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Bounded Variation and the Riemann–Stieltjes Integral

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1 · Prerequisites
Real Analysis
Set Theory
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2 · Logical flowchart

3 · Items

Definition

Bounded variation and total variation on an interval

Lemma

Total variation bounds increments; bounded-variation functions are bounded; zero variation means constant

Lemma

Total variation is additive over adjacent subintervals and decreases under restriction

Definition

Variation function and positive and negative variations

Lemma

The positive and negative variations are nondecreasing and give the Jordan identities

Theorem

Jordan decomposition for functions of bounded variation

Corollary

A bounded-variation function has at most countably many discontinuities, all of the first kind

Lemma

The jumps of a variation function equal the absolute jumps of the original function

Lemma

Homogeneity and subadditivity of total variation

Theorem

Functions of bounded variation form an algebra

Corollary

Every bounded-variation function on a compact interval is Riemann integrable

Definition

Absolute continuity on a compact interval

Theorem

C¹ implies Lipschitz, Lipschitz implies absolutely continuous, and absolutely continuous implies continuous and bounded variation

Definition

Riemann–Stieltjes sums, upper and lower sums, and the Riemann–Stieltjes integral

Theorem

Darboux criterion for Riemann–Stieltjes integrability with a nondecreasing integrator

Lemma

The Riemann–Stieltjes integral is unique

Lemma

Refinement and tag-change estimates for Stieltjes sums

Theorem

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

Corollary

The total-variation bound for a Riemann–Stieltjes integral

Theorem

Linearity and interval additivity of the Riemann–Stieltjes integral

Theorem

Riemann–Stieltjes integration by parts

Corollary

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

Theorem

A bounded function with finitely many discontinuities is Stieltjes integrable against a continuous bounded-variation integrator

Theorem

A continuous function of a Stieltjes-integrable function is Stieltjes integrable for a nondecreasing integrator

Lemma

Every bounded-variation function is uniformly approximable by step functions

Theorem

Two bounded-variation functions with no common discontinuity are Riemann–Stieltjes integrable

Corollary

The identity integrator recovers the Riemann integral

Theorem

A continuously differentiable integrator reduces Stieltjes integration to ordinary integration

Theorem

A countable pure-step integrator evaluates a continuous integrand as the absolutely convergent weighted sum of its values at the jumps

Theorem

Change of variable for the Riemann–Stieltjes integral

Lemma

Young's partition estimate for rational Hölder exponents

Theorem

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

Remark

Conventions and proved scope for bounded variation and Stieltjes integration

Sources