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Fundamental theorem of calculus for absolutely continuous functions
Statement
Assume the Axioms of Countable Choice and Dependent Choice. For , the following are equivalent:
- is absolutely continuous.
- exists almost everywhere, , and
Facts & Assumptions
Given: Countable choice, dependent choice, and a real function on the compact interval .
Proof
Assume (1). By Absolutely continuous functions have integrable derivatives, put a.e.; the first FTC The indefinite integral of an function is differentiable almost everywhere says a.e.
The corollary The indefinite integral of an function is absolutely continuous makes AC, so is AC and has derivative zero a.e. The zero-derivative theorem makes constant; , giving (2).
Conversely, (2) says , and the same corollary makes it AC. The singleton interval satisfies both clauses directly.
Depends on
- The Axiom of Countable Choice ($\mathrm{AC}_\omega$)
- The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain
- The indefinite integral of an $L^1$ function is absolutely continuous
- The indefinite integral of an $L^1$ function is differentiable almost everywhere
- Absolutely continuous functions have integrable derivatives
- An absolutely continuous function with zero derivative almost everywhere is constant
Used by
- The composition of two absolutely continuous functions need not be absolutely continuous Counterexample
- x² sin(1/x²) is differentiable everywhere but not absolutely continuous Counterexample
- Chain rule for an indefinite integral after an absolutely continuous composition Lemma
- Sharp and classical fundamental theorems of calculus agree Remark
- Change of variables for an absolutely continuous map under an absolutely continuous composition hypothesis Theorem
- Change of variables for an increasing absolutely continuous function Theorem
- Countably exceptional differentiability and integrable derivative imply absolute continuity Theorem
- Integration by parts for absolutely continuous functions Theorem
- Lipschitz characterisation within absolutely continuous functions Theorem
- Total-variation function of an absolutely continuous function Theorem
Dependency tree · two levels
22 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
- Christopher Heil, Absolute Continuity and the Banach--Zaretsky Theorem, Theorem 23 (standard reference, not scraped)