicting the necessary condition [L1].

L1∎

Depends on

Used by

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Dependency tree · two levels

37 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, discussion following Theorem 6.10 (standard reference, not scraped)