TheoremStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck 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.

Young's Riemann–Stieltjes existence theorem for rational Hölder exponents

Statement

Let $p,q\in\mathbb Q\cap(0,1]$ with $p+q>1$ . If $f:[a,b]\to\mathbb R$ is $p$ -Hölder and $g:[a,b]\to\mathbb R$ is $q$ -Hölder, then both $\int_a^b f\,dg$ and $\int_a^b g\,df$ exist. They satisfy

$\int_a^b f\,dg+\int_a^b g\,df=f(b)g(b)-f(a)g(a).$

Facts & Assumptions

Given: Hölder functions $f,g$ with rational exponents whose sum exceeds one.

[L1]

The Young partition estimate controls refinement errors by a constant times $\lVert P\rVert^{p+q-1}$ (Young's partition estimate for rational Hölder exponents).

[L2]

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

[L3]

A Stieltjes integral is the common limit of all sufficiently fine tagged sums (Riemann–Stieltjes sums, upper and lower sums, and the Riemann–Stieltjes integral).

[L4]

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

[L5]

Tails of a geometric series with ratio in $(0,1)$ tend to zero (For $|r| < 1$ , $\sum_{k \ge 0} r^k = 1/(1-r)$ , and for $|r| \ge 1$ the series diverges).

Proof

technique · direct

1.1

For the dyadic left sums $L_m$ , the first estimate in [L1] and the geometric-tail fact [L5] make $(L_m)$ a Cauchy sequence. It therefore converges to some $I$ by [L2].

L1L2L5

2.1

Given a partition $P$ , compare it and a sufficiently fine dyadic partition $D_m$ with their common refinement. The second estimate in [L1] bounds the two refinement errors by a constant times $\lVert P\rVert^{r-1}+\lVert D_m\rVert^{r-1}$ . Together with $L_m\to I$ , this shows that every sufficiently fine left-endpoint sum is close to $I$ . Replacing a left endpoint $t_i$ by an arbitrary tag $\xi_i$ changes the $i$ th term by at most $K_fK_g|t_{i+1}-t_i|^r$ ; the total is at most $K_fK_g(b-a)\lVert P\rVert^{r-1}$ . Thus every fine tagged sum tends to $I$ , and [L3] gives $\int f\,dg$ . Interchanging $f$ and $g$ gives $\int g\,df$ .

step 1.1L1L2L3L4

3.1

On every partition, the right-endpoint sum for $f\,dg$ plus the left-endpoint sum for $g\,df$ telescopes exactly to $f(b)g(b)-f(a)g(a)$ . Passing to the two limits established in step 2.1 proves the formula.

step 2.1L3∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 109 results over 24 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

Nourdin, Nualart, and Peccati, The Breuer–Major theorem in total variation: improved rates under minimal regularity, Section 2.2 (standard reference, not scraped)