Alphabeta MathBounded Variation and the Riemann–Stieltjes Integral: Examples and Counterexamples
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1 · Prerequisites
Real Analysis
- Bounded Variation and the Riemann–Stieltjes Integral
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Properties of the Integral and the Working FTC
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Series: Convergence and the Nonnegative Tests
- Suprema and Infima
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Derivative and the Mean Value Theorems
- The Riemann Integral: Definition and Integrability
- Topology of ℝ
Set Theory
2 · Logical flowchart
3 · Items
Example
|x| is Lipschitz and absolutely continuous but not C¹ on [-1,1]
Example
√x is absolutely continuous but not Lipschitz on [0,1]
Example
The Cantor function is continuous and of bounded variation but not absolutely continuous
Counterexample
A continuous function on [0,1] can have unbounded variation
Example
A one-jump integrator evaluates a continuous integrand at the jump
Example
A finite-step integrator gives a weighted sum over its jumps
Example
A Riemann–Stieltjes integrable integrand need not be bounded
Example
The Cantor function defines a nonclassical Stieltjes integrator and ∫₀¹ 1 dc=1
Counterexample
A common jump can destroy Riemann–Stieltjes integrability
Example
∫₀¹ x d(x²)=2/3
Example
A nonlinear reparametrisation leaves a Stieltjes integral unchanged
Example
Young's theorem integrates a Hölder function of unbounded variation against itself
Sources
- Christopher Heil, Absolute Continuity and the Banach-Zaretsky Theorem, Section 2
- Christopher Heil, Absolute Continuity and the Banach-Zaretsky Theorem, Example 3.3
- William F. Trench, Introduction to Real Analysis, Section 3.2
- W. Rudin, Principles of Mathematical Analysis, Ch. 6, Theorem 6.15
- W. Rudin, Principles of Mathematical Analysis, Ch. 6, discussion after Definition 6.1
- W. Rudin, Principles of Mathematical Analysis, Ch. 6, discussion following Theorem 6.10
- W. Rudin, Principles of Mathematical Analysis, Ch. 6, Theorem 6.17
- W. Rudin, Principles of Mathematical Analysis, Ch. 6, Theorem 6.19
- Nourdin, Nualart, and Peccati, The Breuer–Major theorem in total variation: improved rates under minimal regularity, Section 2.2